School of Logic: a Collection of Puzzles involving the Student Sixby Coyotek4ChaptersPuzzle 1: MentorsPuzzle 1 solutionPuzzle 2: Who got the 96?Puzzle 2 solutionPuzzle 3: Pick a NumberPuzzle 3 solutionPuzzle 4: Squirrels and ChipmunksPuzzle 4 solutionPuzzle 5: Buckball GamesPuzzle 5 solutionPuzzle 6: Gemstone JumblePuzzle 7: Silverstream of ConsciousnessPuzzle 8: Chaotic ReasoningPuzzle 8 solutionPuzzle 9 solutionPuzzle 10: QuibbleFic Part I (Bears and Wolves)Puzzle 10 solutionPuzzle 11: Random InferencePuzzle 11 solutionWelcome Back, CoccyxPuzzle 12: Silent PegasiPuzzle 13: Button, Button, Which One's 'Gray Button'?Puzzle 13 solutionPuzzle 14: Balancing ActPuzzle 14 solutionPuzzle 15: Bejeweled and BefuddledPuzzle 15 solutionPuzzle 16: More Buckball GamesPuzzle 16 solutionPuzzle 17: QuibbleFic Part II (The Cursèd Code)Puzzle 17 solutionPuzzle 18: It's 'The Mind'Puzzle 18 solutionPuzzle 19: Double Vision, Part 1Puzzle 19 solutionPuzzle 20 solutionPuzzle 6 solutionPuzzle 7 solutionPuzzle 9: Ocellus is StumpedPuzzle 12 solutionIMPORTANT ANNOUNCEMENTPuzzle 20: Double Vision, Part 2Puzzle 1: MentorsAfter so many months of having her as a professor, the ‘Young Six’ had gotten rather used to seeing Pinkie Pie in a number of strange poses and appearances. This day, upon seeing the pink earth pony enter the room regaled in fortune-teller garb, a crystal ball placed upon the center of her desk, just felt a bit more unusual than others. “Yep,” Gallus begins, “just another normal school day with Equestria’s most abnormal professor.” Yona follows: “Uh, Professor Pinkie Pie? What with weird get-up?” “I’m not getting up, silly; I’m sitting down,” Pinkie cheerfully replies as she takes her seat at her desk, pulling out a blank sheet of paper from a drawer. “Now then,” she continues as she gazes into the crystal ball, “Which of you have the ability to see into the future???” The students exchange quizzical looks with each other. Smolder turns to Sandbar: “Is this another one of those ‘pony’ things we don’t know about?” The latter can only shrug. “Now,” Pinkie starts, “your professors, including yours truly, will be holding one-on-one sessions with all of you.” “Did we do something wrong?” Silverstream asks, fear in her voice. Pinkie laughs. “No, nocreature is in trouble. We just feel … well, actually, it’s Headmare Twilight who feels … that you all may benefit from some private conversations. A chance to share your experiences, goals, and worries, and get some useful toothy … uh … tube-lace? …” “Tutelage?” “Thanks, Ocellus! Yeah, some private tooty-ledge from none other than the Element Bearers themselves. Each of you will get to meet with one of us. Isn’t that great?” “That’s wonderful,” Gallus deadpans. “What does that have to do with fortune-telling?” “And who meet with Yona?” the yak adds. “What an incredible coincidence!” Pinkie declares. “Those questions go hoof in claw in talon!” Once again, the students turn to each other in confusion. Yona opens her mouth to speak, but Gallus pats her shoulder and speaks for her. “Yona, Smolder, Sandbar, Silverstream, Ocellus AND Gallus confused.” “As we speak … well, ‘me’, anyway … Twilight and the others are randomly determining which pony gets which student. We felt … well, ‘Twilight’ felt … that nocreature should feel slighted if he or she really wanted one particular professor, or headmare as the case will be for one of you. This way, everycreature has an equal chance to meet with everyprofessor.” “OK,” Smolder says, “but what about the fortune-telling part?” “That’s the fun part. You’re all going to make predictions as to who each of you get!” Gallus blinks. “I’m sorry, professor, but where does the ‘fun’ come in?” “Predictions can be lots of fun! Trying to guess who’ll win the annual Young Fliers competition, or wondering which evildoer will be next to try to take over all of Equestria … you can make a game about anything!” “I love games!” Silverstream declares, as Gallus puts a talon to his face and groans to himself. “So come on, everycreature, make your predictions!” Pinkie states before grabbing a pencil with her mouth, prepared to jot down their comments. The students turn to each other, none sure how to begin. Finally, Yona speaks up: “Yona predict that Yona not get Headmare Twilight.” Pinkie proceeds to put the comment to paper. Sandbar goes next: “Uh, I’ll predict that Ocellus gets Professor Rainbow Dash.” His comment draws a shy smile from the changeling. Smolder follows: “Well, I’ll predict that I’ll get either Headmare Twilight OR Professor Rainbow Dash.” Gallus snorts in derision. “This really isn’t particularly fun, is it?” Pinkie spits the pencil out from her mouth. “Aw, where’s your playful spirit?” she gleefully chides. “C’mon, what do you predict?” she asks, before again grabbing the pencil by her mouth. “Fine,” Gallus responds: “I predict that Smolder is wrong.” His comment causes the young dragon to stare daggers at him. “What? She wanted a prediction, I gave a prediction.” In an effort to avoid further confrontation, Ocellus quickly speaks up: “I predict that Silverstream gets somepony other than Rarity.” “Oh, oh, my turn!” Silverstream excitedly states. “OK, I predict that Sandbar gets … uh … one of our pegasus professors. So, either Professor Rainbow Dash or Professor Fluttershy.” The pencil falls from Pinkie’s mouth as she jots down the last of the predictions. “Great job, all of you!” she states. Her tail twitches, her ears flap, and she assumes several trance-like poses, before reverting to normal. “And not a moment too soon!” she exclaims as Twilight enters the room. “Wait,” Smolder says, “did you just predict Headmare Twilight coming into the room?” Twilight observes the students, then turns to Pinkie. “Your ‘Pinkie Sense’ went off again, didn’t it?” The grin and closed eyes that forms on Pinkie’s face confirm her belief. Twilight then turns back to address the class. “I trust that Professor Pinkie Pie told you about the meetings we’ll be holding for the six of you.” “Yeah,” Gallus responds, “right before having us make predictions about who gets who.” Twilight again turns to Pinkie, inspecting her outfit. “That would explain the garb,” she states as she eyes the predictions on the desk. Quickly, she scans the predictions. “Hm,” she notes. “This is a fun coincidence.” “Ooo, more fun,” Pinkie replies as she pulls the notes back towards her. “So what’cha got?” “Well, it’s just that exactly two of these six predictions are true … and the students who made those predictions were the ones who got paired with you and me.” The students look towards Twilight with confusion. “Uh, Headmare Twilight?” Sandbar asks, “what’s fun about that?” Pinkie quickly replies: “ ‘Fun’ is in the eye of the beholder, Sandbar. We all have our ideas of what makes something ‘fun’. It’s part of what makes ‘fun’ so special!” She turns to Twilight and starts bouncing in anticipation. “So who did I get, Twilight?” “I already told you,” the alicorn cryptically reply. Pinkie ceases bouncing as she takes in the response, the students looking on in surprise. “Waaaaaait a minute … is this another one of those puzzles where you’re gonna tell me you’ve given me enough information to deduce which pony got assigned which student?” “That’s exactly what this is,” Twilight gleefully concludes. “WOOHOO!!!” … Ocellus’s outburst causes her fellow students, and Pinkie, to turn to face her. “Oh … sorry, just a little excited,” she sheepishly explains. Author's Note Can you figure out which pony got paired with each student? As mentioned before, I ask that anyone who wishes to post a solution to this story uses hidden text. Or just PM me with your solution, if you wish to know before I post the next chapter. The following chapter will reveal the solution. Puzzle 1 solutionFor a brief moment, Ocellus braces herself for a round of insulting laughter at her expense … but the friendly chuckles that ensue merely serve to reinforce the sentiment that she is around true friends. “Always the bookworm,” Gallus dryly states. “More like, book- … changeling,” Silverstream adds, before blushing and sheepishly going quiet. Another round of chuckles follows. “Seriously, though,” Smolder says, “how could anyone figure out which of us got which professor based on those predictions. They were all over the place, and we don’t even know which ones are true.” “It’s all about using logical deduction and reasoning,” Twilight states. “Focus on one or two statements whose veracity can be figured out, and use that information to determine that of the other statements.” “Just like you did on that one trip we took, right?” Pinkie gleefully asks. Twilight nods. “Yona confused. Where Yona start?” “Yona starts with Yona’s prediction,” Twilight answers. Her response leads to further quizzical looks from the young yak. Twilight smiles and addresses Yona directly, as the others listen in. “You predicted that you would not get me as your mentor. That prediction is either true or false. Now suppose it was false; that would mean you did get me as your mentor. Understand so far?” Yona nods. “But, I gave you the additional information that the student who got me made a correct prediction. And THAT is our contradiction.” Yona ponders Twilight’s reasoning. “So … that means Yona’s prediction must be true. So Yona not get Headmare Twilight.” Twilight nods in agreement, which in turn causes Yona to smile. “Yona getting less confused!” “But there’s more,” Twilight continues. “Since your prediction was true, then you must have either myself or Pinkie Pie. And since you correctly predicted that you would not get me …” Twilight pauses, waiting for Yona to reach the logical conclusion: “… then Yona must have Professor Pinkie Pie! YONA GET IT!!!” “WHEEEE … I get to mentor Yona!” Pinkie bounds over to Yona and gives her a big hug, causing Smolder to wince slightly. “Sorry … still not used to all the touchy-feely and such.” “What about when we were in the basement and …” Ocellus stops mid-sentence, upon seeing the dragon stare daggers at her. “Uh … never mind.” Sandbar speaks up: “So, we know that Yona will be mentored by Professor Pinkie Pie. But what about everyone else? Where do we go from here?” “We go to Gallus’s prediction,” Twilight states. Gallus and Smolder both react with surprised looks. “ME??? I didn’t say anything useful at all!” “Yeah, all he said was that I’d be wrong!” “But that makes all the difference,” Twilight states. “Remember, only two predictions were true, and we already know that one of those was Yona’s prediction. So there’s exactly one other true prediction among the remaining five. Now think about Gallus’s prediction for a bit.” The students go quiet as they ponder Twilight’s suggestion. Soon after, Ocellus speaks: “If Gallus’s prediction was true, then Smolder’s prediction would have to be false. And if Gallus’s prediction was false, then Smolder’s prediction would have to be NOT false … it would be true.” “Ooo, ooo,” Silverstream suddenly interjects. “But that means one of them HAD to have the other true prediction!” Sandbar follows: “And that means the rest of us all made false predictions.” Twilight clops her hooves together. “Excellent work, all of you! Yes … that means we know that all the predictions made by Sandbar, Ocellus, and Silverstream are false. Now think about what Ocellus predicted.” Ocellus recollects her prediction. “I said that Silverstream would not get Rarity … and that means Silverstream DID get Rarity.” “And that makes two pairs figured out,” Twilight states. “Can you go further?” Smolder continues the train of thought: “Silverstream’s prediction was false, so Sandbar didn’t get either Rainbow Dash or Fluttershy. And he didn’t get either Pinkie Pie or Rarity, since those two were already assigned. So that leaves either Twilight or Applejack.” “Wait,” Yona interrupts, “but seapony made false prediction, so Sandbar not get Headmare Twilight … Sandbar get Applejack!” she excitedly concludes. “Wow … it’s like unraveling a dress by pulling on a thread!” Silverstream states before quickly backtracking: “Oh … maybe I shouldn’t use that when talking to Professor Rarity.” “Three pairs down, three to go,” Twilight states. “Now back to Sandbar’s prediction.” Sandbar thinks for a minute. “Uh … I said that Ocellus would get Rainbow Dash. That means she didn’t … and since her prediction was false, she didn’t get Twilight either. So Ocellus got Fluttershy as her mentor.” “Very well done,” Twilight again states. “So that just leaves Smolder and Gallus, and myself and Rainbow Dash as mentors.” She turns to Gallus. “Can you figure out which pony you got?” “Uh … no?” Twilight laughs. “Come now, you’re an especially bright griffin. I know you can figure this out.” “I can’t,” he starts with an air of exhaustion. “I mean, I know that either my prediction was true or false. And I know that Smolder’s prediction was either true or false. But how can I know which is which?” “Think back to what Smolder’s prediction was.” Twilight suggests. “Well … she predicted that she would get either you or Rainbow Dash.” Moments later, a revelation comes to him. “But those are the only professors left … so she must have made a true prediction! And that means she got either Twilight or Pinkie Pie … and THAT means Smolder got paired with you!” “And that only leaves Rainbow Dash left to mentor you,” Smolder adds. “And that is the final piece of the solution,” Twilight concludes. “To sum up: I will mentor Smolder, Pinkie Pie will mentor Yona, Rainbow Dash will mentor Gallus, Applejack will mentor Sandbar, Rarity will mentor Silverstream, and Fluttershy will mentor Ocellus.” Brief silence gives way to a chorus of ‘ooo’s and ‘aaah’s from the class. Smolder sighs. “So all this logic and stuff just comes naturally to you?” “It’s something I’ve always been interested in, ever since I was a foal,” Twilight admits. “But we can discuss that in more detail during our one-on-one. And speaking of which,” she states as she beckons the class to rise from their seats, “it’s time for some counseling. Follow me, everycreature.” Puzzle 2: Who got the 96?“Another ‘history of friendship’ quiz in the books,” Gallus exclaims with relief as he passes his paper up along with his classmates. “You’d think saving Equestria would be enough for graduation purposes.” “From what I’ve seen in my days in Ponyville, we’re not even in the Top Ten around here.” Sandbar’s retort brings about a fresh round of laughter. “It won’t take long to grade these,” Twilight asserts, “so you can just chat amongst yourselves as I get these marks in.” Her horn glows as a quill dips into in an inkwell and she peruses the first of the six papers. The students adjust their desks in a circle and begin pondering how they fared: “Yona think Yona did well. Yona think Yona earn score of 97.” “Why 97?” Sandbar asks. “97 prime, like yaks. Yaks renowned for ways in mathematics.” “I’ll have to take your word on that,” Gallus says, “but I feel like I did better than you did.” Yona’s eyes glare at the griffin. “It’s not a knock on you, Yona. I’m not usually confident on quizzes like this, but I think I did well. Besides, professor Pinkie Pie’s been pushing me to make more predictions.” “Well, I’m not as confident as some of you are,” Smolder states, “but I don’t think I got the lowest score of the six of us.” “Then who do you think did?” Ocellus asks. Smolder replies with a shrug and a trite ‘meh’. “So how did you do?” Sandbar asks Ocellus. The latter opens her mouth to reply, but Silverstream breaks in: “Oh, Ocellus always does well on these. I bet her score is one of the top three among the six of us.” The comment draws a slight blush from the changeling. “OK, that’s four predictions so far,” Sandbar recaps. “So, Yona thinks she got a 97, Gallus thinks he did better than Yona, Smolder thinks her score wasn’t the lowest, and Silverstream thinks Ocellus had one of the top three scores. So I guess that just leaves—” “FINISHED!” Twilight’s exclamation causes the students to suddenly turn to face their headmare. “That was quick,” Smolder asserts. “It’s easy to grade papers when they’re all of such high quality,” Twilight replies with a broad smile. “All of you did incredibly well!” “Really?” Gallus reflexively asks. “Yep. In fact, all of you got scores ranging from 94 to 99. But none of you got the same score.” “All right … A’s for everyone!” Sandbar exclaims. “So tell us,” Ocellus prods, “who got each score?” “Well it’s funny,” Twilight starts. “She sure has a strange notion of funny,” Gallus quietly tells the others, as Twilight continues. “I couldn’t help but overhear your predictions while I was grading. As it turns out, anyone who made a true prediction ended up with an even-numbered score, while anyone who made a false prediction ended up with an odd-numbered score.” Silverstream laughs, causing the others to turn to her. “Oh, she said it was funny, so I thought …” Ocellus turns to Twilight, anticipation in her voice. “So does that mean we can figure out who got what score?” “Oh, not at all. At least not with only that information.” Twilight ponders the situation further. “Although … this is actually quite interesting. IF I told you which of you got the score of 96, then you WOULD have enough information to figure out everyone’s score.” “Whoa … now Yona confused.” “So who did get the 96?” Smolder asks. “I don’t need to tell you that anymore,” Twilight replies. “But, I’ll forego homework assignments for a week if you can figure out who got what score.” “DEAL!” ... Everyone turns to Gallus, surprised by his renewed enthusiasm. “What’re you all staring at, let’s solve this!” Author's Note Admittedly, this puzzle turned out FAR harder than originally intended. Nevertheless, there IS a unique solution, as will be explained in the next 'solution' chapter. Puzzle 2 solution“You know, Gallus, if you put as much effort into your studies as you do in ways to get out of doing your studies …” Gallus quickly brushes off Ocellus’s observation. “Yeah yeah yeah, whatever, where do we begin?” The changeling sighs, shakes her head, and chuckles to herself. “I’m happy with getting help on this one,” she finally admits. “Well, first we need to figure out who got the score of 96. All we know for certain is that it’s not Yona.” The yak blinks. “Why Yona not get 96?” “You predicted you would get a 97,” Ocellus answers. “Now if you did get the 97, your prediction would be false; that’s a contradiction. So your prediction was false, but that means you got an odd-numbered score. Since it can’t be 97, you must have gotten either the 95 or the 99.” “Wait … Yona might have gotten high score in class? Yona happy!” Sandbar chuckles. “May have gotten the highest score. We don’t know yet.” “Fifty-fifty better than one-in-six!” Yona declares. “Whoa,” Gallus responds with a tinge of surprise, “you really do know math.” His comment causes Yona to beam. “So anyway,” Ocellus says, returning to the task at hoof, “Yona got either 95 or 99; she could not have gotten 96. Now we need to figure out who did … and the key is, knowing who got 96 must lead to a unique solution.” “Sounds good so far,” Smolder states, her thoughts drifting towards the possibility of no homework. “Now I predicted that I would not get the lowest score.” The young dragon ponders her comment for a moment. “Sooo … if I did get the lowest score, that would be a 94 … but that would mean I made a false comment with an even-numbered score, right?” Ocellus nods with an ‘mmm-hmmm’. “Cool … I guess … so that means I didn’t get the lowest score, and I made a true prediction … so I had to get either a 96 or a 98, right?” “That’s right,” Ocellus states. “One more piece of the puzzle.” “Yeah, but I think we’re out of edge pieces,” Gallus adds. Silverstream starts searching the floor around them, causing the griffin to roll his eyes. “Not literally, Silverstream!” The hippogriff gets back to her seat. “Sorry, sorry,” she says with a meek chuckle. “We still have two predictions to examine,” Ocellus continues. “Now Gallus predicted that he would get a higher score than Yona. Now if that prediction was true, then Gallus would had to have scored an even-numbered score … but it couldn’t be 94, since that was the lowest score. So he either scored 96 or 98, and Yona would have scored 95.” Silverstream picks up where Ocellus leaves off: “But if Gallus had a wrong prediction, then he would had to have scored an odd-numbered score, and lower than Yona. But that means he couldn’t have a 99, so he would have either a 95 or 97, and Yona would have the 99.” “EXCELLENT WORK, SILVERSTREAM!” The students turn to Twilight, who blushes a bit for her outburst. “Sorry … I’ll be quiet from now on,” she states before going silent. “So what about that last prediction?” Sandbar asks. “We can’t get much from that, can we?” “Well, no,” Ocellus admits. “If it’s true, then Silverstream got an even score and I got either the 97, 98, or 99. And if it’s false, then she got an odd score and I got either the 94, 95, or 96. So yes, a lot of possibilities.” “So now what?” Smolder asks. “Now we guess … I guess,” Gallus states. “I’m open to starting with anycreature.” “I say we start with Smolder getting the 96,” Ocellus suggests. “If anyone else got the 96, we’ll know she got the 98.” Sandbar speaks up: “All right … if Smolder got the 96, then … it seems like there’s a lot of ways to assign scores.” “If we then assume my prediction was true,” Gallus says, “then I got the 98 and Yona got the 95.” He ponders a bit further. “Silverstream could then have the 94, and either Sandbar or Ocellus could have the 97 or the 99.” “And if griffin wrong,” Yona adds, “then Yona get 99 and griffin get lower score, like 95. And maybe seapony get 94 again, and Ocellus get 98 and Sandbar 97. That work too, right?” “I could have gotten 97 myself, right?” Silverstream adds. “But that’s the whole point, right?” Ocellus concludes. “Knowing that Smolder got the 96 leads to too many possibilities … so Smolder couldn’t have gotten the 96. She must have gotten the 98, and someone else got the 96.” Upon hearing this comment, Twilight’s horn glows as a quiz paper floats from the desk to Smolder’s seat. The dragon picks up the page. “ALRIGHT, a 98!” she proudly declares. “And now we know one of the six scores,” Sandbar adds. “Where do we go from here?” “I’ll give this a go,” Gallus says. “If I got the 96, then that means Yona got the 95. After that … there’s the 94, 97, and 99 left for Sandbar, Silverstream, and Ocellus. Silverstream could have the 94, and if she did, we still don’t know who got the 97 and who got the 99. So … I guess that means I didn’t get the 96.” “And since griffin not get 98 either,” Yona adds, “that mean griffin wrong and Yona get 99!” Another paper levitates from Twilight’s desk towards the group, landing at Yona’s seat. “YES!!!” she triumphantly declares. Gallus turns to the yak. “Yona, you know I meant nothing personal with my prediction, right?” “All good,” Yona replies with a wide smile. With four scores left to determine, Sandbar is next to speak up. “What if I got the 96? That would leave the 94, 95, and 97 for Gallus, Silverstream, and Ocellus. Then Silverstream could have the 97, and Gallus would have the 95 and Ocellus the 94.” “Sounds good to me!” Silverstream declares. “Wait a minute,” Ocellus counters. “Silverstream could have gotten the 94 instead, and I would have had the 97. So again, there’s no way to know for certain.” “So there’s only two possibilities left for the 96,” Smolder adds. “Right,” Ocellus says as she continues. “Now suppose I got the 96. Then that would mean Silverstream made a false prediction, so she would have either the 95 or the 97, just like Gallus. And Sandbar would have the 94 … only two possibilities, but that’s still one too many. So all that’s left is …” Everyone turns to Silverstream. Seconds pass. “… OH! … Oh, that means I got the 96!” The hippogriff thinks for a few more seconds. “So my prediction is true … so Ocellus got the 97 … and since Gallus didn’t get the 94, he got the 95 and Sandbar got the 94.” “Then that’s it, isn’t it?” Gallus asks. “From highest to lowest, that means Yona got the 99, Smolder the 98, Ocellus the 97, Silverstream the 96, me the 95, and Sandbar the 94.” Twilight nods as the remaining papers fly towards the students, each landing at their appropriate spot. Quickly, the group stomp their appendages in triumph. “Never been so happy to have the lowest score in class,” Sandbar admits. “Great job, everyone!” “Indeed, you’ve all exceeded all expectations,” Twilight adds. “I think after that little exercise, you’ve all earned some time off from homework.” Puzzle 3: Pick a NumberOne-on-one conversations do not come naturally for Gallus. Perhaps this is due to his griffin heritage, or perhaps it’s more of an introvertive nature. Regardless, he often finds comfort in having discussions with his entire group of fellow nonpony students, rather than individual chatter. In light of his recent discovery of Yona’s mathematical prowess, however, he cannot help but initiate a discussion with the young yak, once back in the main foyer of his and his fellow students’ quarters. Thus, he and Yona have a talk, each seated near a table, while most of the others had already retired for the evening to their own rooms; Ocellus remains the exception, reading a tome to herself as the two chat. “So what Ocellus read?” “She mentioned something about a professor Stallyan,” Gallus replies, “and that’s a logic book written by him that Headmare Twilight owns. She said she read parts of it when the two of them took the balloon to parts relatively unknown for that friendship quest they were called for.” Yona turns towards Ocellus and utters a ‘hmm’ in response, the changeling remaining focused on the book. “So,” Gallus continues, “yaks are good at math, huh?” “Yaks best at everything,” Yona proudly responds, “including math. Many yaks renowned for math ability.” “Well I’m not one to brag,” he starts, phased slightly by Yona’s chuckle, “but griffins are also well-known for their abilities in the ways of mathematics.” “That really true?” “Well … it’s not like griffins are known for friendship.” Both he and Yona have a laugh at the comment. “Anyway, I have a simple challenge for you,” Gallus continues. “I’m thinking of a three-digit number, and I want to see how long it takes for you to guess the number.” “There nine hundred numbers like that,” Yona states. “Yona don’t want just guess numbers randomly.” “Let me finish,” Gallus calmly replies as he pulls a bag of gemstones towards them. “You can ask a series of yes-or-no questions to narrow down the possibilities. Smolder was nice enough to let me borrow her snack bag; to keep track of your response, I’m going to use red rubies and green emeralds to signify the answers to your questions. I want to see if you can figure out my number in 5 or fewer gems. So … I hope I haven’t confused you with all that.” “Yona not confused,” the yak insists. “You have number in mind?” “I do … so ask away.” Ocellus keeps an ear towards their conversation as she continues to read from her book. Yona, meanwhile, begins asking questions: “Is number perfect square? Like, number times itself?” “I’m well aware what a perfect square is,” Gallus states as he selects a gemstone from the bag and places it on the table. “OK … is number perfect cube? Griffin know what perfect cube is, right?” she devilishly asks. Gallus sighs and grins. “Yes, Yona, I know what a perfect cube is, too.” He selects another gem from the bag and places it on the table. Yona ponders for a minute. “Does number have same first and last digit?” “Huh … interesting question,” Gallus admits as he chooses a third gem from the bag and sets it alongside the others. “Oooo … Yona close now. Iiiiiiis … number greater than 200?” Gallus takes a fourth gem from the bag and adds it to the collection on the table. “YONA KNOW NUMBER!” “Whoa whoa whoa,” Gallus says with a quiet undertone. “Not so loud, you’ll disturb the others.” “Oh … sorry. Sorry, Ocellus,” Yona calls out across the room. “Not a problem,” Ocellus replies, keeping one ear open to their conversation as she continues perusing her text. Gallus studies the four gems he had placed on the table. “You realize you can’t know the number yet, right? There’s still two possibilities.” “That wrong,” Yona insists. “Griffin wrong. Proof that yaks better at math.” “You’re right about a lot of things, but not this,” Gallus counters … before a thought occurs to him. “Yona, you realize that green means yes, right?” “What???” “I’ve been using the green gems as ‘yes’ responses, and the red gems as ‘no’ responses.” “That make no sense. In yak culture, red always symbol for start of journey or task. Red always mean yes … so green must mean no.” Gallus slaps his head. “Well that explains everything … you took all my ‘no’ answers as ‘yes’ and my ‘yes’ answers as ‘no’. That’s why you think there’s only one solution.” “Well that not sporting of griffin!” Yona grouses. “Look, why don’t we just start over. And I’ll give you a starting hint: my number is greater than 600.” Ocellus promptly blurts out a number, causing the two to turn to her in surprise. ... “What that for?” Yona asks. “Oops … sorry, force of habit,” Ocellus sheepishly admits. “I guess I couldn’t help but listen to your conversation. I thought it an interesting challenge.” Gallus stammers for a response: “But … you didn’t even see any of the gems I placed … you didn’t hear any responses … how in Tartarus could you know the number I was thinking of???” “Well … you said it was greater than 600. That was all I needed to know.” Gallus shakes his head. “You’ll have to do better than that.” Author's Note This one may not be to everyone's taste, as some math knowledge is necessary. That said, there IS enough information to determine Gallus's number … as well as the number Yona thought was the correct answer. Puzzle 3 solution“Seriously, Ocellus, do changelings have the power to read minds as well as transform into any creature?” “That would be cool,” Yona admits. Ocellus puts down her book and trots over to Gallus and Yona. “It’s nothing so out-of-the-ordinary. I just took the information I had and deduced what your responses must have been. “But you didn’t see any of my responses,” Gallus states incredulously. “I mean, with 4 questions, each one of which could be answered with ‘yes’ or ‘no’, that makes for …” “… 2 to the fourth, or 16 possibilities,” Yona finishes. “… OK, I won’t doubt your math anymore.” Gallus turns from the yak to the changeling. “So yeah, 16 possibilities. How could you know the answers I gave?” Ocellus proceeds with her explanation: “The key is the fact that Yona thought all your responses were the opposite of what you intended. That meant there had to be a series of responses so that both the actual AND negated responses led to only a hoofful of possibilities.” Yona and Gallus both look blankly at Ocellus, who chuckles slightly before continuing. “Take Yona’s first two questions: whether or not the number was a perfect square, and whether or not it was a perfect cube. Now if Yona saw what she believed to be two ‘yes’ answers, she would have immediately thought the number was 729 … that’s the only 3-digit number that’s both a perfect square and a perfect cube.” “Wait … how you so sure of that?” Yona asks. “Because any number that’s both a perfect square and a perfect cube must also be a perfect sixth power. And since 1 to the sixth is 1 and 2 to the sixth is 64, both of those are too small. But 3 to the sixth is 729, and anything bigger is just too big.” Gallus turns to Yona. “No offense, but Ocellus might have both of us beat.” The yak nods in solemn agreement. “OK,” Ocellus continues, “so Yona couldn’t have thought the first two answers were both ‘yes’. On the other hoof, she couldn’t have thought the first two answers were both ‘no’ … there would be far too many possibilities, no matter how the other questions were answered. For example: any of 101, 111, 131, 141, and so on all have identical first and last digits, and all are less than 200. Also, any of 202, 212, 222, 232, and so on all have identical first and last digits, and all of those are greater than 200. And if the first and last digits are not the same, that’s just so many more to pick from. And all that’s to say that Yona must have thought the number was either a perfect square or a perfect cube, but not both.” “Yona getting confused … but Ocellus right; Yona did see different colored gems for first two answers.” “That’s still a lot of possibilities, though,” Gallus counters. “I mean, there’s 21 perfect squares with exactly 3 digits.” “True,” Ocellus admits, “but only three of those have identical first and last digits.” “But how would you know that a ‘yes’ answer to that question went with the perfect square?” “Because if it went with the perfect cube, then either Yona would have ‘known’ the number was 343 … that’s the only perfect cube that satisfies that condition … or she never would have been able to figure the number out with only that last question. So instead, there’s only two possibilities at this point: either Yona thought your answers were ‘yes’-‘no’-‘yes’, or she thought your answers were ‘no’-‘yes’-‘no’. In either case, there’s only 3 possibilities … with the first case, the number could only be 121, 484, or 676; with the second case, the number could only be 125, 216, or 512.” “Incredible,” Yona states in disbelief, “Yona did pick one of those numbers. But … how you know which one?” “Well, in either of those cases only one possibility is less than 200 … so that must have been what you thought.” “But that’s still not enough for you, right?” Gallus asks. “I mean, for all you knew, Yona could have answered either 121 or 125.” “Indeed,” Ocellus replies, “but whichever possibility Yona thought, the other one must have an option greater than 600 … because you said as much.” “… I did, didn’t I?” “Yep … so now I know everything: Yona must have read your answers as ‘no’-‘yes’-‘no’-‘no’ and thought the number was a perfect cube less than 200 … specifically, she must have thought the number was 125.” “That WAS Yona’s guess!” she exclaims in amazement. “But instead,” the changeling continues, “Gallus actually responded with ‘yes’-‘no’-‘yes’-‘yes’, indicating a perfect square greater than 200 with identical first and last digits … either 484 or 676. And since you said the number was greater than 600, it must have been …” “… six hundred seventy-six,” Gallus slowly concludes, still stunned at Ocellus’s reasoning. The sound of a door opening causes all three to turn. Smolder enters the foyer and makes her way towards the table. “Sorry Gallus,” she says as she picks up her bag of gems. “Just had a craving for a late-night snack.” She notices three emeralds and a ruby on the table. “So what’ve you all been up to?” “Trust me,” Gallus replies, “wait until morning before asking that.” Author's Note Originally, I posted that Gallus's number was greater than 500. That would have led to an alternate solution: Yona could have been thinking the number was 121 (perfect square, NOT perfect cube, same end digits, less than 200), with Gallus actually thinking 512 (NOT perfect square, perfect cube, different end digits, greater than 200). Puzzle 4: Squirrels and Chipmunks“Yer just in time, sugarcube … uh, Ah mean, Headmare Twilight.” Twilight chuckles at Applejack’s backpedaling. “There’s no need to be so formal with me.” “Well, in front of the students is all … Ah thought a bit more formality would be in order.” Applejack turns from Twilight to the class, who all shrug in indifference. “I’ve never been comfortable with anypony addressing me as ‘Princess’; as it is, I’m still adjusting to ‘Headmare’. Formality can be appropriate, but not among friends.” “Uh … we can still call you ‘Headmare’, right?” Twilight laughs at Ocellus’s question. “Whatever makes you most comfortable,” she replies before turning back towards Applejack. “So what do you have for today’s lesson plan?” “So, for today Ah thought it’d be interesting for the students to see first-hoof how confusin’ it can be when competin’ groups try to be uncooperative towards one another. The point, of course, is to show that honesty is always the best policy.” “I’m with you so far,” Twilight accedes. “So how does this work.” “Ah’ve already assigned each of the students here to one of two groups; one group is called the Squirrels, and the other group is called the Chipmunks.” Twilight ponders the names for a moment. “I think all those friendship quests you’ve been sent on with Fluttershy are starting to have an effect.” The two share a laugh as the class quietly observes. “Well, Ah didn’t want to just use ‘A’ and ‘B’ for this; she may have given me suggestions,” Applejack admits. “So anyway, Ah assigned three of ‘em to be Squirrels and three of ‘em to be Chipmunks. Then Ah laid down some simple rules for the students. They can each talk to only one other at a time, and Ah included a rule what causes confusion when ponies … well, any creature gets all underhoofed.” “And what rule would that be?” “Any statement made toward someone from the ‘other’ group has to be a lie, while any statement made towards someone from the same group has to be honest. So if, say, Gallus and Smolder are both Squirrels, they have to be honest when talking to each other … but if Gallus is a Squirrel and Smolder is a Chipmunk …” The other four students burst out in giggles, as Gallus and Smolder sigh in unison. “… uh, anyway, if Gallus and Smolder are in different groups, then any statement made by one to the other has to be a lie.” “I think I got it,” Twilight states. “So I assume everyone knows their group.” “Yes indeed.” Applejack turns from Twilight to the students. “How ‘bout a demonstration for the Headmare, y’all?” “What are we supposed to say?” Ocellus asks. “Ah dunno … maybe keep things simple, like makin’ comments to one another about yer affiliation.” Gallus sighs a second time, swallows his pride, and turns to Ocellus: “I am a Chipmunk.” He pauses for a moment. “Grampa Gruff can never know I spoke those words in that order, okay?” “Ooo, my turn!” Yona exclaims as she, too, addresses Ocellus: “Yona is Squirrel.” Smolder then beckons Yona to face her: “We are different types.” Sandbar goes next, getting Smolder to face him: “You are a Squirrel.” The comment draws rolled eyes from the young dragon. “Oh, I want to go now!” Silverstream insists as she flutters towards Gallus: “You are a Chipmunk,” she states before moving on towards Sandbar: “Either Yona is a Squirrel or Smolder is a Chipmunk.” “Whoa now, love the enthusiasm there Silverstream, but Ocellus hasn’t even had a chance to speak.” Applejack double-checks her list and thinks about the comments made. “Ah think some of you may have forgotten the rule Ah put in place. Two of those statements wouldn’t’ve been made.” “What do you mean?” Ocellus asks. “What Ah mean is that one of those statements is true when it should be false, and one of those statements is false when it should be true … though the other four statements hold up. It’s alright, Ah know it’s a bit confusin’ at first … that’s why Ah always believe in honesty towards all.” Applejack turns to Twilight, noticing the latter lost in thought. A notion quickly comes to her: “Oh shoot, Twilight … Ah never even told you which group each student was in.” “You don’t need to,” Twilight replies with a smile. … “If this is going where I think this is going,” Gallus says, “can I just take my seat and observe this time? I’m still worn out from the last few puzzles.” Author's Note So … which three students are 'Squirrels' and which three are 'Chipmunks'? Also: which two statements should not have been made? Puzzle 4 solution“Now hold on, Twi … Ah didn’t tell you what student was in each group, you know that some of those six comments made are true and some false, you know that two of those comments are wrong from what they should be … and you still think you got this all figured out?” “Well … yeah.” Applejack takes in Twilight’s simple reply and sighs. “If y’all don’t mind, Ah think Ah’ll just do like Gallus over there and take a seat, and let your headmare do the talkin’ for the rest of the lesson.” She proceeds to her desk and takes a seat, as the others focus in on Twilight’s solution. “So where do you begin with this one, Headmare Twilight?” Silverstream asks. “Well to start, let’s think about Smolder’s comment.” Twilight’s suggestion causes heads to turn Smolder’s way. “What … I just told Yona that we were different types. I thought that was alright.” “But here’s the strange thing,” Twilight explains: “Under the rules Applejack gave, NO ONE could utter that phrase, to ANY creature.” She pauses for the gasps to subside, then gives further details: “Suppose one Squirrel said that to another Squirrel. Then the statement would be a lie, and any statement made between two in the same group is supposed to be true. The same would hold if a Chipmunk said the statement to another Chipmunk.” “But what if a Squirrel told a Chipmunk that?” Sandbar asks. “In that case, the statement would be true … but the rules state that only false statements can be made between two from different groups. So no matter the pair, that statement should never be said.” “Oh,” Smolder utters. “I guess I got a bit confused, like professor Applejack said.” “But there were two statements made in error,” Ocellus reminds the group. “I don’t see how the other statement could be determined … and without knowing that, I don’t know how you could continue.” “The key here,” Twilight continues, “is to focus on two statements … specifically, the two directed at you, Ocellus.” The changeling thinks back to the two comments made to her. “Gallus said that he was a Chipmunk, and Yona said that she was a Squirrel. It seems like either could be true OR false.” “By themselves, either comment could be made … but as you’re about to see, it’s not possible for BOTH comments to be directed towards the same creature under any circumstance.” “Really???” “Let’s start with Gallus’s comment. If true, then he is a Chipmunk, and so is the student he’s addressing. If false, then he’s actually a Squirrel and he’s lying; that means the student he’s talking to is again a Chipmunk. So either way, the recipient of the conversation is a Chipmunk; in this case, that would be you, Ocellus.” “Weird … a comment referring to himself ends up revealing who I am.” “Indeed … and by the same reasoning, Yona’s comment would then force you to be a Squirrel. And that’s where the contradiction lies; you can’t be both a Chipmunk AND a Squirrel.” “Wait,” Yona asks, “did Yona make mistake? Yona not mean to.” “Well either you or Gallus erred, but as of now there’s no way to know who erred. All that’s known is that one of those comments is wrong … and combined with Smolder’s comment, we now know that all the other comments are within the rules. Now we can turn our attention to those comments.” “So, my comment to Smolder was OK?” “It was, Sandbar … and your comment actually forced YOU to be a Squirrel.” Sandbar gapes. “Really? I said that Smolder was a Squirrel.” “Yes, you did. And if your statement was true, that would force you to be a Squirrel so you were in the same group. But if your statement was false, then Smolder would actually be a Chipmunk; since false statements are only made by creatures from different groups, you would again have to be a Squirrel.” “That is incredible!” Silverstream admits. “So if I say that someone else is a Squirrel, that means I’m a Squirrel … and if I say that I’m a Squirrel, that forces the other student to be a Squirrel.” “Exactly; it’s a fun property that exists here.” Twilight catches a slight derisive laugh from Gallus. “Well, I suppose it does depend on one’s idea of ‘fun’.” “So, all that means Sandbar has to be a Squirrel and Silverstream has to be a Chipmunk, right?” Ocellus asks. “It does … and since Silverstream and Sandbar are in different groups, that means her comment to him has to be a lie: it is NOT the case that either Yona is a Squirrel or Smolder is a Chipmunk, so the reality is, Yona is a Chipmunk and Smolder is a Squirrel. And that takes care of four of your affiliations.” “Remarkable,” Sandbar says in awe. “And you’re right about all four of us … but what about Gallus and Ocellus?” “Yeah, couldn’t either of them be the last Squirrel or Chipmunk?” Smolder asks. “There’s still one fact we haven’t used,” Twilight states. “Applejack didn’t just say that two comments were in error. She said that one statement was true that should be false, and one comment was false that should be true. So now go back to your comment.” Smolder recollects what she said. “I said that Yona and I were different types. And now you know that I was a Squirrel and Yona was a Chipmunk … so my statement was true, and should have been false.” “And therefore …” Twilight starts. “… the other wrong statement was false and should have been true,” Ocellus finishes. “And that means the two students involved with one of those statements had to be in the same group.” “But Gallus and Ocellus have to be in different groups,” Silverstream continues. “So it was the other statement that was in error.” Yona looks to the floor for a moment. “Yona not mean to make mistake.” “Well, how about atoning by finishing the solution?” Twilight offers, bringing a smile to the yak’s face. “OK … so, Yona’s comment to Ocellus must be false and we must be in same group … so Ocellus must be Chipmunk.” “And that leaves Gallus to be the last Squirrel,” Smolder adds. “And … that’s it, right?” “That is it,” Twilight concludes. “In summary: Gallus, Smolder, and Sandbar are the Squirrels while Ocellus, Yona, and Silverstream are the Chipmunks. Yona’s comment was false and should have been true, Smolder’s comment was true and should have been false, and the other comments all followed the rules: Gallus lied, Sandbar was truthful, Silverstream lied to Gallus, and she also lied to Sandbar.” … “No matter how often you do that, Twi, Ah’ll never understand how simple you make everything sound,” Applejack states. “Comes from a good chunk of my youth spent working on logic,” Twilight says with a tinge of remorse. “Probably too much … but I’m always happy to impart such knowledge on students such as yourselves.” She glances at the clock in the room. “Oh … class time is almost up! Sorry, Applejack; I didn’t mean to take up all your lecture time.” “Shoot, Twilight, Ah love watchin’ you do yer thing. And hey, less work for me, to be honest,” she adds with a sly smile.” “Less work for me too,” Gallus adds instinctively. His fellow students all turn to him and scowl. “What?” “It’s alright,” Twilight calmly says. “I’m sure Gallus will prove to be up to the next puzzle challenge … whenever that will be.” Puzzle 5: Buckball GamesOf all the classes to miss … why’d it have to be Professor Dash’s? Gallus muses to himself as he works on a paper for another class while holed up in Headmare Twilight’s office. He stretches his left wing for a moment before wincing and pulling it back. “You probably just overextended your wing,” Twilight suggests as she pores over her latest curriculum updates. “You should be fine in the morning, as long as you don’t exert yourself too much this afternoon.” Gallus sighs. “It was supposed to be a friendly flying competition between me and Smolder … I guess I got a little carried away.” “I would have thought you and the others would know better after seeing how some of your professors can get,” Twilight lectures. “In any case, what’s done is done and now you have time to catch up on your studies.” “But Buckball Day,” Gallus pines. “The one day when classwork doesn’t involve any actual learning.” “Teamwork, good sportscreatureship, being there for your teammates during both wins and losses … seems to me that you can learn a lot from activities like that. And who better to teach?” Gallus ponders Twilight’s comment. “Well, thanks for trying to make it sound boring,” he replies before refocusing on his paper. Twilight chuckles at the comment before setting herself back to her tasks. … A knock on the door catches Twilight’s attention. “Come in!” she calls out. The door gently opens and Ocellus flutters into the office, followed by the rest of her classmates. “You ready to head back to the dorm?” she asks. “Just about,” Gallus replies as he puts his work away. “So how’d your buckball games go?” “Pretty well,” she responds. “We were all on five different teams. No names given, just ‘A’, ‘B’, ‘C’, ‘D’, and ‘E’. There wasn’t enough time for every team to play every other, but all teams played either two or three games, always against different opponents. And of course, there were no ties.” “Sounds like it was fun,” Gallus dejectedly points out. “So who won?” Instinctively, all four remaining students begin calling out different game results. “Whoa, whoa, whoa … I don’t have 20 ears, you know! One at a time.” The four students look to each other, before Silverstream continues: “A beat B, but B beat C. Then again, E beat B. Oh, and E beat D, too.” Sandbar speaks next: “A beat C and A also beat E. Oh, and B beat D.” Yona follows: “B beat D, C beat A, and D beat C.” Smolder finishes up: “Well … A beat E but D beat A. Also, B beat D.” Gallus looks from one student to the next. “Those was some strangely random results you all just gave me.” He turns towards Ocellus. “Don’t you have any results to add?” “No, but I do have this to say: Any result given by somecreature who was on one of the teams mentioned is true, but all the other results stated are false.” Gallus’s jaw drops. “So for example,” Ocellus continues, “If Yona was on team A, then C did beat A … but if she was on team B instead, then C didn’t beat A; C either lost to A or didn’t play A at all.” “I understand what you said,” Gallus declares, “but why would you bother giving me the results this way.” “Well …” Ocellus slowly admits, “… professor Rainbow Dash heard about how much fun Headmare Twilight had solving Applejack’s puzzle with the squirrels and chipmunks, so she wanted to give her a puzzle of her own.” “… You’re kidding.” “She may have asked for my assistance in creating it,” Ocellus continues. Gallus slaps his right wing to his head, then turns to Twilight. “Feel free to explain this one, headmare.” Twilight smiles. “… what?” “Remember what I said after the last puzzle? I think you’ll prove to be up for this challenge … and I would prefer you not being dismissed for the day until you’ve solved it.” Author's Note Can you help Gallus out and determine the results of all the games played? Puzzle 5 solution Silverstream: A beat B, B beat C, E beat B, E beat D Sandbar: A beat C, A beat E, B beat D Yona: B beat D, C beat A, D beat C Smolder: A beat E, D beat A, B beat D Gallus examines the statements placed on the portable chalkboard. “I guess I shouldn’t be surprised that you have a chalkboard in your office,” he notes. “I like being prepared,” Twilight replies somewhat defensively. “For what, someone giving you a pop quiz?” Ocellus asks. “One never knows,” Twilight states, envisioning the day Princess Celestia makes an impromptu visit. OK, Gallus, you’re smart, you can figure this out. He stares at the predictions for a few minutes … “So … three of you all said that B beat D,” he starts, talking the problem out aloud to himself as the others listen in. “But you couldn’t all be on teams B or D … so that means B didn’t beat D, and it had to be Silverstream and Ocellus who were on teams B and D, in some order.” “Well started,” Twilight says. “OK … so Sandbar, Yona, and Smolder were on teams A, C, and E, in some order. And both Sandbar and Smolder said that A beat E … and at least one of them had to be on either A or E … so that statement has to be true, and both Sandbar and Smolder are on teams A and E. And that means—” “Yona on team C!!!” The group turn to Yona, who sheepishly smiles. “Sorry … Yona excited.” “Yeah,” Gallus says before continuing: “so anyway, knowing that Yona was on team C means that both of her other statements were true: C beat A and D beat C … and the first of those two contradicts Sandbar’s claim that A beat C, so Sandbar couldn’t have been on team A … he was on team E, and Smolder was on team A.” “Nicely done, dude,” Sandbar says with encouragement. His confidence growing, Gallus proceeds: “Now I know that Smolder was on team A, and that means that D beat A.” He pauses and ponders for several seconds. “Now what?” “Would you like a hint?” Ocellus asks. “No … I need to get there on my own,” Gallus insists. He ponders for several more seconds. “Let me recap,” he says aloud to himself: “A beat E, C beat A, D beat C, D beat A … wait a minute … so A played E, C, and D … but no team played every other team … so A didn’t play B at all.” “I think you’re going to do it!” Silverstream exclaims. “So that means Silverstream was wrong by saying that A beat B … Silverstream couldn’t be on team B, so she was on team D and Ocellus was on team B. That’s all the teams!” “Excellently done!” Twilight states. “Now can you finish the results?” “Silverstream was on team D, so E did beat D. And now … yes … I got it now! D played A, C, and E, so D couldn’t have played B either! But since every team played at least two games, B must have played C and E. And since both of Silverstream’s remaining statements must be false, C must have beaten B and B must have beaten E. And that’s all the results: A beat E, B beat E, C beat A, C beat B, D beat A, D beat C, and E beat D.” “You got them all!” Ocellus proclaims. “Way to go!!!” The student all crowd around Gallus to embrace him. He winces slightly at the pressure put on his left wing, causing the others to back away slightly. “It’s OK, everyone,” he reassures the group, “I’ll be all right. Just, maybe not so much crowding right now.” “Puzzle-solving may not be to everycreature’s taste,” Twilight concludes, “but still, you should be proud of your capabilities. Problem-solving skills in general are useful everywhere, to solve all sorts of problems, friendship-based and otherwise.” “Thanks, Headmare Twilight … but if it’s all the same, I hope that next time I’ll be playing rounds of buckball instead of solving a puzzle about them.” Puzzle 6: Gemstone JumbleIn all their classes, the students receive ample knowledge of the various aspects of friendship. Regarding the personal interests of the professors, however, they got more out of some professors than others … “Professor Rarity,” Smolder speaks up during yet another lesson veering towards fashion. “Does generosity only apply towards dresses?” A tinge of dejection overcomes the unicorn, but she quickly shakes off the feeling. “I admit that my own ventures gravitate towards my profession of choice … but generosity is a vital aspect of friendship regardless of situation. Everything from offering the last slice of pie during a lunch to providing financial assistance to those who cannot afford the finer things in life.” “Is generosity always associated with physical things?” Ocellus asks. Rarity chuckles. “Generosity can go far beyond the mere tangible. One can be generous with their time, volunteering themselves for community service. One can be generous with their emotional state, being there for another when times can get particularly difficult. Why, one can even be generous with one’s rationality.” The sound of Rarity’s final word causes the class to observe their professor with confusion. “What do you mean by being generous with your rationality?” Gallus asks for the group. “Oh … perhaps I should be careful with my choice of words …” Her attempts to backpedal do little to answer Gallus’s question; she sighs and continues: “Obviously, with my line of work, I get all sorts of requests from all sorts of VIPs … Very Important Ponies, of course. And in some cases, those requests can push my sanity beyond tolerance. BUT, one must always work towards the greater good, and not put one’s mental concerns at the forefront.” “Yona confused. Can professor give example?” “Yes, well … I cannot divulge personal information to you, but I suppose I can give you at least a little insight as to what I often have to deal with. Several weeks ago, I got a request from a well-known entertainer.” “Sapphire Shores?” “The Countess?” “An act at Burning Dragon?” “No no no, I simply cannot go into details,” Rarity insists, “but I will say that this particular entertainer, for one particular performance, had five backup performers, and she wished to have each of them made up in a dress consisting primarily of a single type of gemstone. A shiver runs down Gallus’s spine, as he begins to sense the direction the lecture is taking them. “Now again, I cannot reveal details … so let us refer to the backups each by a single letter: A, B, C, D, and E. My client asked that each have a dress made up of one of four gems: either emerald, sapphire, ruby, or diamond. No dress should have more than one gem type, and at least three different gem types had to be used for the dresses, though not necessarily all four gem types.” “Sounds simple so far,” Silverstream states. “Would that it were so simple … but my client had a number of provisions that she simply MUST have; else, she would cancel the entire order.” Don’t ask, don’t ask, don’t ask … “What sort of provisions?” Sandbar’s query causes Gallus to slap his head with a wing and groan to himself. “Is everything all right?” Rarity asks with concern. “Huh? Oh … yeah, everything’s fine,” he replies with as much sincerity that he can muster. “OK then … well to answer your question, Sandbar, my client asked that the gem types used for the backups adhere to strict guidelines that, honestly, still make little sense to me.” “Like what,” Smolder asks. Rarity muses for a moment. “Yes … it’s still fresh enough in my mind … lesse …” “ ‘C must wear sapphire, unless D wears ruby. And D must wear emerald, unless B wears the same gem type as E. And E must wear either sapphire or emerald, with A wearing diamond if E wears sapphire and with A wearing ruby if E wears emerald. Oh, and C should wear ruby, unless B wears either sapphire or emerald, and if B does wear sapphire then C should wear diamond, and if B wears emerald instead then C should wear emerald as well.’ ” … “Now Yona really confused.” “Yeah, it sounds like your client contradicted herself, like, a dozen times,” Smolder adds. “Believe me, it was a lot to process. But the customer is always right, as they say. And after spending a lot of time parsing out her request, I was able to come up with an ensemble that satisfied all her demands.” “Really?” Silverstream asks in surprise. “How?” Rarity opens her mouth to speak, when Gallus suddenly interrupts: “Don’t bother professor; we’ve been through these a lot as of late.” Author's Note So … can you determine which gem type each of the 5 backups wore? Puzzle 7: Silverstream of ConsciousnessQuizzes are never fun … but for whatever reason, quizzes given by Fluttershy never seem as daunting or fearsome as those for Headmare Twilight or the other professors … “I’m sure you’ve all done wonderfully well on this quiz,” the soft-spoken instructor states as she takes the last of the students’ papers. “We have some time left before class is over, so I’d like to introduce you to a most unusual piece of flora.” “It’s not dragonsneeze trees, right?” Smolder cautiously asks. “Oh, nothing like that … but I must ask that you not touch.” The students turn to each other with mixed feelings of curiosity and trepidation as Fluttershy walks over to a closet and carefully clutches a potted plant sporting blue foliage, gingerly bringing it to her desk. “OK, everycreature; come on up and look, but again, do not touch.” The group of six approach the front desk and crowd around the plant. “So what’s so unusual about this?” Gallus asks. “This is ‘poison joke’,” Fluttershy begins to explain. “Don’t you mean ‘poison oak’?” Sandbar interrupts. “No … ‘poison joke’. And it has strange properties, causing reactions that would make Discord proud.” The students exchange more strange glances towards each other. “Yona confused. Plant causes different reactions for different creatures?” “It does. One time, when all your professors once stumbled into a field of poison joke, we all came down with various afflictions. Headmare Twilight lost her magical ability, Pinkie Pie lost her ability to speak, and Rarity came down with an awful case of frizzy mane.” “And what happened to you?” Ocellus asks. Fluttershy chuckles. “My voice became very deep,” she explains, attempting to naturally recreate a baritone sound. “At first I got very depressed … but with the help of Zecora, I and everypony else got better, and we’ve since been able to laugh at the whole incident.” “Uh … professor? …” Fluttershy turns to address Silverstream, who by now had grown rather quiet. “Goodness, you look nervous. Is everything all right?” “Uh, yeah, everything is fine … so … how long does it take for symptoms to appear?” Fluttershy thinks about the question for a moment. “About a day, I guess? We all got a good night’s sleep before coming down with any ailments. Why do you ask?” “Well yesterday I may have taken a stroll around the area, and I may have come across a patch of this ‘poison joke’ … and I may have rolled around in it because I enjoy the feel of plants.” Instinctively, the other students back away quickly from both Silverstream and the desk. “Oh dear … wait here, everycreature, while I get Twilight,” Fluttershy requests before bolting from the room. The other five students examine the hippogriff in awkward silence. “How do you think this ‘poison joke’ affects non-ponies?” Smolder asks Ocellus. “I couldn’t even imagine,” the changeling replies. “How seapony feel?” Yona asks. “I feel fine, I guess. At least I don’t feel any differently but then I usually feel like this like the other day I was telling myself I-feel-great-but-I-could-always-feel-better-if-I-just—” Silverstream abruptly clasps a talon to her mouth, catching the others by surprise as her eyes dart this way and that. “Silverstream, what’s wrong?” Sandbar asks. Carefully, Silverstream removes her talon from her mouth. She opens her mouth: “I-don’t-understand-all-of-a-sudden-I-can’t-help-but-talk-and-go-on-and-on-and-on-but-I-want-to-be-quiet-but-I-just-have-this-impulse-to-keep-going—” She quickly reclasps her mouth with her talon. “Whoa … and I thought dragonsneeze tree were strange,” Smolder admits. “Don’t worry,” Ocellus reassures, “Twilight and the others will be here to cure you soon enough.” Moments later, Twilight and Fluttershy rush into the room. Twilight trots over to Silverstream, noting the talon over her mouth. “Silverstream, how are you feeling.” She uncovers her mouth: “I-feel-fine-except-I-can’t-stop-talking-which-is-weird-since-I-like-having-conversations-but-this-feels-like-too-much-so—” and immediately re-covers her mouth. Twilight nods and turns to Fluttershy. “Can you recreate Zecora’s cure?” Fluttershy nods. “Go ahead and start a bath; let us know when you’re ready.” Fluttershy nods again and heads off. Twilight turns back to Silverstream. “You’ll be fine, it’s just a strange sensation you’re feeling now. One thing we’ve learned is that ‘poison joke’ is easily cured.” Silverstream nods. Twilight looks towards the desk, seeing the quiz recently taken by the students. “Maybe we can take your mind off your illness for a bit. So, how do you think you did on the quiz you just took.” Silverstream again opens her mouth: “Gallus-was-higher-than-Sandbar-but-Sandbar-was-higher-than-Ocellus-while-Ocellus-was-higher-than-Yona-but-then-Yona-was-2-higher-than-Smolder-while-Smolder-was-3-higher-than-Sandbar-and-Gallus-was-4-higher-than-Yona-but-Ocellus-was-5-higher-than-Gallus—” Silverstream clasps her mouth once again, as Twilight and the others take in the long list of declarations. Minutes later, Fluttershy reenters the room. “I think everything is ready for Silverstream.” “That’s wonderful. Let’s go, everycreature …” The last of the daylight having left Equestria, the students settle down in the dorm, grateful that Silverstream is now rid of the bizarre disease. “It was weird,” she relates to the others. “It was like I felt compelled to keep talking, even ahead of breathing. I never really appreciated pauses in conversation before now.” “What important is seapony all better,” Yona declares. The others nod in agreement. A knock on the door causes the group to turn to the entrance. “Come in,” Gallus calls out; moments later, the door opens and Twilight enters the foyer. “I just wanted to check in on all of you, and especially Silverstream.” “I’m feeling better, thank you. And thanks to Fluttershy for that curing bath; it felt invigorating!” Twilight smiles. “You can thank her face-to-face when you see her tomorrow. Oh, and she finished grading your quizzes in case you’re interested …” She chuckles to herself. “What’s so funny?” Smolder asks. “Well, it’s just that … all those statements Silverstream made while infected with the poison joke.” “You mean the ones concerning our scores on Fluttershy’s quiz?” Ocellus adds. “Yeah, it’s just that … well, she made 7 statements regarding how you placed. And every comment where the students mentioned were either both above or both below Silverstream was true, while every comment where one student was above and one student was below Silverstream was false.” Twilight’s horn glows as a quill jots down the statements on a sheet of paper for reference. “Well, I have to go. We’ll see you all tomorrow, and again, I’m glad to see Silverstream isn’t the worse for wear.” With that, the alicorn departs. “Why headmare not tell us order?” Yona asks. “I think she did,” Sandbar replies. Gallus shakes his head. “Unbelievable … now the plants are giving us puzzles!” Author's Note In what order did the students place, 1st through 6th? (Assume no ties.) Puzzle 8: Chaotic ReasoningThe last of their classes completed, the students head down the hallway leading to Starlight’s office. “A non-eventful day for a change,” Ocellus points out. “Just a quick meeting with the counselor, and the rest of the day is ours.” “It’s weird,” Silverstream says, “I got so used to dealing with solving puzzles in class, now it feels strange not to experience one in a class.” “Just as well,” Smolder replies. “My brain could use a cooling down after the last couple days.” “Yeah, it’s nice to just have a normal school day for a change,” Gallus adds as they reach Starlight’s office. He opens the door … as water pours from the office, quickly enveloping the hooves, claws, and talons of the students. “Why do I tempt fate like that?” he asks as he and the other peek into Starlight’s office. They are surprised to see a downpour taking place within the confines of the office, as Starlight stands atop her desk while barking orders towards a familiar former administrator: “FOR THE LAST TIME, DISCORD, STOP THE RAIN!!!” Discord snaps his fingers; immediately, the clouds that had been enveloping the ceiling disappear as the precipitation ceases. “I still don’t see why you’re so irate … you said you had to water the flowers, I thought I’d save you some time,” he replies. “And now that I’ve done you a favor, our friendship dictates that you reciprocate in kind.” Starlight shakes off as much moisture from her coat and mane as possible. “There’s no position available at the present time, Discord.” “Oh, be reasonable,” he insists, drawing a laugh from the unicorn. “And just what does that mean?” “Discord, you are one of the most unreasonable creatures I’ve ever met! And I’m not saying that to be mean; it’s just your nature.” “Why, I am aghast at your insinuation, Starlight. I can be plenty reasonable, much more than you, of course!” Starlight rolls her eyes at the notion. “Uh, Counselor Starlight,” Silverstream meekly asks, drawing the attention of the two in the office. “Is this a bad time, because we can come back later.” An idea pops in Discord’s head. “Actually, my little nonponies, this is a perfect time to demonstrate my superior powers of reasoning.” “Didn’t you say ‘reasonable’?” Sandbar asks. “Six of one … speaking of six, how about each of you six think of a number, and I’ll try to figure out what numbers you’ve thought up.” Smolder delivers a flat ‘No’ to Discord. “We’ve been doing all sorts of puzzles like that all week. We need a break!” “Fine,” Discord responds in a huff. “I guess I’ll create my own puzzle, one that Starlight won’t be able to figure out. And to relieve you of the burden of thinking up numbers, I’ll assign numbers to you. I’ll even direct you as to what to say.” “Nasty trickster can’t put words in Yona’s mouth,” the yak defiantly states. Discord snorts, then snaps his fingers. Moments later, Yona feels a queasiness from her stomach, causing her to groan. “What trickster do to … Yo … na … AAAUGH!” Before finishing her sentence, Yona spits out a scrolled-up parchment onto the floor. Similar groaning sounds cause her to turn to her fellow students, all of which have experienced the same sensation, and all of which now stand before similar parchments. “Blecch,” Silverstream moans. “How does Spike do this on a regular basis.” Cautiously, she picks up her scroll from the ground and unfurls it; the others do likewise. “What does this even mean?” Sandbar asks. “Just read them, all of you,” Discord requests. “And you may want to take notes, Starlight.” “And if we refuse?” Gallus asks. Discord groans. “You want motivation? Fine, here’s my proposal: if your counselor can solve my puzzle, I’ll rescind my request for a more permanent position here.” Starlight rolls her eyes. “Whatever … anything that will get you out of my office.” Her horn glows as a parchment and quill fly from a far, dry corner of the office. “Excellent,” Discord says. “Now there are rules to follow, so pay attention: I won’t be repeating myself. I have assigned each of these six a number from 1 to 60. And because I like things on the odd side, all their numbers are odd, as well as different. All you have to do is figure out their numbers.” Starlight turns to the students. “What do the scrolls say?” Each student, in turn, reads off his or her scroll: Smolder goes first: “No one has a number that is a multiple of 9. I have the largest number.” Gallus follows: “My number is a perfect cube. Yona’s number is a perfect square. Silverstream’s number contains a ‘3’.” Ocellus goes next: “Yona’s number contains a ‘4’. Smolder's number contains a ‘9’. No two of us have numbers within 7 of each other.” Sandbar continues: “My number contains a ‘1’. Silverstream’s number is 21. Gallus’s number is between 16 and 24.” Silverstream follows: “My number is 10 less than Gallus’s number. Sandbar’s number is 27.” Yona goes last: “Ocellus truthful. Sandbar lies. Both Gallus’s and Silverstream’s numbers prime.” “Oh, and one other thing: any statement made by a creature with a number greater than 30 is true, unless the number is prime, in which case those statements are false. In addition: any statement made by a creature with a number less than 30 is false, unless the number is prime, in which case those statements are true. Got it? Good luck … I suspect you’ll need luck more than reasoning!” Discord gives a final laugh before *POP*-ing out of the office. Author's Note Probably my most contrived storyline, but I really like the puzzle I created. I hope you, the reader, enjoy solving the puzzle as much as I enjoyed creating it. Puzzle 8 solution Smolder: “No one has a number that is a multiple of 9. I have the largest number.” Gallus: “My number is a perfect cube. Yona’s number is a perfect square. Silverstream’s number contains a ‘3’.” Ocellus: “Yona’s number contains a ‘4’. Smolder's number contains a ‘9’. No two of us have numbers within 7 of each other.” Sandbar: “My number contains a ‘1’. Silverstream’s number is 21. Gallus’s number is between 16 and 24.” Silverstream “My number is 10 less than Gallus’s number. Sandbar’s number is 27.” Yona: “Ocellus truthful. Sandbar lies. Both Gallus’s and Silverstream’s numbers prime.” Over 30 TRUE … unless prime, in which case FALSE. Under 30 FALSE … unless prime, in which case TRUE. All numbers different, odd, and under 60. Starlight peruses the parchment for minutes. “Twilight taught me a lot of things,” she admits to the students, “but we never got around to logic puzzles. I don’t know where to begin.” “Leave everything to us,” Gallus reassures. Starlight turns to Gallus and blankly blinks. “Long story, we’ll tell you about it later,” he adds before turning to the other students. “Everyone focus on the statements ‘you’ said; divide-and-conquer seems like the best way to start … and I think I can lead us off.” The others refocus their thoughts to their individual statements as Gallus begins the thought process: “I said my number was a perfect cube. Now the only perfect cubes less than 60 are 1, 8, and 27. It couldn’t be 8, because all our numbers are odd. And if it were 1 or 27, my statement would have to be false.” “Wait,” Sandbar asks, “isn’t 1 a prime number?” “1 neither prime nor composite,” Yona explains. “1 special, like Yona,” she adds with a smile. “OK,” Gallus continues, “so the upshot is I could never truthfully state that my number is a perfect cube. That means all my statements are false, so Yona’s number isn’t a perfect square and Silverstream’s number doesn’t contain a ‘3’.” “Wow … I’m impressed,” Starlight states. “You say you’ve been doing this a lot?” “More than I’d like to admit,” he concedes. “Anyone got anything else?” “I … I think I do,” Sandbar says. “Based on what you just said.” “What do you got?” Ocellus asks. “Well … I said that Silverstream’s number was 21 and Gallus’s number was between 16 and 24. If both of those were true, then his number would have to be 17, 19, or 23 … but all of those are prime, which would mean Gallus’s statements would be true. So I must have lied, too. And my number doesn’t contain a ‘1’.” “Oh, oh,” Yona interrupts, “that mean Yona truthful when Yona say Sandbar lied. So Yona truthful, and that mean Ocellus truthful and both Gallus and Silverstream have prime numbers!” “And that means my statements are true, too,” Ocellus adds. “So Yona’s number contains a ‘4’, Smolder’s number contains a ‘9’, and all our numbers are 8 or more apart from each other.” “Remarkable,” Starlight says in awe. “So you already know the veracity of four of your sets of statements.” “We’re very well-versed … city,” Silverstream says. “But do we know any actual numbers yet?” Yona ponders Ocellus’s statement regarding her own number: “Ocellus truthful, so Yona have number with ‘4’ in it … but 4 can’t be last digit, because all numbers odd. So Yona have 40-something.” “But not a prime number,” Smolder adds, “or else you’d be lying. So … that means it’s not 41, 43, or 47.” “It’s not 49, either,” Gallus adds, “because that would mean Yona’s number is a perfect square, making one of my statements true. So that only leaves—” “YONA HAVE 45!!!” the young yak enthusiastically concludes. “Awesome,” Sandbar says. “One down, five to go. So … now what?” Smolder comments on her own statements: “45 is a multiple of 9 … so I guess I’m a liar, too.” “I’d say you’re in good company,” Gallus responds with a grin. Smolder remains unfazed. “What, you don’t get? … whatever. As you were saying?” “Yeah, well … since I’m a liar, I don’t have the largest number. Now, my number has a 9 in it, so the choices are 9, 19, 29, 39, 49, and 59. And it sure isn’t 59, since that’s the largest number possible.” “It’s also not 39 or 49,” Ocellus adds, “since you’d be telling the truth if you have those numbers.” “Same with 19 and 29,” Sandbar adds as well. “So I guess I got 9,” Smolder concludes. “There’s our second number. Now what?” The group studies the parchment for several minutes. “Now it gets difficult,” Gallus concludes. “Let’s go back to your number,” Ocellus suggests. “What could it be?” “A lot of things, right?” “Not really. I mean, we know it’s a prime number, and that mean it’s greater than 30. It can’t be within 7 of 45, so … it can only be 31, 37, 53, or 59.” “Ooo, now I know something!” Silverstream excitedly asserts. “If my statements were true, then my number would be 10 less than Gallus’s number. That would make my number either 21, 27, 43, or 49.” “But Silverstream have prime number,” Yona adds. “So number would be 43.” “And that would make Silverstream a liar,” Sandbar adds as well, “so Silverstream can’t be truthful, and her number isn’t 10 less than Gallus’s number.” “But her number is still prime,” Smolder follows, “so it’s greater than 30 just like Gallus’s. And, her number doesn’t contain a 3. So what does that leave?” Gallus runs through possibilities: “Greater than 30, doesn’t have a 3 in it, so greater than 40 … not within 7 of 45, so at least 53 … but that contains a 3, so …” “Oh, I have 59!” Silverstream finishes. “And that means my number isn’t 59 or 53,” Gallus adds, “so it has to be either 31 or 37.” “And everyone else’s number has to be less than 30,” Ocellus states. “Let’s go back to my number,” Sandbar suggests. “It has to be less than 30, so it’s can’t be prime. It’s not 1, it can’t be within 7 of 9, it can’t be 17 or 19, or 21 or 23. It has to be 25 or 27 … but It’s not 27 either, because of Silverstream’s statement. I have to have 25!” “And that means I can’t have 31,” Gallus notes, “so I have 37.” “So what’s left for my number? Can’t be within 7 of either 9 or 25, can’t be 1 … my number can only be 17.” “That all numbers,” Yona says with finality: “Smolder have 9, Ocellus 17, Sandbar 25, Gallus 37, Yona 45, and Silverstream 59.” … Silence pervades for several seconds. “Did we … just solve that puzzle?” Silverstream asks. “YOU MOST CERTAINLY DID, YOU MEDDLING KIDS!!!” The students all turn to the entrance and eye Discord nonchalantly standing there, eyeing the group right back. “Technically I win the challenge, since your counselor did nothing more than watch and maybe learn. Still … the way you six demonstrated finding reasoning amidst the chaos I put into that puzzle speaks volumes:” He snaps his fingers, and a number of tomes fly off the bookshelves, ‘flapping their covers’ as they speak accolades about the students: “Well done, chaps.” “Jolly good show.” “Much better than some unicorns!” Starlight’s eyes glare as Discord snaps his fingers a second time. The books immediately return to their places along the bookshelves. “When indoors, it’s always proper to have volume control,” he points out. “In any event, I’m late for work, so I’ll let you be … for now.” “WHAT???” Starlight shouts. “Since when do you have a job?” “Since there seems to be little in the way of educational job openings, I thought I’d offer my services elsewhere to pass the time.” “So … exactly what do you do?” Ocellus asks. “Air-traffic control. I’ll be monitoring the hot-air balloons for the foreseeable future. Anyway, well done to the six of you … and considering yourself lucky,” he adds to Starlight before disappearing with a final *POP*. The puzzle now behind them, the students take in one last look around the water-logged office. “You want us to get a mop or something?” Silverstream asks. “You’ve done so much for me already,” Starlight insists. “I am thoroughly impressed what you were able to accomplish this afternoon.” “Because we solve puzzle?” Yona asks. “Because you got rid of Discord,” Starlight corrects. “Hopefully, now we can return to a sense of normalcy around here.” “Maybe,” Smolder says, “but if I’m taking a break near the fountain outside, I’ll still watch for falling balloons.” Author's Note Bonus points if you figured out my 'Shout-Out' at the end. Puzzle 9 solution“I thought you weren’t interested in puzzles,” Smolder says to Gallus. “I wasn’t … but we’ve been getting so many of these lately, I can’t help but grow at least some interest, right?” “So the fact that I can’t solve this is enough to let you solve this?” Ocellus asks. “It’s not just that you can’t solve the whole puzzle … it’s that you can’t figure out anyone else’s number. That’s what gives me your number; I already know both Smolder’s and Sandbar’s.” “Really?” Sandbar asks. “How?” “Smolder said her number was three less than Sandbar’s. That forces their numbers to be either 1 and 4, 2 and 5, or 3 and 6 … but they can’t be 2 and 5, because Yona already said her number was 2. So that leaves either 1 and 4 or 3 and 6.” “OK, so how you know Ocellus number?” Yona asks. “If Ocellus had either 1 or 4, she would know that Smolder and Sandbar had 3 and 6. And if Ocellus had either 3 or 6, she would know that Smolder and Sandbar had 1 and 4. In any of those cases, she would know more than just her own number and Yona’s. The only way she could be completely stuck is if she has 5.” Ocellus shows her number to the others. “You’re right … I do have 5 … and I don’t feel so stupid, now.” “ ‘Stupid’ is the last word I would ever use to describe you,” Gallus flatly states. “Yeah, you’re brilliant,” Silverstream adds. Ocellus smiles at the comments. Yona speaks up: “So … since griffin say he have higher number, he have 6. And that mean dragon have 1 and pony have 4.” “And that leaves 3 for Silverstream,” Ocellus adds. “And that’s all there is to it,” Gallus concludes: “Smolder has 1, Yona has 2, Silverstream has 3, Sandbar has 4, Ocellus has 5, and I have 6 … and considering my baking prowess, it’s just as well I go last.” The group shares one final chuckle before heading to their next class. Puzzle 10: QuibbleFic Part I (Bears and Wolves)Closing her eyes, Silverstream leans back and falls into the waiting hooves of Sandbar. “Wow, so that’s what a trust fall is, huh? That was incredible!” “I’m still uneasy about the whole thing,” Smolder states despite Gallus successfully catching her. “It’s all part of loyalty, I guess,” Ocellus concedes, Yona having caught her moments earlier. “So what now?” she asks Rainbow Dash, noting her professor perusing a collection of hoofwritten notes. Rainbow looks up with a start. “Wha? … oh yeah, go ahead and change places and do it again.” Ocellus turns towards Yona, then back to her professor. “Um … I don’t wish to cause trouble, but …” “What are you looking at?” Gallus interrupts. “You haven’t paid any attention to us since we got here.” Rainbow groans and shoves the papers aside. “I’m sorry, everyone. I’m just trying to do somepony a favor, but it’s proving harder than I thought.” “Are you trying to do someone’s taxes?” Silverstream asks. “I’ve heard those are super-challenging.” “No, it’s nothing like that,” Rainbow says before sighing. “I’m reading fanfiction for a friend.” “What fan-fiction?” Yona asks. “Lemme back up … how many of you are familiar with Daring Do?” Sandbar nods as the others shrug in ignorance. “Well,” she starts, careful not to reveal her favorite author’s true identity, “Daring Do is this pegasus adventurer who goes on quests for rare artifacts and treasures. Her stories are really incredibly awesome,” she adds with a touch of over-enthusiasm. “Sounds interesting … for a pony, anyway,” Smolder concedes. Rainbow continues: “Now many of us … I mean, many other fans are so impressed with these stories that they create their own stories related to the original.” Gallus scoffs. “Really? Ponies can’t make enough of their own lives, so they have to live the life of made-up characters?” Smolder and Yona chuckle at the comment, as Rainbow bites her tongue. “In any case, I have this friend that I met at a Daring Do convention, named Quibble Pants. The other day, I got a message from him: he’s been writing his own Daring Do stories, and he wants my opinion.” “Let me guess,” Gallus says, “his writing is terrible and you don’t know how to break it to him.” “That’s an awful thing to say,” Ocellus chides, before pondering for a moment and turning towards Rainbow. “That isn’t what the problem is, is it?” “His writing is OK, I guess … but he and I have differing thoughts as to what makes for the best stories. I love the action-adventure aspects, but he’s all about the puzzle-solving that was more … uh … there was more of it in the earlier stories.” “You mean it was more prevalent?” Ocellus asks. “Yeah, that … saaay …” Rainbow continues, an idea popping into her head. “Say what?” Yona asks. “I’ve been hearing a lot from the other professors about how well you six solve puzzles … like, thinking puzzles and whatnot … maybe you can help me out with these.” “Seems like our reputation is spreading,” Smolder states. “So this ‘Quibble’ guy, he made puzzles?” “Yeah, kinda like a lot of the earlier stories, but way harder,” Rainbow states. “Like, he’s got a whole chapter where Daring Do is imprisoned by some new villain that he dubs a logitaur.” “A what?” Sandbar asks. “Basically, a minotaur who likes logic puzzles the same way Quibble does. So without going into the whole plot, the logitaur has Daring Do trapped in a room with 3 doors; she has to choose the correct door to make her way to freedom.” “What if she chooses the wrong door?” Silverstream asks. “Then she gets mauled by either a bear or a wolf.” The others exchange funny looks with each other. “I skipped some stuff,” Rainbow admits. “Basically, the logitaur keeps a number of bears and wolves, whose job is to prevent trespassers from escaping.” “Wonderful,” Gallus deadpans. “So what does any of this have to do with puzzles?” “Here, have a look,” she says as she passes a sheet of paper across her desk. The students gather round to inspect the puzzle: Door 1: “This room contains a bear.” Door 2: “This room contains a wolf.” Door 3: “This room leads to freedom.” “According to the story,” Rainbow explains, “the logitaur placed a bear in one room and a wolf in another room; the third room leads towards the final exit. If Daring Do chooses the wrong door, really bad things will happen to her. Each door has writing on it to hint at the correct way forward.” “So why not just take Door 3?” Sandbar asks. “Because the logitaur put in rules regarding the messages: Any room containing a bear has a true statement, and any room containing a wolf has a false statement. The room leading onward could have either a true or a false statement on it.” The group ponders the rules set out for a minute. “Seems like a convoluted way to get the reader to solve a puzzle,” Smolder states. “But still, an interesting puzzle,” Ocellus adds. “Fairly easy though, this one.” “There’s more,” Rainbow states. “After Daring Do solves the first puzzle, she moves on to another room that contains 5 doors. Of those, two contain bears and two contain wolves, with one room leading onward. Here’s what the doors to those rooms say,” she continues as she pushes a second page towards the group: Door 1: “Rooms 2 and 3 both have bears.” Door 2: “Rooms 4 and 5 both have wolves.” Door 3: “Rooms 2 and 4 contain animals of different types.” Door 4: “Rooms 2 and 5 contain animals of the same type.” Door 5: “This room leads to freedom.” “That one look harder,” Yona states. “And Quibble created a third puzzle,” Rainbow states. “After Daring Do solves the second puzzle, she moves on and eventually finds herself in a third room, this time containing 7 doors. Three bears, three wolves, and one room that leads onward.” She pushes a third page across her desk: Door 1: “No two adjacent rooms contain animals of the same type.” Door 2: “An even-numbered room has the exit.” Door 3: “The room with the exit is adjacent to a room with a wolf.” Door 4: “Either this room has a wolf or Room 5 has a bear.” Door 5: “Room 2 has a wolf and Room 6 has a bear.” Door 6: “Room 3 has a wolf.” Door 7: “This room leads to freedom.” “And after that,” Rainbow starts. “Don’t tell me this guy put in a fourth puzzle!” Gallus says with some exasperation. “Now you know how I feel … but no, he has something completely different after this. But still, he expects me to solve all three of these puzzles. Can you believe that?” Yona speaks: “So when professor say class can help out, professor mean …” “Can you solve these?” Author's Note Moving on for the moment from E. R. Emmet to Raymond Smullyan for the next couple puzzles. Those who have read this far and have not already done so may be interested in another of my Fics: The Chalice of Rhinyorgrun. That Fic contains similar puzzles to what I have here, but uses colors rather than words to signify instructions. If you do decide to peruse, let me know what you think. Puzzle 10 solutionThe group take another look at the three pages containing Quibble’s door puzzles. “We can probably solve these,” Gallus states, “but what does solving puzzles have to do with loyalty?” “Oh, well … it has everything to do with loyalty,” Rainbow explains. “You see … one key aspect of loyalty is being there for your friends when they need you the most. And, uh … I need you, right now.” “So … you’re not just our professor, you’re our friend?” Smolder asks. “We’re all friends,” Rainbow earnestly asserts. “So … all we do now is puzzles?” Yona asks. “Sure looks like it,” Sandbar replies. “May as well get to it.” The group focuses on the first puzzle: Door 1: “This room contains a bear.” Door 2: “This room contains a wolf.” Door 3: “This room leads to freedom.” “This one’s pretty easy,” Ocellus states. “The exit must be Door 2.” “Why is that?” Rainbow asks. “If Room 2 had a bear, the statement would be true, meaning the room had a wolf, which is doesn’t. And if Room 2 had a wolf, the statement would be false, meaning the room didn’t have a wolf, but it does.” Rainbow’s head begins to spin. “Could you go over that one more time?” “If statement true, room have wolf, but wolf room lies, so statement false, which mean no bear either, so room lead to freedom,” Yona sums up. “Oh … I think I get it. But what about the other rooms?” Silverstream takes a shot: “Well, Door 3 must be false now, so that’s where the wolf is. And that leaves Door 1 to lead to the bear. Easy-peasy.” “Huh … not bad,” Rainbow admits. “What about the next puzzle?” The students shift their focus to the second puzzle: Door 1: “Rooms 2 and 3 both have bears.” Door 2: “Rooms 4 and 5 both have wolves.” Door 3: “Rooms 2 and 4 contain animals of different types.” Door 4: “Rooms 2 and 5 contain animals of the same type.” Door 5: “This room has the exit.” “So now what?” Rainbow asks. “To start,” Smolder says, “Room 5 can’t have a bear. If it did, the statement would be true and it would lead to the exit, not contain a bear.” “Meanwhile,” Ocellus continues, “Room 1 can’t have a bear, either. If it did, then Doors 2 and 3 would both lead to bears as well, and there’d be at least three bears … but we know there’s only two bears to go with two wolves and one door to freedom.” “So that means Door 1 is false,” Gallus says, “and either Room 2 or Room 3 doesn’t have a bear. And that means Room 4 has a bear.” “Uh, Gallus?” Ocellus says. Gallus pays no heed and continues: “And since Room 4 has a bear, Rooms 2 and 5 have animals of different types … but Room 5 doesn’t have a bear, so it has a wolf and Room 2 has a bear.” “Gallus?” “And since Room 2 has a bear, Rooms 4 and 5 both have wolves. So Room 3 has the other bear, and Door 1 leads to freedom. Nothing to it,” he concludes. “GALLUS!” The griffon turns to Ocellus. “What?” “Door 1 CAN’T be false! You said Room 4 had a bear, but then you said Room 4 had a wolf. It can’t be both!” Gallus blinks … and goes over his reasoning a second time. “Hmmm … if Door 1 was false, then Door 4 would have a bear, so Rooms 2 and 5 would different animals, but Room 5 doesn’t have a bear, so Room 2 has a bear, and then Rooms 4 and 5 have … so Room 4 would have a wolf … so there’s no solution!” “No,” Ocellus insists. “There IS a solution.” “How? Now I know Door 1 can’t be false so it has to be true, but there can’t be a bear in Room 1.” “Right … so therefore …” The realization dawns on the griffon. “Ohhh, Door 1 leads to the exit!” He continues to ponder the possibility: “Lesse … Door 1 is true means Rooms 2 and 3 have bears, leaving Rooms 4 and 5 to have wolves. Do all the statements work? … Huh, I guess they do.” “Whoa … harder than it looked,” Smolder observes. “What about the last puzzle?” The six ponder the final puzzle: Door 1: “No two adjacent rooms contain animals of the same type.” Door 2: “An even-numbered room has the exit.” Door 3: “The room with the exit is adjacent to a room with a wolf.” Door 4: “Either this room has a wolf or Room 5 has a bear.” Door 5: “Room 2 has a wolf and Room 6 has a bear.” Door 6: “Room 3 has a wolf.” Door 7: “This room has the exit.” “Anyone wanna start?” Gallus asks the group. “Room 7 not contain bear,” Yona points out. “Just like last couple puzzles.” “OK,” Gallus continues, “What else?” … A minute of silence passes. … “What about Room 4?” Sandbar finally says. “What about it?” Smolder asks. “It can’t have a wolf, right? Because if it did, the statement on the door would be true.” “Good job!” Ocellus states. “I must be slipping.” “Or we’re getting better,” Silverstream adds. The others, including Ocellus, chuckle at the comment. “All right,” Smolder says, “so Room 4 can’t have a wolf. Can it be the exit?” “Going right for the finish, huh?” Sandbar says. “But then the statement could be true or false.” “So let’s start with ‘true’,” Ocellus suggests. “that would force Room 5 to have a bear, and that would mean Room 2 has a wolf and Room 6 has a bear.” “But Door 2 true,” Yona states. “So Room 2 not have wolf.” “That’s right,” Sandbar concurs. “So if Room 4 was the exit, the statement would have to be false.” “Which means Room 5 doesn’t have a bear,” Ocellus adds. “And since it would have to have an animal, it would have a wolf.” “Room 7 also have wolf,” Yona adds, “since it not exit.” “And based on the statement on Door 6,” Gallus continues, “either that room or Room 3 has the final wolf.” “Meaning Rooms 1 and 2 both have bears,” Smolder states, before pausing. “… but Rooms 1 and 2 are adjacent, so Door 1 would be false, and that can’t be.” “And all of that goes to show that Door 4 doesn’t lead to the exit!” Ocellus concludes. Rainbow eyes the students quizzically. “Wait … how long have you been doing these?” Yona ponders her professor’s question. “Maybe one week?” Rainbow blinks, then shakes her head. “That’s … that’s ...” “So after all that,” Silverstream states, “what does that mean?” “It means,” Gallus replies, “that Room 4 has a bear. Which means the statement on that door is true, and that means Room 5 has a bear as well.” “And that mean Room 6 have bear and Room 2 have wolf,” Yona adds. “No more bears left.” “But we have two wolves, too: Room 2 and 3,” Sandbar says. “All that’s left are Rooms 1 and 7.” “What haven’t we used,” Smolder asks aloud. Several seconds pass. “What about Door 3?” Silverstream finally asks. Ocellus responds: “Door 3 has to be false … which means the room with the exit isn’t adjacent to a room with a wolf. So the exit can’t be Room 1, since Room 2 has a wolf. But it could be Room 7, since Room 6 has a bear.” “And that means it has to be Room 7,” Gallus concludes. “That’s funny,” Silverstream adds: “The room that says it’s the exit really is the exit this time!” “Probably intentional, considering the previous puzzles,” Ocellus suggests. “ ‘Fool me once’, and all that?” “So that’s that,” Smolder states. “Hope that helps, professor … uh, professor?” The group all turn to face Rainbow, noting the spiral formation in her eyes. “Is professor pony OK?” Yona asks. “… why yes, I would love some eggs,” Rainbow replies in a half-daze. Sandbar turns his attention to the clock on the wall. “Whoa, class is almost done for the day.” “Looks like whatever that other puzzle she wanted us to look at is gonna have to wait,” Ocellus states. “You guys wanna hang by the fountain for a bit?” Gallus suggests. The group collectively nod in approval, and quietly file out, leaving Rainbow Dash to stare blankly at the back wall. “… no more eggs, I’m good …” Author's Note Gallus's reasoning in the 2nd puzzle was my own. I knew a bear couldn't be in Room 1, but I forgot that the statement could still be true. Fortunately, the location of the exit was the same either way ... … and then it was pointed out that my original solution was wrong! I rewrote the section containing the error. I originally planned on doing the other 'QuibbleFic' puzzle immediately after, but I'm giving myself time to properly develop that one. Puzzle 11: Random Inference“Just tell me, already!” “Maybe Gallus tell Yona first!” “You first!” “No, you first!” Unable to focus on her reading, Ocellus groans and turns her attention towards her two vitriolic classmates. “Why are either of you making a big deal about this?” “It’s a matter of personal pride,” Gallus explains. “Look, I can accept that Yona scored higher than me on whatever you call that ‘random quiz’ thing Professor Pinkie Pie gave us, but I’d still like to know by how much.” “Then just tell each other what you got,” Ocellus suggests with a tinge of exasperation. “Private information is private,” Yona states, “and should be earned rather than given away.” Ocellus shakes her head and returns to perusing her book, keeping an ear towards the conversation. “How about this,” Gallus suggests: “we each ask a single question about the other’s score, then try to deduce what that score is. That sound fair to you? Up for the challenge?” he concludes with a grin. “Fine,” Yona says. “And Yona ask first.” “Go right ahead.” Yona ponders for a moment before asking: “Is Gallus score multiple of six?” Ocellus perceives a slight pause, assuming the griffon either nodded or shook his head. “My turn,” Gallus counters: “Is your score a triangular number? Oh … do I need to explain what that is?” “Yaks know what triangle numbers are!” Yona flatly states. “OK then … well, is your number triangular?” Ocellus perceives another pause. A smile comes across both Gallus’s and Yona's faces. They announce simultaneously: “I know your number.” / “Yona know Gallus number!” “Well here,” Gallus says as he pushes a piece of scrap paper towards the yak. “You write down your guess, and I’ll write down my guess.” “Yona not guess,” she replies in kind. She and Gallus write down their guesses and simultaneously show each other. Moments later, Ocellus catches a pair of cries of “HEY!!!” She puts down her book and turns to the others. “What now?” “YOU LIED TO ME!” “GALLUS LIE TO YONA!” “You two really deserve each other,” Ocellus states. Gallus and Yona look at each for several seconds, before bursting into laughter. “This really did spiral out of control, didn’t it?” Gallus admits. “Professor Applejack not approve,” Yona states. “All right,” Gallus adds. “I’ll just tell you first, and then you tell me.” “That quiz was out of 20 points, right?” Ocellus asks. Yona responds with a simple ‘uh huh’. “Then it’s settled,” Ocellus says … before stating both their scores. Gallus stumbles for a response: “How … how could you possibly know our scores???” “Yeah, and how you know Yona score?” “Well to be honest, I only knew for certain that one of those scores was correct … but apparently, your scores differ by more than one, and that’s all I need to know for certain.” Author's Note So, what score did Gallus and Yona each receive? Also, what score did each think the other got? Apologies for the delay; I was a bit under the weather, and was also in pure 'Netflix binge' mode. Also, I admit this puzzle is fairly simple … but I've struggled in the past with solving puzzles that require inference, and wanted to make my first such creation simple just to be sure I didn't err. My next puzzle should have a bit more meat on it. Puzzle 11 solutionAuthor's Note The original version of the puzzle contained incorrect information, as mentioned in the 'Comments' section for that chapter. Those who were stuck the first time may wish to re-read that chapter before continuing with the solution below. Puzzle 11 solution “If those were just guesses,” Gallus states, “I need to introduce you to the Griffonstone Lottery.” “Or maybe trip to Las Pegasus,” Yona adds. “They weren’t pure guesses,” Ocellus insists. “I knew that at least one of those numbers had to be; the luck was that your scores differed by more than one, which forced you to have those scores.” “But how could you know?” Gallus reiterates. “Your head was buried in that tome the whole time; you never saw us nod or shake our heads.” “Yeah, so how you tell what we answer?” Yona follows. “It’s all about inference,” Ocellus says as she begins her explanation. “The key is that each of you thought you knew the other’s score. So, start with Yona’s question to Gallus. Yona, you asked whether or not Gallus’s score was a multiple of six.” Yona nods. “Well suppose Gallus said ‘no’.” “Gallus not say anything,” Yona corrects. “Gallus just nod head.” Ocellus sighs. “You know what I … never mind. Suppose Gallus shook his head to mean ‘no’. Then how could you possibly know his score? I mean, your score could be 8, or 10, or 13, and you wouldn’t be able to figure much out about Gallus’s score. From your view, he could have a score of 1 or 2 or 3, right?” Yona ponders Ocellus’s statement. “Yona guess that make sense.” “Now it could be possible for you to know Gallus’s score … but only if your own score was 2, in which case you would know his score was 1. Anything higher, and you couldn’t know if his score was 1, 2, or possibly something else. And if your score was 2, Gallus would have a score of 1 and he would NEVER know your score after his one question.” “OK, OK, you got me,” Gallus admits. “Yes, I nodded to Yona’s question: I did claim that my score was a multiple of six.” Ocellus smiles. “All right, so now I figured out what Yona’s score could be.” Yona interrupts: “How? Yona not say whether number triangular or not.” “True, but again, you claimed to know Gallus’s score. So suppose your score was 13 or higher. How could you know whether Gallus got a score of either 6 or 12?” Yona struggles to respond. “Exactly … you wouldn’t know. And that means your score has to be between 7 and 12, and you thought Gallus’s score was 6.” Yona stares at Ocellus for a moment, then flips over a scrap of paper near her, revealing a guess of ‘6’ for Gallus’s score. “Yona impressed.” “Me too,” Gallus concedes. “But you narrowed it down further, didn’t you.” “Yes, because now I can focus on your question to Yona. You asked if her number was triangular; if she shook her head to say ‘no’, then you would have no way of knowing whether her score was 20 or 19, or possibly other values, unless your own score was 19 … and In that case, Yona would have a score of 20, which we now know cannot be.” “Yeah, Yona did nod her head, just like I did with her question to me.” “So now you’re thinking ‘Yona’s score is a triangular number.’ … what could that number be? The triangular numbers less than or equal to twenty are 1, 3, 6, 10, and 15.” “You know triangle numbers?” Yona asks in surprise. “I do … I know a lot about numbers,” Ocellus admits with a tinge of embarrassment. “I suppose I really shouldn’t be surprised,” Gallus states. “OK,” Ocellus continues, “so how can you think to know her score? If your own score was 9 or less, you couldn’t know whether her score was either 10 or 15 … so your score has to be between 10 and 14, and you must have thought Yona’s score was 15.” Gallus picks up the scrap containing his guess and shows it to Ocellus, confirming her belief. “Right again, you little detective.” Ocellus smiles at the comment. “So to sum up: Yona must have a score between 7 and 12, thinking Gallus scored 6, and Gallus must have a score between 10 and 14, thinking Yona scored 15. But Gallus’s score is less than Yona’s, so the only possible combinations are 10 and 11, 10 and 12, or 11 and 12. No matter what, either Gallus scored 10 or Yona scored 12, so that’s what I guessed … and, well … I got lucky with both, I guess.” ... “Wow … it’s like you’re in our heads,” Gallus states. “Yona prefer head privacy … but that was incredible.” “Thank you both,” Ocellus says. “I’ll just, uh, get back to my book, then.” As she turns to return to the couch, Gallus catches her with a closing question: “So are all changelings as smart as you with logic?” Ocellus chuckles. “Funny you should ask … Headmare Twilight pulled me aside after classes today to tell me to expect a visitor tomorrow. That’s kinda the reason why I’m rereading this,” she adds as she indicates the tome on the couch.” “Ooo, Ocellus have family visit? Can we meet with Ocellus family?” “Actually, Headmare Twilight says he’ll be a special guest lecturer for us tomorrow.” “Really?” Gallus asks. “Who’s coming to speak to us?” “My uncle Coccyx.” Welcome Back, Coccyx“Good morning, everycreature,” Twilight addresses the group of six. “I have a special announcement for you today.” “Special announcement?” Silverstream repeats with great anticipation. “Ooh, are we taking another field trip? Are we finally going to learn basic plumbing?” Confusion spreads across Twilight’s face, but she quickly shakes off the feeling and chuckles. “No plumbing lessons today, Silverstream.” “Aww,” she replies with a tinge of sadness. “Instead, you’ll have a special guest lecturer today!” Twilight cheerfully continues. Sandbar’s eyes prick up, as the others gaze on in mild interest. Slightly put off, Twilight addresses Gallus: “I thought there would be more enthusiasm from all of you.” “Sorry, Headmare, but we kinda already know who’s coming … Ocellus told me and Yona last night.” “Really,” Twilight sternly responds as she walks over to Ocellus. “You know, it’s not very friendship-like to spoil a surprise for others.” “I, oh, um … I’m sorry,” Ocellus nervously replies. Twilight stares down Ocellus for several seconds … then bursts into laughter. “I’m kidding; it’s all right if you told your friends already!” The news causes the young changeling to breathe a sigh of relief. Twilight returns to Gallus. “OK then, why don’t you tell the class who’s visiting today?” “Ocellus said her uncle Coccyx is visiting … but that’s all I know about it.” “I see,” Twilight continues with a wry smile. “So you don’t even know what this ‘Coccyx’ individual looks like, do you?” “No, she didn’t say. But he’s a changeling, so he could take on any appearance.” “He could, couldn’t he …” Gallus takes a second look at Twilight, as an idea dawns on him. “Wait … you’re Coccyx, aren’t you!” His declaration draws interest from the others. “Whoa,” Sandbar admits, “that’s impressive. You fooled me, anyway.” “You fooled all of us,” Smolder adds. “Nicely done.” ‘ Twilight’ peruses the students. “So, just between us … how do you feel about your real headmare?” “Headmare Twilight is awesome!” Silverstream quickly exclaims. “She and the others have taught us all so much, and we’ve even saved Equestria because of her lessons!” “Well, just between you and us,” Gallus starts … before catching himself. “Yes?” ‘Twilight’ prods. “Waaaaait a minute … what if you really are Headmare Twilight? Are you just trying to figure out what all of us really think about you?” “What? Don’t be silly,” ‘Twilight’ states with a nervous chuckle. “I mean, as if I really needed to … I mean, as if your headmare would really …” “Is everything all right?” The class turns to the door as Twilight … another Twilight … trots into the room. The second Twilight takes a look at the first Twilight, then Ocellus, then back to the first Twilight. “What’s going on?” she asks the group. Ocellus looks from one Twilight to the other. “I really want to give my uncle a big hug … but which one do I hug?” “I can’t be certain,” Smolder starts, “but I’d say the first Twilight is the real Twilight.” “What makes you say that?” the two Twilights ask in unison. “Well for one, you got pretty jittery when Gallus started questioning which one you were.” The first Twilight blushes at the comments. “And second, you knew Silverstream’s name.” The first Twilight turns to the second. She nods her head, and the second Twilight morphs before everyone’s eyes into a yellow changeling with bulbous green eyes and fluorescent tan wings. “Uncle!” Ocellus cries out as she approaches and hugs her fellow changeling. “It’s great to see you again,” Coccyx says as he looks around the room. “So these are your friends. A most eclectic group if I’ve ever seen one.” “So what was with the whole ‘pretending to be someone else’ bit?” Gallus asks Twilight. “Just trying to hone my sense of humor,” she sheepishly replies. Gallus utters a flat ‘uh huh’ in response as Twilight composes herself. “In any case, I’m sure Coccyx has much to relate to you, and I have more planning to do at the present time. Enjoy!” She turns and heads out the door, leaving the elder changeling with the class. Author's Note The next couple puzzles revolve around the 'twin towns' of Utopia and Paradise, as well as the 'third town' of Elysium. It will not be absolutely necessary to read my earlier Fics to solve these upcoming puzzles, but for those who are interested: Twilight's Logic Puzzle Adventure Twilight's New Logic Puzzle Adventure Puzzle 12: Silent Pegasi“So Ocellus,” Coccyx starts, “care to introduce me to your fellow classmates.” “Oh … well,” Ocellus replies, “these are Sandbar, Smolder, Silverstream, Yona, and Gallus.” Each student nods upon hearing his or her name. “And … those are my classmates … and my friends.” “It’s a pleasure to meet all of you,” Coccyx states. “Do any of you have any questions about me?” “From what Ocellus has told us,” Gallus says, “you’ve been spending time in some pony village that served as inspiration for some logic puzzles. That seems … odd.” Coccyx laughs. “Yes, the twin towns of Utopia and Paradise … now the triple towns, including Elysium. It’s not as foreboding for travelers as it used to be, at least from what I’ve heard of their history. However, they still very much embrace what made their towns special.” “Oh? What would that be?” Silverstream asks. “It would take too long to delve into what I know of their history, so I’ll keep it simple. Utopia and Paradise were founded by unicorns and pegasi, and over the years a quirk developed regarding the inhabitants of those towns: in Utopia, all the unicorns tell the truth and all the pegasi lie; in Paradise, it’s the opposite: all the unicorns lie and all the pegasi tell the truth.” “That’s … really weird,” Sandbar admits. “I’m only giving a summary, but I’ve been told that the twin towns were the basis of a number of logic puzzles. As a simple example: suppose you came across two unicorns, either of which could come from either town. One unicorn states that they both come from Paradise. Could you tell which town each come from?” The group ponder the question for a few minutes. Soon after, Yona responds: “First unicorn say both from Paradise … but you say all unicorns from Paradise lie, so it can’t be that both unicorns from Paradise.” “On the other hoof,” Sandbar adds, “the first unicorn lied, which means she is from Paradise.” “But since the statement was false,” Silverstream continues, “they can’t both be from Paradise … so the second unicorn must be from Utopia!” “Very impressive,” Coccyx admits to the class. “And a very quick response, no less,” “We’ve had some practice as of late,” Gallus states. “But what about earth ponies?” Smolder asks. “Ah, that’s where more recent history comes into play. Initially, earth ponies were not even part of either town … they were left behind and forced to form a third town, away from the others; that town, they dubbed ‘Elysium’.” “Wait,” Yona says, “so ponies not always friends?” “I think that was the basis of Hearthswarming,” Ocellus replies. “Long ago, there was animosity among the three pony types.” Sandbar nods in agreement. “And from what I know,” Coccyx continues, “that animosity existed in some form when these towns were formed. Now, I can tell you that things aren’t that way anymore. All three towns now sport all pony types … but in a nod to the old traditions that made their towns unique, they still adopt their truthfulness regarding place of residence to the old ways.” Gallus thinks for a moment. “So what that means is …” “In terms of where each come from, the only truth-tellers are Utopian unicorns, Paradisian pegasi, and Elysian earth ponies. The unicorns from either Paradise or Elysium lie, as do the pegasi from Utopia or Elysium, and the earth ponies from either Utopia or Paradise.” “That’s … even weirder,” Silverstream states. “I know … it’s weird to me, too. But it does make for interesting puzzles; for example, it would render the previous puzzle as having multiple solutions: now the first pony could be from Elysium and the second pony could be from any of the three towns. Or for that matter, the first pony could still be from Paradise, but now the second pony could be from Elysium.” The class silently takes in the explanation. “Here’s another, simpler example: if a unicorn from one of the three town stated that he was from Paradise, what could you conclude?” After a brief pause, Smolder offers a quick response: “The unicorn can’t be from Paradise, since then he’d be telling the truth. But he can’t be from Utopia either, since all Utopian unicorns are truthful. So that means he must be from Elysium.” “Very good,” Coccyx states. “Maybe you guys need a bigger challenge,” he adds with a sly grin. “What sort of challenge?” Sandbar asks. “Recently, I and a number of fellow changelings have acclimated ourselves to the customs of the triple towns. The elder statesponies of the towns insist that we changelings understand and abide by these customs, and as such have created a number of challenges to keep us on our hooftips. One such challenge was presented to me just this past week; I’m curious if you are able to solve this puzzle.” Gallus snorts. “Friendship is logic, I guess. Go ahead.” Coccyx chuckles. “OK then. As a test, I was introduced to six ponies: two unicorns, two pegasi, and two earth ponies. I was told in advance that of each pair, one would tell the truth and one would lie, in accordance with the rules stated beforehoof. For simplicity’s sake, I’ll refer to the unicorns as A and B, the pegasi as C and D, and the earth ponies as E and F. They made the following statements: A said “None of the others have the same hometown as me.” B said “F and I have the same hometown.” E said “C is from Utopia.” F said “D and E have different hometowns.” … “What about the pegasi?” Ocellus asks. “For some reason neither one said anything,” Coccyx explains. ‘Pega-shy’ was more like it, I guess.” “Well it’s not like we don’t know a shy pegasus,” Gallus admits. “So did that invalidate the test?” Coccyx smiles. “On the contrary, there’s enough information to figure out all six ponies’ hometowns.” Author's Note So … what are the hometowns of each of A, B, C, D, E, and F? Puzzle 13: Button, Button, Which One's 'Gray Button'?“Fake wings?” “Fake horns?” Coccyx nods his head at both phrases. “All in the name of pony type anonymity. Quite a fascinating backstory, actually; by not revealing their true nature while adhering to a higher moral code, the Alicorn Society was able to bridge the gap between unicorns and pegasi … and later, earth ponies.” “But wouldn’t that make puzzle solving, like, waaay harder?” Silverstream asks. Coccyx chuckles. “Indeed … though before the inclusion of earth ponies and the town of Elysium, it actually led to some fascinating logical truths.” Hearing a chorus of ‘Oh?’s, he continues: “Here’s a simple example: suppose you met one of these ‘alicorns’, and I give you the knowledge that she is not an earth pony and is not from Elysium. The alicorn states that she is, in fact, a unicorn. What could you conclude?” The class goes silent for several seconds. Then, Sandbar speaks up: “If she’s telling the truth, then she must be from Utopia. If she’s lying, then she’s not a unicorn, so she must be a pegasus … and she can’t be from Paradise, so she must be from Utopia.” “Exactly,” Coccyx concludes. “An alicorn that claims to be a unicorn must be from Utopia, but could be either pony type, unicorn or pegasus. And at the same time, an alicorn that claims to be from Utopia must, necessarily, be a unicorn from either Utopia or Paradise. In short: stating that she is a particular pony type reveals her hometown, and stating her hometown reveals her pony type. I really think there’s an elegance to the symmetry, don’t you?” Ocellus, Sandbar, Silverstream, and Yona nod in agreement, while Smolder and Gallus each shrug off the perceived beauty with a ‘meh’; the latter two cause Coccyx to laugh again. “I suppose such things are not for every creature,” he admits. “But … symmetry gone with Elysium and earth ponies?” Yona asks. “Indeed … without the knowledge I gave you, the previous example would have multiple possibilities. Now if she were telling the truth and was a unicorn, then of course she would still be from Utopia; but now, lying doesn’t mean she’s from Utopia: she could be a pegasus from either Utopia or Elysium, or she could be an earth pony from Utopia or Paradise. So now there’s five possibilities, and further information would be needed to reduce this number.” “I’m afraid to ask how much more complicated the puzzles can get with these ‘alicorns’,” Smolder states. “You don’t have to,” Coccyx replies. “Recently, I was given a particularly daunting challenge that revolved around a quintuplet of alicorns.” A sly smile spreads across his face. “Let’s see how you fare on this one.” The class murmurs to itself in excitement, save for Gallus. “It beats the usual history lessons,” he eventually states. “I’m game.” “Wonderful … now before we begin, I do want to remind you of an important distinction: if an alicorn, for example, states that she is a Utopian pegasus, that is different from the alicorn stating that she is from Utopia, and then stating that she is a pegasus in a separate statement.” “What’s the difference?” Silverstream asks. “In the former case, you know she’s lying but she could be any of the following: Utopian earth pony, Paradisian unicorn, Paradisian earth pony, Elysian unicorn, Elysian pegasus. In the latter case, she still must be lying, but now she could only be either a Paradisian unicorn, Paradisian earth pony, or Elysian unicorn.” “So, in one case she could be a Utopian earth pony, but not the other?” “Exactly. In the first case, all we know is that she is not the specific specified combination; she could be either half of the combination. In the second case, she cannot be either half.” “Got it … I think,” Silverstream replies with a tinge of confusion. “Excellent. Well then … I was introduced to five gray alicorns. I was informed that from this group of five, every pony type and every hometown was included at least once. I was also informed that no two of them had the same combination of pony type and hometown, so there couldn’t be two Utopian pegasi, for example. One of these five, I was told, went by the name ‘Gray Button’; my task was to determine the pony type and hometown of all five, as well as identify which one was, in fact, Gray Button.” Ocellus begins calculating: “3 possibly pony types and 3 possible home towns for each alicorn, so 9 times 9 times 9 times 9 times 9 is … 59,049 possible combinations. Not accounting for which one is Gray Button.” “But alicorns all different,” Yona corrects, “so actually 9 times 8 times 7 times 6 times 5, or 15,120.” “Then times 5 for Gray Button,” Gallus adds, “so actually 75,600.” “Uh, guys?” Sandbar asks. “Can we slow down with the ‘math’ part?” “Oh … sorry, Sandbar,” Ocellus says in apology. “I see your math skills are high as well,” Coccyx admits. “But I assure you, there’s only 1 possibility once the statements are taken into account. I’ll continue to use ‘A, B, C, D, E’ for simplicity. These were the statements made: A: “I am a unicorn. I come from Elysium. Gray Button comes from Paradise.” B: “I am NOT an earth pony. I come from Paradise. Three of us are the same pony type.” C: “I am an earth pony. I do NOT come from Elysium. At most one of us comes from Utopia.” D: “I am NOT a unicorn. I do NOT come from Paradise. Gray Button is a pegasus.” E: “I am a pegasus. I do NOT come from Utopia. Three of us come from the same town.” “And from those statements, along with the information I provided, you have enough to figure out the pony type and hometown of all five alicorns, and state which one is Gray Button.” “So,” Coccyx concludes, “Can you figure this one out?” Puzzle 13 solution A: “I am a unicorn. I come from Elysium. Gray Button comes from Paradise.” B: “I am NOT an earth pony. I come from Paradise. Three of us are the same pony type.” C: “I am an earth pony. I do NOT come from Elysium. At most one of us comes from Utopia.” D: “I am NOT a unicorn. I do NOT come from Paradise. Gray Button is a pegasus.” E: “I am a pegasus. I do NOT come from Utopia. Three of us come from the same town.” Having written the statements on the chalkboard, along with the information that there at least one of each pony type and hometown among the five, with no two being the same for both … Coccyx waits for several minutes as the class ponder the information. “Yona confused,” the yak admits as she looks around the room, “but Yona not alone in confusion.” “You got that right,” Gallus adds. “Where do we even begin?” “Where did you begin,” Ocellus asks her uncle. “Well you can’t identify the pony type or hometown from any of the five at the onset,” Coccyx explains, “but you can at least narrow down the possibilities. Try starting there.” “Hmmm.” Ocellus focuses on the first of the five alicorns. “Well ‘A’ must be a liar; if not, she would be an Elysian unicorn, and that contradicts what we know of them. And that means that ‘A’ must be either a pegasus or earth pony, and from either Utopia or Paradise … but not a Paradisian pegasus, since they’re truthful. But that still leaves three possibilities.” “That’s fine,” Coccyx assures. “File away that information and move on.” “Ooh, let me go next,” Silverstream calls out.” “Go ahead,” Ocellus replies. “OK, so about ‘B’ … if she’s telling the truth, then she’s from Paradise so she’d be a pegasus. If she’s lying, then she is an earth pony but not from Paradise … and not from Elysium either, ‘cause that would make her truthful. So she must be either a Paradisian pegasus or a Utopian earth pony.” Smolder follows: “What about ‘C’? If she were telling the truth, then she’d be an earth pony who’s from either Utopia or Paradise … but in either case, she’d be a liar. So she must be a liar, and she’s not an earth pony but is from Elysium. She must be either a unicorn or pegasus.” “Yona look at ‘D’. Uh … ‘D’ either truthful or liar. If ‘D’ truthful, then ‘D’ not unicorn and not from Paradise … so ‘D’ must be earth pony from Elysium. And if ‘D’ liar, then ‘D’ unicorn from Paradise.” Sandbar picks up the final alicorn: “ ‘E’ could be truthful or lying, too. If she were truthful, then she would be a pegasus, and would have to be from Paradise. And if she were lying, then she would be from Utopia and not a pegasus … and not a unicorn, either. So she would be an earth pony in that case.” “Wait a minute,” Gallus interrupts. “Didn’t we conclude that ‘B’ was either a Paradisian pegasus or a Utopian earth pony?” Coccyx chuckles. “You did … and well done, all of you. You’ve done the dirty work that needed to be done. Now you just need to put the pieces together.” “But how,” Ocellus asks, the others looking on with similar confused looks. “Maybe this will help.” Coccyx picks up a piece of chalk and draws a 3x3 figure onto the board, adding letters above and to the left: -- upe U: - - - P: - - - E: - - - “The lowercase letters represent the pony types: unicorn, pegasus, earth pony. The uppercase letters represent the towns: Utopia, Paradise, Elysium. And if you recall, no two alicorns can share both pony type and hometown. Now … who can go where?” “Well,” Ocellus starts, “we know that ‘B’ and ‘E’ could each be either a Paradisian pegasus or a Utopian earth pony. And since they can’t be the same, that means that one of them has to be a Paradisian pegasus and the other one has to be a Utopian earth pony. But I can’t tell which is which.” “It doesn’t matter,” Coccyx says, “you now have two spots filled, at least in some order. Now what about the others?” Yona follows: “ ‘A’ must be Utopian pegasus or Paradisian earth pony or … wait, ‘A’ not Utopian earth pony, if ‘B’ or ‘E’ has to be.” “What about ‘C’?” Smolder adds. “She has to be Elysian, and has to be either a unicorn or a pegasus.” “And ‘D’ has to be either a Paradisian unicorn or an Elysian earth pony,” Sandbar adds. “Oh, let me add this to the board,” Silverstream says as she flutters to the board. Picking up the chalk, she fills in the possibilities: -- u p e U: - A * P: D * A E: C C D “I used a star for ‘B’ and ‘E’,” she clarifies as she returns to her seat. “So where do we go from here?” Gallus studies the letters. “No Utopian unicorns, I see. Wait, we know that ‘A’ and ‘C’ are liars, right?” “Good catch,” Ocellus says. “I think we should wait on ‘A’s comment until later, but what about ‘C’? She said that at most one of them came from Utopia, so in fact, at least two of them do.” “And there aren’t any Utopian unicorns,” Smolder adds, “so there has to be both a Utopian pegasus and a Utopian earth pony. Well the ‘earth pony’ slot is either ‘B’ or ‘E’; we already knew that.” “But now we know that ‘A’ has to be a Utopian pegasus,” Sandbar states. “That’s the only way we can get at least two Utopian ponies.” “We got one!” Silverstream triumphantly asserts. “Now … now what? “What about ‘B’ and ‘E’ comments?” Yona asks. “One truthful and one liar, right?” “You’re right,” Ocellus replies, studying each of their third comments. “One of them says that three of them are the same pony type, and the other says that three of them have the same hometown. So one of those facts has to be true. Which one?” “What can we have three of?” Smolder asks. “We don’t have any Utopian unicorns or Paradisian earth ponies,” Gallus states. “That rules out four of the six categories. There has to be either three pegasi or three Elysians.” “But there can’t be three Elysians,” Ocellus realizes. “ ‘C’ is the only one would could be an Elysian unicorn, and she’s also the only one who could be an Elysian pegasus. She can’t be both.” “That just leave three pegasi,” Yona concludes, “so ‘B’ truthful and ‘E’ lie. That make ‘B’ Paradisian pegasus and ‘E’ Utopian earth pony.” Sandbar speaks up: “If three of them are pegasi, then ‘C’ has to be an Elysian pegasus.” “And since at least one of the five has to be a unicorn,” Smolder adds, “ ‘D’ must be a Paradisian unicorn.” “BRAVO!” Coccyx calls out. “You got the pony types and hometowns of all five alicorns! All that’s left is to figure out which one is Gray Button.” “There’s only two comments to work with,” Gallus says, “and both of them are lies. Gray Button isn’t from Paradise, and she isn’t a pegasus. And the only one who fits both of those statements is … ‘E’. ‘E’ must be Gray Button.” “Wonderful,” Coccyx gushes. “Top notch; top notch! I’m starting to think there’s no puzzle the six of you can’t solve.” Other than why we’re solving so many of these, Gallus thinks to himself. . . . “Uncle Coccyx?” “Yes, Ocellus?” “I really enjoy these puzzles … but none of them seem to have anything to do with you?” Coccyx chuckles. “I’m sure you have concerns since you and Twilight left, but I assure you, relations between the townsfolk and the visiting changelings are very much cordial. It’s been made easier since they’ve made allowances for our presence.” “What do you mean by ‘allowances’?” Silverstream asks. “When we first got there, some of us may have made comments that caused … friction. But we’ve come to some agreements that those in the Alicorn Society have signed off on, and those agreements have made our presence more than tolerable; they now embrace our presence.” “Whoa,” Yona says with surprise. “Maybe yaks take lessons from changelings.” “Dragons, too,” Smolder adds. “And Griffons,” Gallus concludes. “So how did you get them to embrace you so much?” “Simple, really … our presence allowed for even greater complexity of puzzles. Let me explain …” Puzzle 14: Balancing Act“ ‘Logic bomb’? So, these ponies would black out over a sentence?” Coccyx nods as Smolder processes the answer to her question. “Nothing personal, but I couldn’t see any dragon passing out over a bunch of words.” “A by-product of their way of reasoning,” Coccyx assures. “From the beginning, Utopia was created solely for unicorns and Paradise for pegasi. Even after unicorns began settling in Paradise, their pride and sense of loyalty persuaded them to maintain an allegiance to the town of their ancestors’ origin. Hence, no unicorn would ever admit to coming from Paradise, and vice versa. From there, notions of truth and falsity spread organically, up to the point when Twilight first made contact.” Yona asks for clarification: “So, if unicorn claim to be from Paradise, that caused fainting?” “Before Elysium came into the picture, yes,” Coccyx replies. “At that time, all unicorns claimed to come from Utopia and all pegasi claimed to come from Paradise. For that matter, once one factors in the faux-alicorns, all Utopian claimed to be unicorns and all Paradisians claimed to be pegasi.” “But Elysium changed all that, right?” Sandbar asks. “In many ways … now a unicorn could claim to be from Paradise, if in fact she was from Elysium. But no unicorn would claim to be from either Paradise or Elysium.” Coccyx’s last statement draws quizzical looks from the class. “You just said a unicorn could claim to be from Paradise if she was from Elysium,” Gallus says. “And I assume a unicorn could also claim to be from Elysium, if she was actually from Paradise.” “Yes, and yes,” Coccyx replies. “But it’s not possible for a unicorn to claim to be from either Paradise or Elysium? That makes no sense!” “I realize it sounds weird, but it’s true. Suppose a unicorn did say ‘I come from either Paradise or Elysium.’ Assuming that unicorn came from one of the three towns, which one could it be?” Gallus mulls over the question. “… if the unicorn was from Utopia, then she would be truthful, but her statement would be a lie. So the unicorn would actually have to be from Paradise or Elysium, making her statement true … but unicorns from those towns always lie … huh … that’s weird.” “And believe me, I found out the hard way,” Coccyx continues. “While posing as a pegasus, I commented to a townspony that I wasn’t from Paradise; that pony dropped like a rock.” He smiles as he continues. “But they don’t have that problem anymore.” “Why not?” Silverstream asks. “Because of changelings like myself,” he proudly proclaims. “Now that the townsponies are aware of our presence, they can rationalize that such a comment could only come from a changeling. Rather than fear our presence, the knowledge of our being there prevents similar occurrences from happening.” “Wow,” Ocellus admits, “it’s a logic-based symbiotic relationship.” “In a sense, yes. We fill the gaps that would normally come from certain statements. Really, we complete the towns.” “Like imaginary numbers!” Yona asserts. Sandbar, Smolder, and Silverstream turn to Yona with curious looks. “It math thing,” she continues. Ocellus re-addresses her uncle: “So, the changelings are still in good standing with the Alicorn Society?” “Indeed … in fact, new tests have been created that incorporate our presence. Recently, one particularly difficult challenge was created for new recruits. I would like to see how you as a class respond to it.” “Harder than the last one?” Silverstream asks. “Well … you be the judge,” Coccyx says before explaining the challenge: “You are introduced to four alicorns. You are informed that one of them is a changeling with no particular hometown. The other three alicorns consist of one unicorn, one pegasus, and one earth pony; in addition, each one comes from a different one of the triple towns. Each alicorn will make one or two statements, with the usual rules of truth and falsity applying: only a Utopian unicorn, Paradisian pegasus, or Elysian earth pony will make truthful statements, while ponies of the other combinations will lie.” “As for the changeling, he or she will either be truthful or lie … whichever is less prevalent among the others. So if two or three of the others are truthful, the changeling will lie; if two or three of the others lie, the changeling will be truthful. Think of the changeling as attempting to balance out the veracity of the group.” “I will refer to the four alicorns as A, B, C, and D, as before. They make the following statements:” A: C is a pegasus. B: A is from Utopia. C: D is a changeling. A is a unicorn. D: Only one of us is a liar. B is from Paradise. Ocellus ponders the statements. “That doesn’t seem like enough to go on.” “It’s not,” Coccyx replies. “You are also given two additional facts:” A recruit who took the test before you was then told which alicorn was from Elysium, but that recruit still did not have enough information to solve the puzzle. Another recruit who took the test before you, who knew nothing of the first recruit, was then told which alicorn was the earth pony; that recruit also has insufficient information to solve the puzzle. “And now YOU have enough information to solve the puzzle! Which alicorn is the changeling, and what are the pony types and hometowns of each of the others?” Puzzle 14 solution A: C is a pegasus. B: A is from Utopia. C: D is a changeling. A is a unicorn. D: Only one of us is a liar. B is from Paradise. A recruit who took the test before you was then told which alicorn was from Elysium, but that recruit still did not have enough information to solve the puzzle. Another recruit who took the test before you, who knew nothing of the first recruit, was then told which alicorn was the earth pony; that recruit also has insufficient information to solve the puzzle. “Where to begin, where to start, what to do, what to think, …” Silverstream rambles to herself for several seconds, before Ocellus turns to her. “Maybe we should start with ‘D’s first statement. What if it’s true?” Gallus responds: “If it is, then 3 of them are truthful and 1 is a liar. Could all 3 truthful alicorns be ponies?” “That would force them to be a Utopian unicorn, Paradisian pegasus, and Elysian earth pony,” Sandbar replies. “Also, the changeling would be a liar.” “That not work,” Yona counters. “That make ‘A’ and ‘C’ both liars.” “Why so?” Yona answers Ocellus’s query: “ ‘C’ say ‘D’ changeling, so if ‘D’ truthful, then ‘C’ must be liar. And ‘D’ say ‘B’ from Paradise, so ‘B’ must be pegasus, make ‘A’ liar.” “She’s right,” Smolder states. “So the 3 truthful ones can’t all be ponies. It would have to be 2 truthful ponies and a truthful changeling.” “That can’t work either,” Silverstream says. “The changeling isn’t ‘balancing’ in that case.” Ocellus adds: “Besides, given what we know there can’t be exactly 2 truthful ponies; the third would have to be truthful, assuming there’s one of each pony type and one from each hometown.” Gallus concludes: “All of which means ‘D’ is lying, as well as at least one other. And I guess that means ‘B’ isn’t from Paradise … what now?” “What about ‘C’?” Smolder asks. “Could ‘C’ be truthful?” “If so,” Ocellus answers, “that makes ‘D’ a lying changeling. And that would mean either 2 or 3 truthful ponies. It still can’t be exactly 2, and if it were 3, ‘D’ would be truthful; we already ruled that out.” “So ‘C’ is a liar, too,” Smolder concludes. “And ‘A’ isn’t a unicorn. Anything else?” “Well at least one of the others is truthful,” Sandbar states. “Could both of them be?” “Can’t be,” Yona responds. “If both truthful, then ‘A’ truthful Utopian, would have to be unicorn. That not work if ‘C’ liar.” “All right,” Ocellus says. “So far we know that exactly one of them is truthful, and it’s either ‘A’ or ‘B’.” … “We might be stuck now,” Gallus laments. “You give yourself too little credit,” Coccyx insists. “Well done, all of you! You’ve come farther than most. But now I suggest you start figuring out possible solutions, and then use the additional information I provided.” “Possible solutions, eh?” Gallus ponders the suggestion. “OK then, we know that either A or B is the only truthful one, and whichever one she is is also the changeling. So I guess we just ‘pick a path’.” “ ‘A’ or ‘B’?” Ocellus asks the group. “May as well start with ‘A’,” Smolder recommends. “If ‘A’ is the changeling, then ‘C’ is the pegasus … but could be from either Utopia or Elysium.” “If ‘C’ is from Utopia,” Sandbar adds, “then ‘B’ has to be from Elysium; she can’t be from Paradise, since ‘D’ lied.” “That make ‘D’ from Paradise,” Yona continues. “And if ‘C’ Utopian pegasus, then … ‘B’ must be Elysian unicorn and ‘D’ Paradisian earth pony.” “Maybe we should put this on the board,” Silverstream suggests as she flutters over to the front of the class and picks up a piece of chalk. “So, Case 1: ‘A’ changeling, ‘B’ Elysian unicorn, ‘C’ Utopian pegasus, ‘D’ Paradisian earth pony” “OK … but what if ‘C’ was from Elysium instead?” Ocellus asks. “ ‘B’ still can’t be from Paradise,” Gallus points out, “so she has to be from Utopia. And ‘D’ would have to be from Paradise.” “And ‘B’ can’t be the unicorn then,” Smolder adds, “so she’s the earth pony and ‘D’ is the unicorn.” “Got it,” Silverstream states as she writes down the next possibility: “Case 2: ‘A’ changeling, ‘B’ Utopian earth pony, ‘C’ Elysian pegasus, ‘D’ Paradisian unicorn” “That’s all the possibilities with ‘A’ as the changeling,” Sandbar notes. “What about ‘B’?” “If ‘B’ changeling,” Yona starts, “ ‘A’ from Utopia but still not unicorn. Either pegasus or earth pony.” “If we assume ‘pegasus’,” Ocellus says, “then ‘C’ and ‘D’ have to be a Paradisian earth pony and an Elysian unicorn, in some order.” She ponders the original statements for several seconds. “But there’s no way of knowing which is which.” “Looks like Silverstream’s got double-duty, then,” Gallus states. “Can you add those cases?” “I’m on it,” Silverstream assures: “Case 3: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Paradisian earth pony, ‘D’ Elysian unicorn … and Case 4: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Elysian unicorn, ‘D’ Paradisian earth pony” “Only one path left to take,” Ocellus states, “If ‘A’ is an earth pony.” “ ‘C’ can’t be a pegasus in this case,” Smolder adds, “so she’s the unicorn and ‘D’ is the pegasus.” “Then ‘D’ not from Paradise,” Yona concludes, “so ‘D’ from Elysium and ‘C’ from Paradise.” “Last one,” Silverstream echoes as she writes the fifth possibility down: “Case 5: ‘A’ Utopian earth pony, ‘B’ changeling, ‘C’ Paradisian unicorn, ‘D’ Elysian pegasus” Silverstream flutters back to her seat, and the students take in the five possibilities: Case 1: ‘A’ changeling, ‘B’ Elysian unicorn, ‘C’ Utopian pegasus, ‘D’ Paradisian earth pony Case 2: ‘A’ changeling, ‘B’ Utopian earth pony, ‘C’ Elysian pegasus, ‘D’ Paradisian unicorn Case 3: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Paradisian earth pony, ‘D’ Elysian unicorn Case 4: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Elysian unicorn, ‘D’ Paradisian earth pony Case 5: ‘A’ Utopian earth pony, ‘B’ changeling, ‘C’ Paradisian unicorn, ‘D’ Elysian pegasus “We’ve used everything,” Smolder states. “No we haven’t,” Gallus corrects, the realization dawning on him. “We were told that two recruits before us were given additional information, and that information wasn’t enough in either case.” “So what?” Yona asks. “So consider which alicorn could be from Elysium,” Gallus responds. “In order, the cases state that the Elysian is either B, C, D, C, or D. But it can’t be ‘B’ … or that first recruit would have had enough information!” “But if that recruit was told that either ‘C’ or ‘D’ was the Elysian,” Ocellus adds, “that wouldn’t be enough. We can rule out Case 1!” “That still leaves 4 cases,” Sandbar points out. “But now focus on who the earth pony could be,” Smolder states, the final piece coming into place for her as well. “Going in order again, the earth pony is either D, B, C, D, or A … the only one that doesn’t immediately give away the answer is if ‘D’ is the earth pony.” “But didn’t we rule out Case 1?” Yona asks. “But that’s just it,” Ocellus answers: “We know it’s not Case 1, but the second recruit doesn’t know that! If Case 4 is the actual solution, then the first recruit being told that ‘C’ was the Elysian would still allow the possibility that it’s Case 2 … and the second recruit being told that ‘D’ was the earth pony would still allow the possibility that it’s Case 1.” “So … we’re done, right?” Silverstream asks. “It’s Case 4, isn’t it?” “It is!” Coccyx admits: “Indeed, the final solution is that ‘A’ is a Utopian pegasus, ‘B’ is a changeling, ‘C’ is an Elysian unicorn, and ‘D’ is a Paradisian earth pony. EXCEPTIONAL WORK, EVERYCREATURE!” “Sounds like I missed some good work.” Coccyx and the others all turn to the doorway, observing Twilight as she observes them. “What are you doing here?” Ocellus asks. “More like ‘what are you all still doing here?’ … you’ve been here for an hour and a half.” The group turn to the clock on the wall, noting that indeed, ninety minutes had elapsed since Coccyx entered the room. “Wow,” Gallus says, “I don’t think we’ve ever had a class go by so fast.” “I guess time is relative,” Twilight replies, “but I’m afraid you do have other classes to get to.” She turns to Coccyx: “Thank you again for taking time out of your busy schedule to visit our school.” “The pleasure was all mine,” Coccyx insists. “Your students work exceedingly well together … something I suppose I should expect from true friends.” Silverstream, Yona, Sandbar, and Ocellus smile and blush, as Smolder and Gallus simply sigh and smile. “Well, I should be heading back,” Coccyx tells the class. “It was great seeing you again, Ocellus … and it was a real pleasure to meet your classmates.” “Take care, uncle Coccyx … hope everything continues to go well at the twin … er, the triple towns.” Coccyx chuckles. “Hey, maybe a field trip is in order at some point.” He again turns to Twilight. “What do you say?” “There’s a lot on our schedule,” Twilight sadly replies … “but we’ll see. Maybe next semester.” “Well then … until next time ...” Giving one final bow, Coccyx turns and flies out of the room. Twilight scans the chalkboard. “Wow … sorry to have missed whatever puzzle this came from.” “Don’t sweat it,” Gallus says, “these days, the next one’s probably just around the—” *OOF* “… corner?” The class gather around Twilight and get her back on her hooves.” “Oh dear, I am so sorry,” Rarity pleads. “I was just looking for … oh, there you all are!” “Our apologies, Miss Rarity,” Sandbar says. “We kinda went over our time with the last class.” “Oh, that’s quite all right,” Rarity insists. “But come; there’s fashion to be done!” Satisfied that Twilight is no worse for wear, Rarity heads back out the doorway, with Twilight and the students following in tow … Puzzle 15: Bejeweled and BefuddledI can’t believe what I’m about to say, but can we get back to solving puzzles? Gallus knew better than to voice his thoughts to his professor. In fact, he has become rather burned out at the whole notion, even as adept as he and the others had become. But more to the point: fashion never really inspired him the way it did others, such as his professor. And given a choice, he’d rather toil through another puzzle than— “Gallus?” The griffin looks up with a start, now aware of Rarity staring back at him. “Uh … could you repeat the question?” “How much impact would you say fashion has had on griffin culture?” “Fashion? Uh …” Gallus’s eyes dart around the room as he searches for an acceptable response. “Well? … er …” He sighs before concluding: “Not much.” Rarity frowns at the response. “Surely there must be some emphasis made.” “Griffinstone isn’t much for finery,” he sheepishly replies. “Well perhaps we can change that. Come on up and let me take your measurements: we’re going to design an outfit specifically for griffins!” Gallus gulps inaudibly as he gets up and walks towards the front of the class. He catches a glimpse of various photos on the teacher’s desk. “So what are these?” he asks, pointing out the pictures. Rarity sighs. “A failed attempt at pleasing the unpleasable, mixed with a tinge of revenge, no doubt.” Her response draws the attention of the rest of the class. Gallus takes a second look at the photos. “Really? All I see are scarves … six of them, each with some strange markings … are those exes and ohs?” Rarity’s horn illuminates as the photos separate and spread themselves atop her desk. The students get up and gather around the pictures, each of which is of a scarf trimmed with five gems of various colors. “Who are all these for?” Smolder asks. “Chroma-chaotica, the avant-garde fashion designer,” Rarity replies. “Every year, the stallion gets in his head *THE* color combination that is ‘in’, and every year designers from all of Equestria follow his lead. This year, I had hoped to get a jump on the crowd.” Ocellus takes a closer look at one of the photos. The scarf displays a series of five gemstones: diamond, ruby, another diamond, onyx, and a third diamond. In the bottom-right corner, she discerns an ‘x’ and an ‘o’. “Miss Rarity … what do these symbols at the bottom mean?” “I had my trusted Manehattan assistant Coco take six samples to Chroma’s representatives; these photos bear those samples. For each sample, she received feedback as to what truly worked, what worked in a minor sense, and what didn’t work at all.” “Is that what ‘x’ and ‘o’ for?” Yona asks. “The ‘x’ signifies that Chroma fully agrees with the color and placement of exactly one of those gems,” Rarity explains. “So either he thinks a ruby should be in the second spot, an onyx should be in the fourth spot, or one of the diamonds is correctly placed.” “And the ‘o’?” Sandbar asks. “The ‘o’ means Chroma likes the idea of one of the gems, but not its placement. So maybe he believes the ruby should come first, instead. Or maybe last. Or maybe one of the diamonds should be fourth. Unfortunately, there’s no way to know for sure.” “But you have six of these photos,” Silverstream notes. “That’s enough information, isn’t it?” As the hippogriff speaks, Ocellus picks up a piece of chalk and notes each photo’s gem ensemble and associated marks on the board: D R D O D – x o D R S R R – o o o O S S E O – x x R O O S T – x o o S E T R E – o o T S O T D – x o Gallus scans the board. “I guess ‘D’ stands for diamond. What about the other letters?” “ ‘R’ is for ruby, ‘O’ is for onyx, ‘S’ is for sapphire, ‘E’ is for emerald, and ‘T’ is for topaz,” Ocellus replies. “Really pretty colors: white, red, black, blue, green, and yellow.” “So … this like puzzle,” Yona points out. “That means we can solve.” “If only it were that simple,” Rarity sullenly responds. “I’m not particularly adept at figuring things like this out, as you have seen.” “We are,” Smolder asserts. “Yes, the whole school has witnessed your capabilities, but I’m afraid there’s an added issue, for which you can thank Suri.” “Suri?” Ocellus asks. “Suri Polomare … and old rival of mine. Shortly after receiving these photos, I received a letter from Coco that Suri caught wind of my plans. She was able to gain access to the photos and changed the comments on one of them, and I don’t know which one.” “Why just one?” Gallus asks. “Why not just alter all of them?” “It’s her way of toying with me, like baiting a cat with a piece of string. In any event, I daresay the situation is now beyond even your abilities. Only five of these six photos can be trusted as having useful information, and there’s no way to figure out which photo is the wrong one.” “Some might consider that a challenge,” Gallus retorts. “No, no, no … I appreciate all you’ve shown myself and the others, but even if it were possible to solve this conundrum, I’m afraid it would take too much time out of our already abbreviated day. But, if you wish to stay late and inspect the photos on your own time, they’re of little use to me now. Now then, let’s discuss proper griffin regalia …” As early evening comes over Ponyville, the students reflect upon another day’s work … “Miss Rarity can really work fast under a deadline.” Silverstream notes. Ocellus turns to Gallus. “So what do you think Griffinstone will make of your new modeling duties?” she asks with a chuckle. He replies with a laugh. “I don’t want to be the one to disillusion our resident fashion guru, but her first trip to Griffinstone would probably be her last. It’s not exactly Manehattan where I’m from.” “Maybe she’ll get the idea to create outfits for Yakyakistan next,” Sandbar says as he turns to Yona. “Yona not think other yaks interested in attire … but then, Yona not think other yaks interested in others at all, so first for everything?” “Ooh, maybe we’ll design outfits for dragons next,” Silverstream notes. “By the way … shouldn’t Smolder be back by now?” Ocellus muses for a bit. “She said she wanted a few minutes to take a longer look at those photos. Honestly, I’m surprised she wanted to; she never showed a big interest in puzzle-solving before.” “Even if she did,” Gallus adds, “there’s no way she’d still be at it. I just assume she caught up with Spike afterwards. Not like any of us have fellow students of our own species around here.” Sandbar opens his mouth to speak, when the door to their foyer bursts open. The group turn to face Smolder, her eyes wide open and twitching with retinas bulging. “Smolder,” Yona exclaims. “We just talk about you!” “Where’ve you been,” Ocellus adds. “It’s been hours since we last saw you.” “You were with Spike, right?” Gallus asks. “I mean, it’s not like you spent all that time on Miss Rarity’s photos … did you? …” Smolder replies stiltedly: “Four hours … thirty-seven minutes … twenty-five seconds … but I did it.” “You did what?” Sandbar asks. “I figured … it … out …” Author's Note So basically, a solitaire 'Mastermind' puzzle with the added twist that one of the six clues is bogus and needs to be ignored. I'll admit, this proved harder than I imagined … and I suspect it may not get solved without further hints. I do have a logical solution, but it's long and makes assumptions that lead to contradictions. If anyone comes up with a simpler explanation (and NOT just a guess-and-check solution), let me know. If anyone wants a starting point, send me a private message and I'll reveal my first few steps. Puzzle 15 solution D R D O D – x o D R S R R – o o o O S S E O – x x R O O S T – x o o S E T R E – o o T S O T D – x o [one clue wrong] The students note the six clues scrawled into the upper-left portion of the chalkboard; the remainder of the board filled with an alphabet soup’s worth of letters, squiggles, arrows, and other assorted ‘short-claw’ notation. Atop the desk are papers upon papers of additional scribblings. Their attention shifts from the desk and chalkboard to Smolder herself, still wide-eyed and twitchy as she explains her motives to the others. “Dragons competitive, must be best, so I must be best at solving, and pretty fashion, I like pretty fashion, so I put two and two together to get four, or maybe two more for six, so six clues, but I figured out which one wasn’t really a clue, and I figured out the whole thing. Yes, oh yes I did!” Gallus reaches out a talon to console Smolder. “It’s okay … we came here to see all you’ve done, and we’re very, uh, impressed, with the work you put in tonight. Maybe come back to the dorm and get some rest, and tomorrow you’ll be—” “DON’T TOUCH!” Smolder snaps, as Gallus quickly recoils. “Sorry, sorry, out of line there, but really, it’s all so simple now, I just have to explain it, while it’s all still fresh in my mind and spread out here. Allow me this, and everything will be revealed.” The others turn to each other, then back to Smolder. “Go ahead,” Ocellus calmly beckons. “Good, good. So anyway I said to myself, I said ‘You have to start somewhere’ and so I did, I started by making a simple assumption that some clues were right and see where that led, and what better ones than the ones that give the most information, so I assumed that clues 3 and 4 were correct, because one places two gems exactly and the other places three gems with one correct, so I started there, you see?” “Yona see,” the yak replies apprehensively. “Great, I knew I could count on you seeing this, so anyway if clue 3 is correct then exactly two of those five gems are correct and in the right place but which ones … yes, which ones, indeed … so I have to just go through all of those possibilities and see where each led. So it could be that both onyxes are right, or both sapphires, or one of each, or maybe the emerald with exactly one of the others, so I went through all of those while assuming clue 4 was also correct. Got it?” “Uh, yeeeah, we got it,” Silverstream carefully replies. “Good, so start with both onyxes which would then be in spots 1 and 5, while there also wouldn’t be any sapphires or emeralds, but then by clue 2 or clue 5 there would have to be a ruby, since they’re not both false, but if there’s a ruby and two onyxes then by clue 4 there’s no topaz, and that means that clue 5 is invalid. But that forces clue 2 to be valid, and that means there has to be a diamond since there’s no sapphire and three rubies can’t all be in incorrect places, but that means there’s a ruby, an onyx, and a diamond in the solution and that contradicts clue 1 so there can’t be two onyxes. See, we’re on our way!” “Yes we are,” Gallus states. Right to a padded cell … “So it can’t be two onyxes, but what about two sapphires that would have to be in spots 2 and 3, while there then wouldn’t be any onyxes or emeralds? Glad you asked: by clue 4, there would then have to be a ruby and a topaz, which means clue 5 is invalid, but then clue 1 is valid and there’s a diamond somewhere and that contradicts clue 2 so there can’t be two sapphires, either. Now what about having exactly one onyx and one sapphire. Here’s where it gets complicated!” The group blinks in unison. “This is where it gets complicated?” Sandbar asks incredulously. “Oh my, yes, because now all I know is that there’s no emerald, as well as no second onyx or second emerald. But since either clue 1 or clue 2 has to be valid, there has to be either a diamond or a ruby in the solution. So we break it down into cases: Case 1 and Case 2. Now for Case 1, assume there’s a diamond in the solution, so no ruby. Well then of course clue 2 is invalid, but then so is clue 6 since now there can’t be a topaz but clue 4 implies now that there is a topaz, so we can clearly disregard Case 1.” “Are you sure you don’t want to take a break?” Ocellus soothingly asks. Smolders pays no mind as she continues. “But what about Case 2? Well now there’s no diamond and still no topaz, to go with no emerald, no second onyx, and no second sapphire. All that’s left is to have three rubies to go with the one onyx and the one sapphire, so clue 2 has to be invalid. All I had to do now was check all the combinations that existed.” “You mean permutations, right?” Gallus corrects. “Yes of course I did but thank you anyway so now there’s RRROS and RRRSO and a bunch of other ones but that would take forever right? Of course it would but then I remember clues 3 and 4 which combined force a ruby in spot 1 and then by clue 1 I know the onyx can’t be in spot 4 so another ruby must be in spot 2 and that means the sapphire is in spot 3 and the onyx is in spot 5 leaving the last ruby for spot 4, so RRSRO.” Silence pervades for several seconds. Gallus opens his mouth to speak, when— “I KNOW, RIGHT? That CAN’T be right since now clue 6 is false too, so all that proves that there can’t be exactly one onyx and one sapphire! So now there has to be an emerald, and that makes things a lot easier.” Everyone nods and murmurs agreement. Satisfied, Smolder continues on. “So now that there’s an emerald, can there also be an onyx? If so, there’s no sapphire and if clue 2 were valid there would have to be two rubies and a diamond but that goes against clue 4 so clue 2 has to be the invalid one now, but meanwhile by clue 4 either the ruby is in spot 1 or the topaz is in spot 5 and in either case an onyx has to be in the other spot while an emerald has to be in spot 4. Now the first of these would force a second ruby into spot 2 because of clue 1 but then by clue 6 the only spot left that could be correct is spot 3 which would be a second onyx which we know can’t be, so that rules out the ‘ruby in spot 1’ case, but what about the ‘topaz in spot 5’ case? Well now clue 6 forces either a sapphire into spot 2 or a second onyx into spot 3, but we know there’s no sapphire or second onyx. And just like that, there can’t be an onyx and an emerald.” “What about a sapphire and an emerald? With no onyx, clue 4 forces there to be a ruby and a topaz, so clue 5 has to be invalid, and if clue 1 is valid, there has to be a diamond too, and that goes against clue 6, too, so there can’t be a sapphire and an emerald. So you see what all this means, right?” … “IT MEANS CLUES 3 AND 4 CAN’T BOTH BE VALID!!!” … “How long did it take you to get this far?” Sandbar asks. “Oh about three hours, but now everything gets a lot simpler, because now I know that clues 1, 2, 5, and 6 are all valid, and in particular clue 2 is valid so there’s at least one or two rubies in the solution. Could there be only one ruby? Then there would also be a diamond and a sapphire and by the other ‘valid’ clues there wouldn’t be any emerald, onyx, or topaz, so clue 4 would be invalid and clue 3 would have to be valid with both sapphires correct, but that is counter to clue 2, so there has to be two rubies in the solution, but not three since they couldn’t all be incorrectly placed.” “Yona worried for Smolder sanity.” “Why? I’m perfectly fine, great even, because now I know that there’s two rubies AND I know they’re in spots 1 and 3. And since clue 1 has one gem correctly placed, it’s either the onyx in spot 4 or the diamond in spot 5. And now I can prove that clue 3 is the false one: if not, then there would have to be a sapphire in spot 2 and either an emerald in spot 4 or an onyx in spot 5, so either RSROO or RSRED. The first case can be ruled out by clue 3 itself: it has a sapphire and two onyxes, which is one gem too many; the second case can be ruled out because it goes against clue 5. And that means clue 3 is the false clue!” “It’s all downhill from here. From clues 5, there’s can’t be a sapphire and a topaz to go with the ruby, so from clue 4 there has to be an onyx, and now from clue 1 there can’t be a diamond, so by clue 2 there has to be a sapphire. And since the ruby in clue 1 is in the wrong place, the onyx must be in spot 4, and now by clue 6 the sapphire must be in spot 2. All that’s left is spot 5, and it can’t be a diamond, emerald, onyx, ruby, or topaz; it must be a sapphire. And that is the final solution:” “R-S-R-O-S” Smolder takes a number of deep breaths upon completing her reasoning. “And now I can go to sleep.” She then collapses to the ground. “Wha … where am I?” The young dragon’s eyes flicker open. She lifts herself up from her bed and looks around, finding herself in her sleeping quarters, a pale moonlight giving slight illumination to the room. “How did I end up …” She pauses upon seeing her fellow classmates asleep on the floor, atop a number of blankets and cushions. “Oh …” she says to herself, before turning back in. “Dragon feeling normal?” Yona asks the following morning. “Yeah … thanks again, everyone, for putting up with … whatever that was last night.” “No worries,” Sandbar states. “But I’m still curious … why?” “I dunno for sure … I guess with all the puzzle-solving we’ve been doing, and particularly what I’ve seen from the rest of you, I felt like I hadn’t really proven my own abilities. I kept reminding myself of Dragonlord Ember, and how she became ruler through intelligence and savvy rather than brute force. But she’s still competitive, like all dragons … and I wanted to prove to others, and to myself I guess, that I was capable of solving something like this.” “But why this puzzle?” Ocellus asks. “Well … you don’t all know this, but there’s a part of me that really … likes fashion and girly stuff. I wanted to impress professor Rarity with this, really blow her away.” “You blew all of us away,” Gallus states. “Even if I thought I could solve this one, I never would have had the patience to see it through to a conclusion.” “Yeah … thanks, everyone. But had I known, I probably wouldn’t have chosen this puzzle,” she says with a laugh. “Hopefully today will be less strenuous.” “I think professor Rainbow Dash has more buckball planned for today,” Silverstream says. “Lot of running and flying around, getting the heart pumping.” Yona chuckles. “Yep. Good break from strenuous activity!” Author's Note Mea culpa: this puzzle got away from me. I wanted to do a 'Mastermind' puzzle with the twist of a single false clue, but I never thought it'd be so difficult to solve (short of a guess-and-check method that wouldn't disprove an alternate solution). Next puzzle will be more along the lines of earlier ones, I promise. Puzzle 16: More Buckball GamesHaving missed the previous ‘buckball day’, Gallus was secretly thrilled to be able to participate in the next one. Not that anyone else had to see this enthusiasm. “Looking forward to professor Dash’s class today?” Sandbar asks as the group make their way down the hallway. “Oh? What’s so special about today?” “Oh please,” Smolder retorts. “Don’t act like you don’t know what today is; we flat-out mentioned it this morning!” “Oh, right …” Gallus chuckles as they continue making their way. “Yeah, I want to make up for lost time. Show you all my mad skills as a defender.” “Griffin get real mad after Yona score on him!” The group laughs at the young yak’s counter. “Talk is cheap, Yona. You’ll have to prove it in competition.” “Bring it on!” Ocellus turns to Silverstream. “You think some creatures are getting the wrong lessons from professors Rainbow Dash and Applejack?” “Eh … I’m still waiting on that plumbing lesson we’re owed. Who do you think would teach us that?” Sandbar looks at Silverstream with a wry smile. “Maybe headmare Twilight needs to open a Friendship Vo-Tech school for you.” The group reaches the gymnasium, but they’re surprised to see Applejack waiting for them instead of Rainbow Dash, her focus on a large piece of paper held in her forehooves. “Come on in, y’all,” she calls out to the group, her eyes still glued to the paper. “Professor Applejack,” Ocellus calls out, “where’s Rainbow Dash? She usually runs the buckball games.” “We’re still playing buckball today, right … RIGHT???” Applejack quickly alleviates Gallus’s concerns. “Yeh, ye’re all still playin’ buckball, just like the last group. Ah agreed to take over for Rainbow Dash for this period.” “Did she take another trip to Las Pegasus?” Sandbar asks. Applejack chuckles. “Not this time, Sandbar. She muttered somethin’ about another chapter by some friend of hers that she met at the Daring Do convention. Said she needed some time to figure out things …” She pauses for a moment. “Actually, Ah think she might be askin’ fer yer help about this.” Gallus rolls his eyes, now aware of what’s to come later in the day. “Just to be clear,” he says, “we are playing buckball today, right?” “Yes yes, of course …” Applejack dismisses as she turns to her attention back to the paper. “Uh, professor?” Silverstream asks, “What’s that in your hoof?” “Huh … OH, sorry, everycreature. Ah’m just tryin’ to make sense of these here results.” “Results?” Ocellus asks. “The previous group of students what come down here for some buckball. Rainbow Dash was here for that group, and Ah got here just before they all headed out. It was a big enough class that she split ‘em up into four teams and had ‘em play each other. RD told me every team played at least once and no two teams played each other twice. When they were all done, she had a member of each team summarize the results and asked me to record all the game results right before she took off. Unfortunately …” “May I?” Silverstream asks as she reaches for the paper. Applejack obliges, and the students peruse the paper. ….GP...W L T...GF GA A....3.....1 0 2.....4 4 B....2.....0 1 1.....2 4 C....2.....1 1 0.....2 1 D “I take it ‘GP’ means ‘Games Played’,” Ocellus says, “with Wins, Losses, and Ties … but what do ‘GF’ and ‘GA’ stand for?” “They stand for ‘Goals For’ and ‘Goals Against’,” Applejack replies. “She gave me summary totals, but she didn’t give me actual scores. Shoot, I don’t even know who played who?” “Where are D’s results,” Sandbar asks. “Ah guess all the members of that team left before Rainbow did. So I got nothin’ regardin’ their scores, and that ain’t even the worst of it.” “Oh?” Yona asks. “What worst of it.” “Last thing Rainbow said before leavin’: she said one team was havin’ a prank at her expense, and all the figures in that team’s row are all off by 1. Guess that prank is on me now. Dangit, Rainbow!” “All right, so just so we’re all clear on all this,” Smolder states, “These results are missing anything regarding D’s games, one of the three rows is completely wrong with every number off by 1, and professor Rainbow Dash expects you to still recreate the results of every buckball game.” “Eeyup,” Applejack replies, mimicking her brother’s verbal tick. “Professor?” Silverstream asks, “Don’t buckball games usually go until some team scores 5 goals?” “Usually, but due to time constraints, Rainbow only had teams play until someone got to 3 goals. Further, she had a time limit in place, so some games ended before any team got to 3.” “Okay,” Silverstream says as she studies the paper a bit longer. “… aaand that’s it.” Applejack blinks. “What’s ‘it’?” “I figured out all your scores,” the hippogriff cheerfully replies. Author's Note This one's a bit easier than usual, but after that last one, I didn't want to go crazy. Simple question: which teams played each other, and what were the scores in each game? Puzzle 16 solution ….GP...W L T...GF GA A....3.....1 0 2.....4 4 B....2.....0 1 1.....2 4 C....2.....1 1 0.....2 1 D Applejack takes the sheet of summary results back and peruses it. “Seriously? You gone figured out the whole thing based on this? Even with the missing row and mistake row?” “Uh huh,” Silverstream replies. The other students turn to each other and exchange odd looks. Finally, Gallus speaks: “I don’t mind sitting this one out; anyway, I came to play buckball, not go on about scores.” “And I still need a break after last night,” Smolder adds. “Anyone else mind if Silverstream solos this one?” The remaining students shrug, and Silverstream beams at the chance of demonstrating her knowledge. Applejack sighs. “Guess I shouldn’t be surprised anymore. Still, where do you even begin with this one?” “With row ‘A’,” the hippogriff replies. “That one has to be the ‘prank’ one.” “Why’s that?” “Because it’s not possible for ‘A’ to have the same number of goals for as goals against if they have at least one win and no losses; they would have to have more goals for. So all those numbers are off by one, and that means ‘A’ played two games, with no wins, one loss, and one tie.” “Well Ah’ll be …” “And more! Since ‘A’ has a loss and no win, they have to have more goals against than goals for … so they must have three goals for and five goals against.” “Huh … hold on while Ah jot this down.” Applejack picks up a pencil from the ground and fills in the correct information: ….GP...W L T...GF GA A....2.....0 1 1.....3 5 B....2.....0 1 1.....2 4 C....2.....1 1 0.....2 1 D “So Ah take it that means ‘D’ played two games as well?” “It has to be an even number, and it can’t be zero or four, so yes: ‘D’ played two games. And since the other teams combined for one win and three losses, ‘D’ has to win both its games.” “This is really helpful,” Applejack admits as she jots down the further information. “But that still don’t give us the final scores.” “Well for that, look at ‘A’s goal totals. They had one tie and one loss, so their loss had to be by two goals. Their two games had to have scores of either 0-0 and 3-5, 1-1 and 2-4, 2-2 and 1-3, or 3-3 and 0-2 … but the first two of those can be ruled out since no team scored more than three goals, and the last case is out since a team that got to three goals won immediately. So ‘A’s games had to be scores of 2-2 and 1-3. And since only ‘B’ tied as well, ‘A’ must have played ‘B’ to a 2-2 tie.” Applejack scribbles notes as fast as Silverstream recites her solution. “Now ‘C’ didn’t score three goals in total, so they didn’t beat ‘A’; ‘D’ must have, so ‘A’ lost to ‘D’ by a 1-3 score and ‘A’ didn’t play ‘C’ at all. That means ‘B’ didn’t play ‘D’, and their other game must have been a loss to ‘C’ by a 0-2 score. That leaves ‘C’ with one goal against them, so ‘C’ lost to ‘D’ by a 0-1 score, and that’s all the games played.” “Was anyone timing that?” Gallus asks. Sandbar looks up at a clock hanging on the gymnasium wall. “That was, what, 3 minutes?” “Sounds like a record,” Ocellus points out. “Good job!” “Ah’ll say … that was remarkable.” Silverstream beams. “Seriously, if there’s anything Ah can do to repay you …” “Maybe professor give Silverstream one-on-one lesson on plumbing,” Yona suggests. The students laugh, while Applejack ponders the request. “Sounds strange … but Ah do know mah way around a toolbox. Shoot, Ah helped out the spa ponies not too long ago. Ah’ll talk to Twilight about settin’ somethin’ up.” “Really? That’d be super-awesome!!!” Silverstream does aerial somersaults in celebration. “Hey now, save some of that energy for buckball. Alright, everycreature, split up an’ lets get yer bodies a workout for a change …” Puzzle 17: QuibbleFic Part II (The Cursèd Code) Having mastered the puzzles of the bears and wolves, Daring Do found herself in the final chamber before escape and total freedom. However, it was the most heinous of traps, which nopony before her had ever solved: the Chamber of the Cursèd Code. Sandbar looks up from the hoofwritten outline and turns to Rainbow Dash. “Quibble Pants wrote all this?” “All sixty-four pages,” Rainbow laments. “I can appreciate the effort, but there’s a lot better work out there?” “Like yours?” Gallus asks “Exactly … I mean, yeah, there’re so many others that … you know … are better handled to …” Not wanting to continue watching her professor writhe in mental agony, Ocellus refocuses back to the story, reading aloud for those unable to crowd around the desk: Daring observes the room, her body shivering at the shattered bones of so many previous adventurers strewn about the room. Shaking off the macabre images of those unfortunate souls, she catches sight of a large bin containing … children’s blocks? ‘A strange medium’, she thinks to herself when musing about the inner-workings of the mind of the logitaur. ‘What to do with these’ … but the notes she had acquired had already prepared her for this final challenge. In a recess of the far wall, she makes out some sort of contraption, with a shelf capable of holding a number of the small blocks, but an ominous nozzle from some sort of weapon pointing directly at where one would have to stand to place said blocks. ‘Looks like the others also had only one attempt at this’, she ponders as she pulls the notes from her bag. “If X represents any sequence of blocks, then LXR will lead to X. Further, if X leads to Y, then note the following: KX leads to YL, QX leads to YR, DX leads to Y ‘doubled’, and CX leads to Y ‘curtailed’ … find the sequence that leads to itself.” ‘Find the sequence that leads to itself’ … she repeats the phrase over and over, pondering how to crack this infernal code … Ocellus looks up. “That’s it? Where’s the rest of it?” “Quibble didn’t send me the rest … all he cares about right now is that this final puzzle he created has a unique solution.” The students turn to each other quizzically. “What would he care about that for?” Smolder asks. “Probably still bitter about the last time he came across a puzzle like this.” “You mean, like in another book or something?” “Uh, yeeeah, something like that,” Rainbow replies to Sandbar, careful not to mention the true ‘something’. “He’s something of a picky perfectionist. So anyway, can you guys help?” The students take another look at the page containing the puzzle’s description. “Yona confused.” “I have some experience with this, so I can demonstrate.” “Really?” Gallus asks. “You read the same puzzle books as this friend of yours?” “Uh, again, something like that,” Rainbow carefully replies before going to the board and fitting a hoof-glove with a piece of chalk cradled within, allowing her to speak as she writes. “So ‘X’ and ‘Y’ each mean any sequence of letters. If we take the word PONY, for example, the first line means that LPONYR would return PONY. It’s like a machine, where LPONYR is the thing you put in and PONY is the thing you get out.” “So it’s basically a function, where LPONYR is the input and PONY is the output,” Ocellus states. “Yeah, sure, if that helps. So like, that works for anything that starts with ‘L’ and ends with ‘R’. So LYAKR would return YAK, LDRAGONR would return DRAGON, and so on.” “OK, Yona less confused.” “What about the other lines,” Silverstream asks. “The other lines all begin with knowing some in-and-out pair, like LPONYR and PONY. Putting a ‘K’ before the former returns the latter with an ‘L’ at the end, so KLPONYR would return PONYL. Also, KLYAKR would return YAKL, and … you get the idea.” “And the same for ‘Q’, right?” Ocellus asks. “So, QLPONYR would return PONYR?” “Hey, you catch on quick … guess I shouldn’t be surprised about that by now, should I?” “What about the other lines?” Gallus asks. “ ‘Doubled’ means repeated, so DLPONYR would return PONYPONY. And ‘curtailed’ means cutting off the end, so CLPONYR would return PON.” “And you can combine those later lines, right?” Ocellus continues. “So, QKLPONYR would return PONYLR, and DQKLPONYR would return PONYLRPONYLR, and CDQKLPONYR would return PONYLRPONYL, and …” She catches herself before going any further. “So what’s the point of all this?” Smolder asks. “ ‘Find the sequence that leads to itself’ … Daring Do has to find a sequence that returns the same sequence you put in.” “So, the output matches the input: it’s a ‘fixed point’, right?” Ocellus asks. “Exactly!” Rainbow exclaims, hiding her own confusion regarding those terms. “Professor?” Gallus asks, “Who would want to pick up an adventure novel just to solve puzzles?” Rainbow shrugs. “It takes all kinds, I guess. So can you guys help?” The group ponder the puzzle for some time … Author's Note Truth be known, this puzzle is a re-skinning of Quibble's fix of the final puzzle in the 'Chalice' fic. As no one had posted a solution to that one, I felt it OK to re-use it here. I had a similar puzzle in mind … but by pure coincidence, my wife found the Smullyan book that had gone missing ('Puzzles of Scheherazade'), and the puzzle I created was a pure re-skinning of the puzzle in that book. I was inspired by that puzzle (and similar ones from the 'Lady or the Tiger' book), but I didn't want to just re-use that exact puzzle. Puzzle 17 solutionThe group ponder the puzzle for some time … “If X represents any sequence of blocks, then LXR will lead to X. Further, if X leads to Y, then note the following: KX leads to YL, QX leads to YR, DX leads to Y ‘doubled’, and CX leads to Y ‘curtailed’ … find the sequence that leads to itself.” … and after some time had passed, little has been accomplished. “Yona hope letters don’t appear in nightmares after this.” Sandbar ponders the yak’s comment. “Not sure what a nightmare involving letters would look like.” “With Headmare Twilight, the letter ‘F’ would probably keep her awake.” The group chuckles at Smolder’s insight, Rainbow Dash included. “So, this one’s really got you stumped, huh?” Rainbow asks the students. Ocellus continues to peruse the descriptions on the board. “Since LXR leads to X, I feel like we need some sort of X that leads to LXR. But after that, I’m stumped.” “It looks like everything begins with LXR,” Gallus adds, “so maybe we need to focus on what comes before LXR.” He picks up a piece of chalk and writes on the board: LYR --> Y ………… XLYR --> ??? “So ‘X’ is basically any sequence of letters that can act on a given string,” he explains. “Since the only other letters that do anything are K, Q, D, and C, ‘X’ probably has to be some sort of combination of those letters. But what do we want to get from them?” “Well we want to get back what we start with,” Smolder adds, coming to the board to replace the question marks: LYR --> Y ………… XLYR --> XLYR “Wait a minute,” Silverstream says as she approaches the board. “ ‘Y’ is anything right … but it’s a fixed anything. Like, we can put it anywhere on the right, so what if …” She erases one of the X’s and replaces with another Y: LYR --> Y ………… XLYR --> YLYR Yona studies the new line. “Soooo … if X is Y, then XLYR is YLYR, and we get solution, right?” “You’re right,” Sandbar adds. “So now it’s like a construction puzzle. How do we build YLYR from just Y?” “There’s two Y’s,” Ocellus notes, “so we’ll need a D somewhere. But not right away, because there’s an L between them.” “So add the L first,” Smolder adds. “That turns Y into YL, and if we then double, we get YLYL.” “And now we cut last L off,” Yona adds. “And replace with R!” “It looks like we have all the tools we need,” Gallus concludes. “Let’s see … add the L, then double, then cut off the last L and add the R. In order, those correspond to K, D, C, and Q … but they have to be written from right to left, right?” “Wait, left or right?” Silverstream asks. Gallus puts a palm to his face. “Let me rephrase that last part: they have to be written from right to left, correct?” “Oh … yeah, right … I mean, correct. That sounds right … I mean …” “We get it!” Smolder exclaims in slight exasperation. “So from right to left, that’s QCDK. That’s our X, right?” “And that means we should use QCDKLQCDKR,” Sandbar concludes. “Only one way to find out,” Gallus states as he writes down a check on the board: LQCDKR --> QCDK KLQCDKR --> QCDKL DKLQCDKR --> QCDKLQCDKL CDKLQCDKR --> QCDKLQCDK_ QCDKLQCDKR --> QCDKLQCDKR ... Rainbow Dash inspects the work shown on the board. “Wow … it looks like you did it! You guys are as good at this as Quibble was in that chamber.” The students turn to their professor. “What chamber?” Sandbar asks. “The marble chamber underneath the trap door in the Spectrum Catacombs in the library of the Crystal Empire where we …” Too late, Rainbow Dash catches herself before saying any more, aware that she had said too much already. “Uh, saaay, look at the time. Guess we’re done our lessons for today!” “You mean the friendship lessons that we seem to have stopped having ever since we started getting these puzzles?” “Exactly,” Rainbow answers Gallus. “Gotta run!” Quickly, she turns and flies out, leaving a rainbow-colored trail in her midst. The students look towards each other. Soon after, Smolder sighs and turns to the door. “Where Smolder going?” Yona asks. “To track down Rainbow Dash and tell her that Ember already shared with me her story regarding the Chalice.” Smolder proceeds out the door, leaving the others with still more questions. “If this about Daring Do,” Yona adds, “yaks already know she exist. Daring spotted outside of Yakyakistan in past.” “Griffinstone, too,” Gallus adds. “Not sure what the big deal is.” Silverstream and Sandbar look at the two with wonder in their eyes. “Wait,” the latter says, “Daring Do is real? Like, a real pony?” “Uh oh … maybe Yona say too much.” Gallus sighs. “Just … maybe, keep it to yourself for now.” He turns to Ocellus. “So have the changelings ever been paid a visit?” Ocellus pays no mind, her focus on the chalkboard. “Ocellus? You OK?” Her concentration broken, Ocellus turns to Gallus. “Huh? Oh, sorry, just that …” “I know that look,” Gallus says. “What did you find?” “It’s just … there’s multiple solutions to this puzzle.” “Really?” Silverstream asks, essentially speaking for the group. “Well, yeah. I mean, you could cut off and replace the last letter as often as you like, so the letters C and Q could be repeated in X. That means a string like QCQCDKLQCQCDKR would also lead to itself. Or QCKCDKLQCKCDKR, which adds an extra L before cutting it off for another R. You could even add and cut off a letter before doubling, so QCDKCKLQCDKCKR and QCDKCQLQCDKCQR work too.” “We should probably tell Professor Rainbow Dash to tell her friend about that,” Sandbar says. “At least if only one solution is intended.” “Story could ask for shortest solution,” Yona adds. “That would fix, right?” “Good thing there’s no actual machine to punish a different answer, then,” Silverstream states. “So far as we know,” Gallus adds. “Anyway, may as well head to our next class.” “You think someone else is gonna ask us to solve something?” Sandbar asks. Gallus turns to face his pony friend. “At this point, I’d be surprised not to have anyone ask us to solve something.” Puzzle 18: It's 'The Mind'Seconds of dead silence pass by, with both Sandbar and Yona nervously looking at each other. Finally, Yona plays a card face-up on the table. It shows the number 97. “Aaah, nuts,” Sandbar exclaims, revealing a 95. The others groan in exasperation. “Yona knew Yona shouldn’t play,” the yak bemoans. “I had a high number,” the pony replies in defense. “I thought it’d be higher than yours. Smolder groans again. “Why did Headmare Twilight recommend this game to us. It’s hopeless!” “I think it’ll just take some more plays to calibrate our minds,” Ocellus answers. “I actually like this game.” Silverstream picks up the cardboard box that houses the cards. “ ‘The Mind’,” she says. “Weird name for a card game.” “As in, ‘Does anyone mind playing random cards to the table?’ ” Gallus says. “Where do ponies come up with these ideas?” “I like the idea,” Sandbar says. “It’s simple but thought-provoking: cards numbered from 1 to 100, everyone gets one card, and everyone tries to play their cards in order.” “Our headmare likes games that mess with our heads, doesn’t she,” Smolder states. “Yaks not like headgames,” Yona pouts. “Well I’ll try another round,” Silverstream says as she shuffles the cards anew. She deals one card facedown to everyone. “Yona sit this round out, if OK with others.” “Meh,” Gallus replies. “I’ll just take your card.” The five students each look at their own cards, and all wait for another to play the first. Seconds pass … “Look like everycreature got small cards,” Yona comments. “This is ridiculous,” Smolder states as she reaches for everyone’s cards, save the two in Gallus’s talons. She inspects the four cards now in her claws. “Yeah, we all had small numbers; this would’ve taken a while to start.” “So much for that round,” Sandbar says with a slight sigh. Smolder continues to inspect the cards. “Actually, this is interesting. Ocellus, your card was the sum of these other three cards: mine, Sandbar’s, and Silverstream’s. You’re a smart changeling; that give you enough to figure out our three cards?” Ocellus takes back her own card. “Too many possibilities; I have no idea.” Smolder takes a second look. “Well these three cards multiply to 360. That should do it, right?” “Maybe,” Ocellus replies as she ponders the new information … before shaking her head. “No … there’s still multiple possibilities. Gallus ponders the back-and-forth conversation between Smolder and Ocellus. “Huh … you know, this is interesting.” “What is?” Silverstream asks. “If I said what either of my cards was, Ocellus would have enough information to know all three of your cards. Maybe that gives you enough information.” “Hmmm …” Ocellus muses for a minute, before … “YONA KNOW ALL YOUR CARDS!!!” Author's Note So … what are Ocellus's card, Gallus's two cards, and the other three cards. For more information on 'The Mind', go here: https://boardgamegeek.com/boardgame/244992/mind Puzzle 18 solutionGallus and the others turn to Yona, with Gallus in particular observing the young yak through skeptical eyes. “Really,” he deadpans. “Uh huh.” “You know all six cards, despite not seeing any of them.” “Yaks good at math.” “You get all these cards right, and I’ll spend next Yikslirbirtfest with you in Yakyakistan.” Yona’s eyes enlarge at the possibility. “Ah ah ah … I want proof that you know these, not some lucky guesses.” “Griffon on!” Yona declares. Having pondered her own card during the conversation, Ocellus looks up and turns to Gallus. “You may want to bring a scarf.” “And no help!” Gallus adds. “All right … what’s your reasoning.” “OK, well … first, nocreature have ‘1’, or somecreature would have played immediately.” The fivesome turn to each other and nod. “Can’t argue that,” Smolder comments. “So now, three cards multiply to 360, so for example, cards could be 2, 3, and 60, but then Ocellus card 65 and then she know three cards since no other group of three cards add to 65 and multiply to 360. Instead, Ocellus must have card where two groups of three cards exist that add to Ocellus card and multiply to 360.” “She’s right,” Ocellus adds, careful not to reveal her card. “I found two possibilities … but is my card the only card that leads to multiple possibilities?” “No … Yona do math in head and figure out three possible sums: 23, 25, and 29. Hang on, Yona write down.” The young yak grabs a pencil in her mouth and jots down several numbers: 4+9+10=23 and 5+6+12=23 3+10+12=25 and 4+6+15=25 2+12+15=29 and 3+6+20=29 Yona spits out the pencil. “So now Yona know Ocellus have either 23, 25, or 29; all other sets of three numbers that multiply to 360 have unique total when added.” Gallus blinks in astonishment. “You … figured all that out. In your head.” Yona shrugs. “Like Yona say, yaks good at math.” “You sure are,” Smolder admits. “But now what?” “Now Yona remember Gallus comment, that Ocellus know everything if she know either card of Gallus. But that can only be case if Gallus cards are 6 and 12.” “Why is that?” Sandbar asks. “Because 6 and 12 are only numbers that appear in all three rows. So if Ocellus know Gallus has 4, and Ocellus have 23 or 25, then Ocellus know other cards, but not if Ocellus have 29. Or if Ocellus know Gallus has 3; then Ocellus know if she have 25 or 29, but not if she have 23.” “Huh … Guess I’ll need that scarf after all,” Gallus admits, revealing his two cards to the others: 6 and 12. “And since only one combination have neither 6 nor 12, Ocellus must have 23 and other cards must be 4, 9, and 10.” Ocellus reveals her card, the 23, and the other three show their cards, confirming the solution Yona pointed out of 4, 9, and 10. “You never cease to amaze me,” Smolder admits. “Wait a minute,” Silverstream says, an idea popping into her head as she turns to Gallus. “The only way for this puzzle to make sense is for you to know all this when you made your comment … you did all that math in your head, too?” Gallus shrugs, his eyes darting around the room. “Well … yeah. I’m good at math; I just don’t like to brag about it. Remember when Trixie gave that guest lecture about her first visit to Ponyville?” Sandbar turns to Ocellus. “And when you commented that Gallus might need a scarf … you figured this all out, too, didn’t you?” Ocellus blushes. “I just like puzzles,” she meekly replies. Smolder’s gaze turns from Gallus to Ocellus to Yona. “I suppose when we all graduate, the three of you could turn around and open a school for mathematical logic.” “Don’t go selling yourself short,” Gallus replies. “I think we’ve all shown a lot over the past several weeks.” “Mostly the rest of you,” Sandbar says with a tinge of regret. “I feel like I’ve been more of a follower than a leader during most of these.” Ocellus ponders Sandbar’s comment. “If you feel that strongly about it, there’s still some other ordering tests that the rest of us took that you haven’t heard about.” Her words elicit a groan from the others. “What?” “You want to dredge up those meaningless tests?” Smolder asks. “Those things were completely pointless.” “But not useless,” Ocellus corrects. “I think I could create an interesting puzzle based on some of those.” Author's Note Ocellus alludes to this previous one-off logic puzzle, which spawned this collection: A Question of Rankings Puzzle 19: Double Vision, Part 1Daylight shines through the window, a natural alarm clock for Sandbar as he rises from his bed and goes through his daily before-school routine. Entering the kitchen area, Sandbar mumbles greetings to his fellow non-pony students. “Mornin’, Silverstream. Mornin’, Yona, Mornin’, Gallus. Mornin’, Silverstream. Mornin’ …” His eyelids snap open as a realization hits him. He turns around to see a giggling Silverstream, then back to his front to eye the hippogriff chuckling at him. “What gives?” he suddenly bellows, as all around him his friends break out into laughter … soon causing Sandbar himself to join in. “It’s a challenge,” the Silverstream in front of him states. “Can you figure out which Silverstream is me, and which one is … oh wait … shoot!.” This Silverstream smacks her head with a claw as the other one flutters over to her. “Don’t worry about it,” the other hippogriff states before emanating a flash; an instant later, Ocellus hovers next to her friend. “I meant to say ‘which one of us is Silverstream’,” the ‘real’ Silverstream notes. “Don’t sweat it, that wasn’t the test anyway,” Ocellus replies. “Test? What test?” Sandbar asks. “You said you wanted a chance to prove your mental acuity, right?” Gallus states. “Wait, you’re saying Sandbar cute? Don’t we all know that?” Yona asks, causing the griffon to suppress a laugh. “He means Ocellus created a challenge for Sandbar … well, several challenges, I guess,” Smolder corrects. Sandbar turns to Ocellus. “What about those ‘ordering tests’ you mentioned last night?” Ocellus blushes. “I just couldn’t think of anything to come up with that we hadn’t already done. And believe me, I thought about those tests for hours … but it occurred to me that I could create a whole new test for you, based on a test that Thorax gave our professors.” “Wait,” Silverstream interrupts, “our professors were given tests?” “Thorax told me about a ‘Spot the Changeling’ challenge he did, where he would pick one of the six ponies and take her place; the others had to figure out which of them was Thorax in disguise.” “Couldn’t they just ask each other questions that only the professors would know?” Gallus dryly asks. “They put conditions in to prevent that. In any case, I thought we could do that, but on a smaller and more logic-based scale.” Ocellus flutters out to the foyer, returning with the deck of ‘Mind’ cards from the previous night. She quickly searches the deck, pulling out four cards: the 1, 2, 3, and 4. “Here’s how this will work,” Ocellus starts. “You’ll close your eyes, and when you open them you’ll see two Silverstreams, one of which will be me. Each of us will make a number of statements, and then you have to figure out which of us is Silverstream and which of us is me, Ocellus.” “Seems simple,” Sandbar says, “but what’s with the cards?” “This is where the ‘logic’ part comes in. Each of us will have one of these cards in front of us. Any statement made by somecreature with an even-numbered card in front of her will be true, and any statement made by somecreature with an odd-numbered card in front of her will be false. In other words, having the 2 or 4 means true statements and having the 1 or 3 means false statements. And you have to figure out which card we each have as well.” “Whoa … that sounds a lot harder,” Sandbar notes. “We’ll do five rounds … let’s see how you do. Ready?” Sandbar closes his eyes. “Ready.” Ocellus turns to Silverstream. “Are you ready?” “I hope so,” the hippogriff says somewhat apprehensively. “I did try to remember what you told me to say.” Ocellus smiles. “You’ll do fine,” she assures her friend. The sole pony’s eyes now shut, Ocellus changes her appearance to mimic that of her hippogriff friends. Each takes a card, and the rounds begin: [In each round, one Silverstream appears on Sandbar’s left (L) and one appears on his right (R).] [Round 1] L: “I am Silverstream. I have the 1.” R: “I am Ocellus. I have the 2.” [Round 2] L: “I am Silverstream. I do not have the 3.” R: “I am Ocellus. I do not have the 4.” [Round 3] L: “I am Silverstream. Both of us have even-numbered cards.” R: “I am Silverstream. My card’s number is the smaller of the two.” [Round 4] L: “Either I am Silverstream or I have the 1.” R: “Ocellus has the 1. Neither of us has the 4.” [Round 5] L: “Either Silverstream has the 1 or Ocellus has the 3. Neither of us has the 2.” R: “Either Silverstream has the 3 or Ocellus has the 4. Neither of us has the 1.” Author's Note Sorry for the delay; I had an idea for a more traditional Fic and spent time on that before coming back to these. The puzzle is original, but the idea is not: it's a combination of the 'Tweedledum & Tweedledee' chapter of "Alice in Puzzleland" by the late, great, Raymond Smullyan and the Fic "Spot That Changeling" by Inspector Brown. A link to the latter is provided, in case you missed it: Spot That Changeling Puzzle 19 solutionThe sole pony’s eyes now shut, Ocellus changes her appearance to mimic that of her hippogriff friends. Each takes a card, and the rounds begin: [In each round, one Silverstream appears on Sandbar’s left (L) and one appears on his right (R).] [Round 1] L: “I am Silverstream. I have the 1.” R: “I am Ocellus. I have the 2.” Sandbar ponders the two sets of statements. “Oh wait,” he quickly says, pointing a hoof towards the one on his left. “You have to by lying; nocreature would claim to have the 1. So you’re Ocellus, and you can’t have the 1, so you have the 3.” He turns to the one on his right. “And that means you’re Silverstream, and you’re lying too so you have the 1.” One flash of light later, the ‘Silverstream’ on his left reverts to her natural, changeling form. “Good job,” she remarks. “But a pretty easy round, right?” Gallus states. “They’ll get harder,” Ocellus promises. “Close your eyes again.” Sandbar obliges, and the second round starts: [Round 2] L: “I am Silverstream. I do not have the 3.” R: “I am Ocellus. I do not have the 4.” Sandbar considers the two sets of statements, but quickly replies: “This is like the last one, but in reverse, right?” He turns to the one on his right. “Nocreature would lie and claim to not have the 4, just like nocreature would truthfully claim to have the 1. So you’re telling the truth, and you’re Ocellus with the 2.” He turns to the one on his left. “And that makes you Silverstream, and you’re truthful as well so you have the 4.” Another flash of light reveals Ocellus as the ‘Silverstream’ on his right. “Right again,” she comments. “Still pretty simple,” Gallus reiterates, drawing a glare from Sandbar. “Whaaat?” “You could be a little more supportive, you know.” The dragon points out. “Ready for Round 3?” Ocellus asks. “Can I just check the note cards you gave me one more time?” Silverstream asks. “These later rounds get harder to remember.” “Go ahead,” the changeling replies. “Ooo, ooo, Yona want to play too!” She turns to Sandbar. “Unless you want to keep doing these yourself. Yona understand if you do.” “I don’t mind,” Sandbar responds as he shuts his eyes. Smiling, Yona does likewise; moments later, the third round begins: [Round 3] L: “I am Silverstream. Both of us have even-numbered cards.” R: “I am Silverstream. My card’s number is the smaller of the two.” Sandbar turns to Yona. “What do you think?” “Well, Silverstreams can’t both be Silverstreams, right? So one liar.” “Sounds good to me … and that means the one on the left is lying when she says they both have even-numbered cards. So the one on the right is the real Silverstream.” “But what numbers?” Yona asks. “Oh wait, right-hoof Silverstream truthful, so her card 2 or 4 and left-hoof Silverstream has 1 or 3.” “But the one on the right has the smaller number,” Sandbar reasons. “So Ocellus is on the left with the 3 and Silverstream is on the right with the 2.” Ocellus [on the left] reveals her true self once again. “Not so easy this time, was it?” she comments, her eyes focused on Gallus. “I’ll admit that there was more to that one,” the griffon says. He turns to Smolder: “You wanna try to solve the next one?” “Sure, why not?” the dragon replies. She and the others all shut their eyes as the fourth round commences: [Round 4] L: “Either I am Silverstream or I have the 1.” R: “Ocellus has the 1. Neither of us has the 4.” Yona looks at Gallus. “OK, smart guy. You know answer?” “Uh …” he stammers, before admitting: “All right, this one has got some kernels on the cob.” “What do you think,” Smolder asks of Sandbar. The young pony thinks about the statements made by the two Silverstreams. “Well, suppose the Silverstream on the left was lying. That would mean she wasn’t Silverstream and she didn’t have the 1; she would have to be Ocellus, and she would need to have the 3. And that would like the other Silverstream’s first statement false, so they’d both be liars … but then her second statement would be true, and that can’t be!” “I’m impressed,” Gallus says with conviction. Yona picks up where Sandbar leaves off: “That mean Silverstream on left truthful, so she can’t have 1; it must be real Silverstream, and one on right Ocellus.” “And since nocreature would claim to have the 1,” Smolder adds, “Ocellus is lying and must have the 3. And that means one of them does have the 4; it must be Silverstream.” Ocellus [on the right] changes back to her usual form. “Well done!” Silverstream turns to Ocellus. “We still have one more round, right.” “We do … and it’s the hardest round. All of you ready?” “I hope so,” Gallus says, hoping to redeem himself in the eyes of the others. The group of four shut their eyes and the final round plays itself out: [Round 5] L: “Either Silverstream has the 1 or Ocellus has the 3. Neither of us has the 2.” R: “Either Silverstream has the 3 or Ocellus has the 4. Neither of us has the 1.” “Whoa,” Sandbar quietly says. “Where do we begin?” “We gotta begin somewhere,” Gallus starts. “Let’s just say the one on the left is telling the truth. What would that lead to?” Smolder continues: “Well if the Silverstream on the left is telling the truth, then that one has to have the 4 since neither would have the 2. And the Silverstream on the right would have to have either the 1 or 3, making that one a liar.” “And since Silverstream on right liar, that one have 1 and not 3,” Yona follows. Sandbar puts the information together: “So the one on the left is truthful with the 4 and the one on the right is lying with the 1.” He ponders for a minute … “That doesn’t work!” The other three turn to Sandbar, awaiting an explanation. “If Silverstream is on the left, then her first statement is false: she can’t have the 1 and Ocellus couldn’t have the 3 since she would have the 1. But on the other hoof, if Ocellus is on the left, then Silverstream’s first statement would be true since Ocellus would have the 4.” “OK,” Gallus says, “that’s proof enough for me that you’re as adept at this as the rest of us. “So what all that mean?” Yona asks. “We do all this for nothing?” “Not for nothing,” Smolder corrects: “Now we know the one on the left is lying. And that means the one on the right has the 2.” “And that means the one on the left has the 3, since neither can have the 1,” Gallus adds. Sandbar concludes the process: “Silverstream can’t be on the right, ‘cause that would make her first statement false. Silverstream must be on the left with the 3, and Ocellus might be on the right with the 2.” At long last, the ‘Silverstream’ on the right morphs back into Ocellus. “Great job! she exclaims. “I really didn’t think you were going to get that.” “I had good help,” Sandbar replies somewhat bashfully. “Sandbar stop selling self short,” Yona pleads. “Sandbar smart, like all Yona’s friends.” “And Yona herself,” Gallus adds. “That looked like fun for you guys,” Silverstream states. “Can I play along too?” Ocellus ponders the hippogriff’s request. “I do have some more challenges, but I thought those might be a bit too much. Besides, if you’re playing, then who am I copying?” Smolder raises a claw … Puzzle 20 solution“OK,” Ocellus begins, “for these five rounds, you’ll each see two Smolders, one of which will be me. Each of us will have a card from 1 to 4, and of course, we won’t have the same card. You’ll have to figure out which of us is really Smolder, as well as which card each of us has.” The foursome look at each other quizzically, before Silverstream addresses the changeling: “Uh … how is this any different from what we just did?” “One small change, but it makes a big difference: this time, if I have the 1 or 3 then my statements will be false, and if I have the 2 or 4 then my statements will be true.” Ocellus’s comment is met be more strange looks. “That exactly how first five rounds went,” Yona points out. “Oh … yeah, that part’s the same.” Ocellus begins to blush as she continues: “What’s different is—” “What’s different,” Smolder interrupts, “is that if I have the 1 or 3 then my statements will be true, and if I have the 2 or 4 then my statements will be false. It’s the opposite of Ocellus.” “That … really doesn’t sound all that much more challenging,” Gallus states. “Hey, I’m game either way,” Sandbar replies. “Go ahead and start.” The foursome collectively close their eyes, and the rounds begin: [In each round, one Smolder appears on the group’s left (L) and one appears on their right (R).] [Round 1] L: “I am Smolder. I have the 1.” R: “I am Smolder. I have the 2.” It takes only a few seconds for Silverstream to respond: “Well … one is lying and one is telling the truth. I’m sure of that.” “Shouldn’t be hard to figure out which is which,” Gallus adds. “Which one said something that couldn’t be said?” Sandbar ponders for a moment … “The Smolder on the right said she had the 2 … but the real Smolder wouldn’t truthfully say she had the 2, so that has to be Ocellus. And, she has to have the 1 or 3.” “But then Smolder on left truthful,” Yona adds, “so Smolder on left have 1. That only leave 3 for Ocellus on right.” In a flash, the Smolder on the right changes back into Ocellus. “Not bad. Let’s see how you do on the next one.” Eyes close, next round begins … [Round 2] L: “I am Smolder. Ocellus has the 3.” R: “I am Ocellus. Smolder does not have the 4.” Silverstream again begins the thought process: “They each to be somecreature different, so they’re either both truthful or both lying this time.” “But if both truthful, Ocellus have 3,” Yona points out. “That not right.” “So both are lying,” Gallus concludes. “Ocellus is on the left and Smolder is on the right. Now as to their numbers …” “They both lied,” Sandbar states, “so Smolder does have the 4. And since Ocellus can’t have the 3, she has the 1.” Another flash, and the Smolder on the left reverts to her natural form. “It feels weird to say ‘Ocellus has’,” she admits. “Meh,” Smolder says. “I know dragons who like to refer to themselves in the third-creature. No big deal.” “Just big egos,” Gallus says to himself, before quickly backtracking upon noticing Smolder’s death-glare. “Sorry, sorry … old habits … my apologies!” “Wellllll, you may not be too far off with certain dragons.” “We’re all cool, right?” Sandbar asks. “It’s fine,” Smolder says. “Let’s just get back to the rounds …” [Round 3] L: “I have the 1. The other card is not the 2.” R: “I have the 3.” “Wait,” Yona says, “neither one claimed to be Smolder or Ocellus. This round actually have solution?” “There has to be a solution,” Gallus states, “… and it begins with the fact that Ocellus is lying.” “How do you know that?” Silverstream asks. “Ocellus would never truthfully state she has an odd number, either 1 or 3.” “Good point,” Sandbar says. “Does that mean Smolder is telling the truth?” Yona ponders for a moment … “No! Smolder could still be liar; all we know is Ocellus definitely liar.” “All right,” Silverstream continues, “so which one is Smolder? Could she be on the left?” Gallus thinks the possibility over. “Suppose she was on the left and telling the truth. Then she does have the 1 … and if Ocellus is lying, she would have to have the 3.” “But that would make her statement true,” Sandbar adds. “That doesn’t work.” “So now say she’s lying,” Gallus continues. “Then the ‘other’ card is a 2, which would mean Ocellus is telling the truth … that doesn’t work, either.” “So that mean Smolder not creature on left,” Yona concludes. “Ocellus on left, and both statements false; that mean other cards is 2.” Silverstream excitedly wraps up the solution: “That means Smolder is lying and has the 2, and Ocellus being on the left means she has the 3!” The Smolder on the left changes back into Ocellus as the one on the right comments: “Wow, you guys really are getting good at this.” “Two more rounds,” Ocellus points out as they begin anew … [Round 4] L: “The sum of our numbers is 5. My number is the smaller of the two.” R: “One of our numbers is double the other. My friend’s card is not the 1.” Silence pervades for a full minute. Finally … “The one on the left is lying,” Gallus states. “How do you figure that?” Sandbar asks. “If she was truthful, their cards would have to be one even and one odd, and that would force both of them to be telling the truth. But that’s impossible, since the only ways to have one card be double the other is with the 1 and 2, or the 2 and 4 … and neither of those add up to 5.” “Whoa,” Silverstream says, “that is some high-level thinking there!” “So where we go from here?” Yona asks. “What if both of them were lying?” Sandbar asks. “Then they’d still have one even and one odd card, right?” Gallus nods. “But,” Sandbar continues, “we already know their cards don’t add up to 5, so the cards would have to be either 1 and 2, or 3 and 4, right?” “Sounds good to me,” Gallus replies. “Wait,” Silverstream says, “1 and 2 are out, right? ‘Cause, that would make the ‘double’ statement true.” “That leave 3 and 4,” Yona adds, “but that make other statement true, about friend card not being 1.” “So all that means the statements from the Smolder on the right can’t be false; they must be true,” Gallus concludes. “That means the numbers are either both even or both odd,” Sandbar states. “They can’t be both odd, since one is double the other, so they must be 2 and 4.” “And since the Smolder on the left lied,” Silverstream adds, “her number is the bigger of the two. And that means—” “Smolder on left with 4 and Ocellus on right with 2,” Yona concludes. Ocellus (on the right) reveals herself. “That was amazing! I really thought I had you stumped on that one.” “That just leaves one final round,” Smolder says. “Let’s see if you can go five-for-five …” [Round 5] L: “Either I am Smolder or I have the 3.” R: “I am Smolder and I do not have the 4. Neither of us has the 2.” “I don’t like conjunctions,” Silverstream admits. “They make my head hurt with these puzzles. “We gotta start somewhere,” Gallus states. “What if the one on the left is the real Smolder?” “That mean one on left truthful,” Yona replies. “Also, one on right lying if that one actually Ocellus.” “And if she’s lying,” Silverstream adds, “then one of them has to have the 2.” “But that’s not possible,” Sandbar insists. “Either Smolder would have the 2, and that would force her to lie, or Ocellus would have the 2, and that would force her to be truthful. Neither of those works with these assumptions.” “So much for figure out which one is Ocellus,” Gallus says. “The one on the left is Ocellus.” Rather than wait for the final solution, Ocellus (on the left) changes back. “Wow … I really thought this was the hardest one.” “They still need to figure out the cards,” Smolder points out. “But if Ocellus truthful, she would have 2 or 4,” Yona says. “That not work here, so Ocellus lying and have 1 or 3 … but not 3, since that make statement true. So Ocellus have 1.” “Meanwhile,” Silverstream adds, “the only way Smolder could by lying is if she has the 2 or 4. If she had the 2, her first statement would be true, and if she had the 4, her second statement would be true. So Smolder must be telling the truth, and her card has to be the 1 or 3 … but not the 1, since Ocellus has it, so she has the 3.” Ocellus and Smolder each reveal their cards, confirming the final piece of reasoning. “That was actually a lot of fun!” Silverstream states. “So what’s next?” Ocellus shrugs. “There’s not much more I can do with this format? I mean, I could make it so one of us is truthful with 1 and 4 and the other is truthful with 1 and 3, or some other combination, but really, it’s all pretty much the same.” “Maybe we need two pairs, like two Sandbars and two Galluses,” Yona says. The others chuckle at the thought. “Maybe we should just have two of all of us,” Gallus says. “That would mess with someone’s mind.” “But if we did that, who’d be left to actually solve the puzzle?” Sandbar asks. Ocellus takes all the comments in as her mind races … Puzzle 6 solutionRarity gives Gallus an odd look. “I beg your pardon?” Gallus pauses for a few seconds before responding. “Professor, how long did it take for you to figure out an appropriate ‘ensemble’ for your client?” “Oh … I don’t remember, really … maybe, a couple hours?” “I see,” Gallus replies. He turns to his fellow students. “Ocellus, did you write down the requests made by Rarity’s client?” Ocellus looks up from her sheet of notebook paper, gives a sheepish smile, and nods. “Professor, may I come to the board?” “Well … uh … yes, that would be OK.” Gallus takes the paper from Ocellus’s desk and casually writes down a number of sentences on the board: No dress should have more than one gem type, and at least three different gem types had to be used for the dresses, though not necessarily all four gem types. C must wear sapphire, unless D wears ruby. D must wear emerald, unless B wears the same gem type as E. E must wear either sapphire or emerald, with A wearing diamond if E wears sapphire and with A wearing ruby if E wears emerald. C should wear ruby, unless B wears either sapphire or emerald, and if B does wear sapphire then C should wear diamond, and if B wears emerald instead then C should wear emerald as well. “That’s all the information, correct?” Gallus dryly asks his fellow students. They nod in unison. “Ocellus, care to begin?” Ocellus responds quickly: “The last request indicates that C should wear either ruby, diamond, or emerald. That means that C should not wear sapphire. But then there’s the request that C must wear sapphire unless D wears ruby … since C cannot wear sapphire, that means that D must wear ruby.” “Wow … I’m very impressed with your acumen, Ocellus. And yes, D did indeed wear ruby. But then—” “Sandbar,” Gallus continues, “what do you see?” Sandbar studies the requests for another minute. “Ocellus said that D wore ruby. But that one requests says that D should wear emerald, unless B and E wear the same gem type. And that means that B and E have to wear the same gem type.” Gallus turns to Yona. “Your turn.” “What? Oh, yeah, Yona ready. Uh … B and E wear same gem, but E wear either sapphire or emerald, so that mean B also wear either sapphire or emerald. And if B wear sapphire, then C wear diamond, and if B wear emerald, then C wear emerald.” “Smolder?” The young dragon sighs. “If B and E both wear emerald, then C wears emerald as well. And if E wears emerald, then A wears ruby. But D already wears ruby, so everyone would be wearing either emerald or ruby, and that can’t be since at least three gem types have to be used.” “Silverstream? Care to finish?” The hippogriff enthusiastically continues: “That means B and E must both wear sapphire instead, and since B wears sapphire that means C wears diamond, and since E wears sapphire that means A wears diamond.” “So in summary,” Gallus concludes as he strolls back to his seat, “A wears diamond, B wears sapphire, C wears diamond, D wears ruby, and E wears sapphire.” Rarity gapes in astonishment before sputtering with a response. “What? … but … I mean … that’s all correct, but … I spent hours on … and you just …” “We’ve had a lot of practice as of late,” Gallus casually admits. “Now I can’t believe what I’m about to say, but can we get back to discussing fashion?” Puzzle 7 solution (1) Gallus was higher than Sandbar (2) Sandbar was higher than Ocellus (3) Ocellus was higher than Yona (4) Yona was 2 higher than Smolder (5) Smolder was 3 higher than Sandbar (6) Gallus was 4 higher than Yona (7) Ocellus was 5 higher than Gallus “Did I really blurt all these statements out?” Silverstream asks as she peruses Twilight’s notes. “In about 5 seconds, yeah,” Smolder replies. “So, Ocellus, care to explain?” Ocellus opens her mouth to answer, but Gallus puts a wing on her shoulder. “How ‘bout someone else take the lead for a change?” He turns to Smolder. “Why don’t you begin this one?” Smolder groans. “What does it matter? For that matter, why don’t we just ignore this and hear our order from Fluttershy tomorrow?” “If you haven’t noticed, we’ve been getting involved in a lot of these puzzles as of late, almost as if the universe wants us to keep our minds honed. I’m just saying we shouldn’t be overly reliant on a single creature to handle all the mental heavy lifting.” The others give a quizzical look towards the griffin … before he and the others burst into collective laughter. “Couldn’t keep a straight face with that one,” Gallus mirthfully admits. “OK, OK, Ocellus, care to start us off.” “Actually, I do want to begin,” Smolder insists. “I mean, you never know, right?” she adds with a grin. After a further round of chuckles dies down, Smolder inspects the list of sentences. “It was nice of Headmare Twilight to number the sentences for reference. So, they’re only true if both students mentioned are either above or below Silverstream at the same time, right?” “That sounds right,” Ocellus says. “OK then … where to begin …” Smolder inspects the statements for a few minutes before coming to a realization: “Wait … if sentence 7 was true and Ocellus was 5 higher than Gallus, then Silverstream would have to be between them. And that can’t be … so that statement is false.” “Nice,” Sandbar states. “Can I give it a go?” “Go ahead,” the young dragon replies as Sandbar peruses the statements. “So … I see Smolder’s name in sentences 4 and 5, so if both were true, then Yona would be 5 higher than me … but that can’t be either, so at least one of those statements is false. But that’s all I got. What now? Anyone else want to jump in?” “Yona give shot,” the yak replies. “… false statement mean students on opposite side of seapony. That mean Ocellus and Gallus on opposite sides, and Smolder opposite either Yona or Sandbar. So at least two above and two below seapony … Silverstream either third or fourth!” “That’s brilliant!” Ocellus states. “Honestly, I was a little stuck there. OK, since Silverstream is either third or fourth, then statement 6 can’t be true, because that would force Gallus and Yona to be on the same side, which could only happen if Silverstream was either first or last. And that forces Gallus to be opposite Yona.” “OK, let’s take another look at all those false statements,” Gallus suggests. “The last two statements are false; we know that now. That means that I’m on the other side of both Yona and Ocellus … so they have to be on the same side of Silverstream. And that means that statement 3 is true: Ocellus is higher than Yona.” “And if statement 4 was true as well,” Ocellus adds, “then there would be four of us on the same side of Silverstream: myself, Yona, Smolder, and whoever was between Yona and Smolder … but then Silverstream couldn’t be third or fourth in that case. So statement 4 is false, and Yona is opposite Smolder.” “But statement 5 can’t be true, either,” Sandbar states. “That would also force Silverstream to be somewhere other than third or fourth. And that means Smolder is opposite me.” “Hey, can I work on this?” Silverstream asks. “They were my statements, right?” The others laugh. “Our apologies,” Gallus states with a wry smile. “By all means.” “OK, OK … so all the last four statements are false, so Ocellus is opposite Gallus who’s opposite Yona who’s opposite Smolder who’s opposite Sandbar. So … it’s like, every other student is on the same side, right? Because two wrongs make a right, or something like that?” “Sounds about right to me,” Smolder admits. “Cool! … so that means that Ocellus, Yona, and Sandbar are all on one side of me while Gallus and Smolder are on the other side. That means that Gallus and Sandbar are on opposite sides, so statement 1 is false; Sandbar must be higher than Gallus … and that means that Ocellus, Yona, and Sandbar are one, two, three in some order, I was fourth … aww, I was only fourth? … and Gallus and Smolder are five, six in some order.” “We’ve all been doing well on all these quizzes,” Ocellus reassures. “I’m sure we all passed with flying colors.” “I hope so,” Smolder adds with a tinge of derision. “Calm down, everyone,” Sandbar says. “Let’s finish this, huh?” “Yona finish,” the yak replies. “So, Sandbar, Ocellus, and Yona all above Silverstream … so statements 2 and 3 are true, so Sandbar first, Ocellus second, and Yona third.” “That just leaves Smolder and myself,” Gallus states, “… but since Yona was third, Smolder couldn’t be fifth since statement 4 is false. That means I was fifth, and Smolder was sixth.” He turns to Smolder. “No offense meant.” “Meh … none taken,” Smolder reassures. “I don’t think professor Fluttershy has ever given a student a bad grade.” Ocellus muses for a moment. “About what I’d expect from the bearer of the element of Kindness.” “So … that’s it, right?” Silverstream asks. “I mean, we did it, didn’t we.” “Looks like it,” Gallus answers. “In order: Sandbar-Ocellus-Yona-Silverstream-me-Smolder. I hate to say it, but I’m starting to enjoy doing these.” “Wow,” Yona says, “so which professor bearer of element of Miracles?” The group shares one final laugh before eventually retiring for the night. Puzzle 9: Ocellus is Stumped“Now everycreature take a number,” Pinkie instructs the class, “and starting tomorrow, you’ll all take turns baking cupcakes for the class.” “I’ve never baked anything before,” Sandbar reminds his professor. “I’m not sure my batch will turn out OK.” “It’s not about the quality, silly!” Pinkie reassures. “It’s about the act of doing something for your friends. And if they don’t come out exceptionally well, then you have a fun story to share.” “I’m with Sandbar on this,” Gallus states. “What if my ‘fun story’ gets told in a hospital ward?” Smolder dismisses Gallus’s concern: “Meh … I can taste-test if everyone else wants. We dragons have strong stomachs; it takes a lot to make us sick.” “Is that a challenge?” Gallus slyly asks. The group laughs at the veiled threat. “See … even this discussion is fun,” Pinkie declares. “Now come on up, one number per student.” Each student, in turn, randomly selects a number from Pinkie’s desk. Each peruses his or her own number, without seeing any other number selected. “Okie-dokie, that’s all for today! See you bright and early tomorrow!” Having concluded classroom proceedings for the day, Pinkie pronks her way out the classroom door. “At least she’s not going on about the walls again,” Silverstream states. “So when everyone else go?” Yona asks. “Yona second,” she adds as she shows off the ‘2’ to the others. Smolder leans over to look at Sandbar’s number. “My number is three less than Sandbar’s,” she states. Silverstream opens her mouth to speak, but Gallus stops her and turns to Ocellus. “OK, you’re the puzzle-lover here. I’m guessing you can figure out everyone’s number just based on those statements, right?” Ocellus looks at her number and ponders the previous statements. “I … I …” “Is everything OK?” Sandbar asks. “I … I don’t know.” “You don’t know if you’re OK?” Silverstream asks. “No … I don’t know everyone else’s number. In fact, I don’t know anyone else’s number, besides my own and Yona’s. That’s not like me.” “ ‘Huh’,” Gallus says, taking a second look at his own number. “If that’s the case, then I know everyone’s number.” “YOU DO???” the other students ask in unison. “I do … and if I add that my number is higher than Ocellus’s, everyone can now figure out who has what number.” Author's Note So … what number does each student have? (All numbers are from 1 to 6.) (Admittedly a bit on the easy side ... I'll try to make the next one more challenging.) Puzzle 12 solution Unicorn A said “None of the others have the same hometown as me.” Unicorn B said “F and I have the same hometown.” Earth pony E said “C is from Utopia.” Earth pony F said “D and E have different hometowns.” The students ponder the four statements, aware that the two pegasi, C and D, said nothing, and also aware that of each pair of like-type ponies, one was truthful and one lied. “There doesn’t seem to be much to go on,” Silverstream states to the group. “When I solve puzzles like this,” Ocellus states, “I usually start with statements one makes about one’s self.” “Like what B said?” Yona asks. “Exactly. B said that she had the same hometown as F. Let’s start there.” The group ponder in silence for a minute, before Sandbar speaks up. “If B was truthful, she’d be from Utopia … and so would F.” Smolder continues: “And that would make her a liar. It would also force E to be a truth-teller, and that would mean E was from Elysium.” “But if F was a liar,” Silverstream adds, “then D and E would have the same hometown … they would both be from Elysium.” “And if E truthful,” Yona says, “C from Utopia.” Gallus concludes the train of thought: “So C would be from Utopia and D would be from Elysium … neither one would be a truth-teller, and that’s contrary to what we were told from the outset, that one of the two pegasi is truthful.” “OK then,” Ocellus states, “that means that B is a liar and A is truthful. So A is from Utopia … the only pony from Utopia, as it turns out.” “But if no one else is from Utopia,” Sandbar says, “then E lied. That means F is truthful, so F is from Elysium.” “But B lied,” Smolder adds, “so B isn’t from Utopia. And if F is from Elysium, B isn’t … B must be from Paradise.” “And since E lied,” Silverstream states, “that means E isn’t from Elysium, right? But no one else is from Utopia, so E must be from Paradise, too.” “F told the truth,” Gallus adds, “so D and E do have different hometowns. If E is from Paradise, D isn’t … and D isn’t from Utopia either, since A was truthful. That means D is from Elysium … and would have lied if she said anything.” “Then C truthful,” Yona says, “so C from Paradise.” “So that’s the answer, right?” Ocellus asks. “A is from Utopia, B is from Paradise, C is from Paradise, D is from Elysium, E is from Paradise, and F is from Elysium.” Coccyx clops his hooves together in applause. “BRAVO, everyone! That was some incredible deductive reasoning you demonstrated. I am most impressed!” “These towns really do sound weird,” Smolder states. “I can’t speak for other dragons, but I get the feeling most wouldn’t care about these ponies’ oddball behavior.” “Still, it’s incredible that ponies would go to such lengths for any reason,” Sandbar adds. Ocellus adds: “You haven’t even heard about their alicorns.” “ALICORNS???” The group turn to Ocellus, then to Coccyx. “These out-of-the-way villages in some remote corner of Equestria have princesses?” Gallus asks incredulously. “Not princesses, alicorns,” Coccyx corrects. But not in the way that you think.” “Could you tell us more about these ‘alicorns’?” Sandbar asks. Coccyx chuckles and smiles. “This, my friends, leads to a whole new level of logic puzzles.” IMPORTANT ANNOUNCEMENTThank you to everyone who has read this collection of MLP-themed logic puzzles. Due to copyright infringements, I will be deleting this thread effective at 8pm EST. I only learned about this shortly before heading off to work, so I will be unable to take action until I get home later today. More details to come … . . . . . . . . . . . OK, about that announcement: (Seriously, though, if anyone can let Raymond Smullyan know that his work inspired me to write pony-themed puzzles …) I'm going to get to at least 20 before all is said and done. After that … we'll see ... Puzzle 20: Double Vision, Part 2After several minutes of prepping in a separate room, Ocellus asks Smolder one last time: “You know what you have to say?” “I got it,” the young dragon insists. Satisfied, Ocellus returns to the dining nook and explains the next challenge to the others, with Smolder following close behind. “OK,” she begins, “for these five rounds, you’ll each see two Smolders, one of which will be me. Each of us will have a card from 1 to 4, and of course, we won’t have the same card. You’ll have to figure out which of us is really Smolder, as well as which card each of us has.” The foursome look at each other quizzically, before Silverstream addresses the changeling: “Uh … how is this any different from what we just did?” “One small change, but it makes a big difference: this time, if I have the 1 or 3 then my statements will be false, and if I have the 2 or 4 then my statements will be true.” Ocellus’s comment is met be more strange looks. “That exactly how first five rounds went,” Yona points out. “Oh … yeah, that part’s the same.” Ocellus begins to blush as she continues: “What’s different is—” “What’s different,” Smolder interrupts, “is that if I have the 1 or 3 then my statements will be true, and if I have the 2 or 4 then my statements will be false. It’s the opposite of Ocellus.” “That … really doesn’t sound all that much more challenging,” Gallus states. “Hey, I’m game either way,” Sandbar replies. “Go ahead and start.” The foursome collectively close their eyes, and the rounds begin: [In each round, one Smolder appears on the group’s left (L) and one appears on their right (R).] [Round 1] L: “I am Smolder. I have the 1.” R: “I am Smolder. I have the 2.” [Round 2] L: “I am Smolder. Ocellus has the 3.” R: “I am Ocellus. Smolder does not have the 4.” [Round 3] L: “I have the 1. The other card is not the 2.” R: “I have the 3.” [Round 4] L: “The sum of our numbers is 5. My number is the smaller of the two.” R: “One of our numbers is double the other. My friend’s card is not the 1.” [Round 5] L: “Either I am Smolder or I have the 3.” R: “I am Smolder and I do not have the 4. Neither of us has the 2.”
Puzzle 1: MentorsAfter so many months of having her as a professor, the ‘Young Six’ had gotten rather used to seeing Pinkie Pie in a number of strange poses and appearances. This day, upon seeing the pink earth pony enter the room regaled in fortune-teller garb, a crystal ball placed upon the center of her desk, just felt a bit more unusual than others. “Yep,” Gallus begins, “just another normal school day with Equestria’s most abnormal professor.” Yona follows: “Uh, Professor Pinkie Pie? What with weird get-up?” “I’m not getting up, silly; I’m sitting down,” Pinkie cheerfully replies as she takes her seat at her desk, pulling out a blank sheet of paper from a drawer. “Now then,” she continues as she gazes into the crystal ball, “Which of you have the ability to see into the future???” The students exchange quizzical looks with each other. Smolder turns to Sandbar: “Is this another one of those ‘pony’ things we don’t know about?” The latter can only shrug. “Now,” Pinkie starts, “your professors, including yours truly, will be holding one-on-one sessions with all of you.” “Did we do something wrong?” Silverstream asks, fear in her voice. Pinkie laughs. “No, nocreature is in trouble. We just feel … well, actually, it’s Headmare Twilight who feels … that you all may benefit from some private conversations. A chance to share your experiences, goals, and worries, and get some useful toothy … uh … tube-lace? …” “Tutelage?” “Thanks, Ocellus! Yeah, some private tooty-ledge from none other than the Element Bearers themselves. Each of you will get to meet with one of us. Isn’t that great?” “That’s wonderful,” Gallus deadpans. “What does that have to do with fortune-telling?” “And who meet with Yona?” the yak adds. “What an incredible coincidence!” Pinkie declares. “Those questions go hoof in claw in talon!” Once again, the students turn to each other in confusion. Yona opens her mouth to speak, but Gallus pats her shoulder and speaks for her. “Yona, Smolder, Sandbar, Silverstream, Ocellus AND Gallus confused.” “As we speak … well, ‘me’, anyway … Twilight and the others are randomly determining which pony gets which student. We felt … well, ‘Twilight’ felt … that nocreature should feel slighted if he or she really wanted one particular professor, or headmare as the case will be for one of you. This way, everycreature has an equal chance to meet with everyprofessor.” “OK,” Smolder says, “but what about the fortune-telling part?” “That’s the fun part. You’re all going to make predictions as to who each of you get!” Gallus blinks. “I’m sorry, professor, but where does the ‘fun’ come in?” “Predictions can be lots of fun! Trying to guess who’ll win the annual Young Fliers competition, or wondering which evildoer will be next to try to take over all of Equestria … you can make a game about anything!” “I love games!” Silverstream declares, as Gallus puts a talon to his face and groans to himself. “So come on, everycreature, make your predictions!” Pinkie states before grabbing a pencil with her mouth, prepared to jot down their comments. The students turn to each other, none sure how to begin. Finally, Yona speaks up: “Yona predict that Yona not get Headmare Twilight.” Pinkie proceeds to put the comment to paper. Sandbar goes next: “Uh, I’ll predict that Ocellus gets Professor Rainbow Dash.” His comment draws a shy smile from the changeling. Smolder follows: “Well, I’ll predict that I’ll get either Headmare Twilight OR Professor Rainbow Dash.” Gallus snorts in derision. “This really isn’t particularly fun, is it?” Pinkie spits the pencil out from her mouth. “Aw, where’s your playful spirit?” she gleefully chides. “C’mon, what do you predict?” she asks, before again grabbing the pencil by her mouth. “Fine,” Gallus responds: “I predict that Smolder is wrong.” His comment causes the young dragon to stare daggers at him. “What? She wanted a prediction, I gave a prediction.” In an effort to avoid further confrontation, Ocellus quickly speaks up: “I predict that Silverstream gets somepony other than Rarity.” “Oh, oh, my turn!” Silverstream excitedly states. “OK, I predict that Sandbar gets … uh … one of our pegasus professors. So, either Professor Rainbow Dash or Professor Fluttershy.” The pencil falls from Pinkie’s mouth as she jots down the last of the predictions. “Great job, all of you!” she states. Her tail twitches, her ears flap, and she assumes several trance-like poses, before reverting to normal. “And not a moment too soon!” she exclaims as Twilight enters the room. “Wait,” Smolder says, “did you just predict Headmare Twilight coming into the room?” Twilight observes the students, then turns to Pinkie. “Your ‘Pinkie Sense’ went off again, didn’t it?” The grin and closed eyes that forms on Pinkie’s face confirm her belief. Twilight then turns back to address the class. “I trust that Professor Pinkie Pie told you about the meetings we’ll be holding for the six of you.” “Yeah,” Gallus responds, “right before having us make predictions about who gets who.” Twilight again turns to Pinkie, inspecting her outfit. “That would explain the garb,” she states as she eyes the predictions on the desk. Quickly, she scans the predictions. “Hm,” she notes. “This is a fun coincidence.” “Ooo, more fun,” Pinkie replies as she pulls the notes back towards her. “So what’cha got?” “Well, it’s just that exactly two of these six predictions are true … and the students who made those predictions were the ones who got paired with you and me.” The students look towards Twilight with confusion. “Uh, Headmare Twilight?” Sandbar asks, “what’s fun about that?” Pinkie quickly replies: “ ‘Fun’ is in the eye of the beholder, Sandbar. We all have our ideas of what makes something ‘fun’. It’s part of what makes ‘fun’ so special!” She turns to Twilight and starts bouncing in anticipation. “So who did I get, Twilight?” “I already told you,” the alicorn cryptically reply. Pinkie ceases bouncing as she takes in the response, the students looking on in surprise. “Waaaaaait a minute … is this another one of those puzzles where you’re gonna tell me you’ve given me enough information to deduce which pony got assigned which student?” “That’s exactly what this is,” Twilight gleefully concludes. “WOOHOO!!!” … Ocellus’s outburst causes her fellow students, and Pinkie, to turn to face her. “Oh … sorry, just a little excited,” she sheepishly explains. Author's Note Can you figure out which pony got paired with each student? As mentioned before, I ask that anyone who wishes to post a solution to this story uses hidden text. Or just PM me with your solution, if you wish to know before I post the next chapter. The following chapter will reveal the solution.
Puzzle 1 solutionFor a brief moment, Ocellus braces herself for a round of insulting laughter at her expense … but the friendly chuckles that ensue merely serve to reinforce the sentiment that she is around true friends. “Always the bookworm,” Gallus dryly states. “More like, book- … changeling,” Silverstream adds, before blushing and sheepishly going quiet. Another round of chuckles follows. “Seriously, though,” Smolder says, “how could anyone figure out which of us got which professor based on those predictions. They were all over the place, and we don’t even know which ones are true.” “It’s all about using logical deduction and reasoning,” Twilight states. “Focus on one or two statements whose veracity can be figured out, and use that information to determine that of the other statements.” “Just like you did on that one trip we took, right?” Pinkie gleefully asks. Twilight nods. “Yona confused. Where Yona start?” “Yona starts with Yona’s prediction,” Twilight answers. Her response leads to further quizzical looks from the young yak. Twilight smiles and addresses Yona directly, as the others listen in. “You predicted that you would not get me as your mentor. That prediction is either true or false. Now suppose it was false; that would mean you did get me as your mentor. Understand so far?” Yona nods. “But, I gave you the additional information that the student who got me made a correct prediction. And THAT is our contradiction.” Yona ponders Twilight’s reasoning. “So … that means Yona’s prediction must be true. So Yona not get Headmare Twilight.” Twilight nods in agreement, which in turn causes Yona to smile. “Yona getting less confused!” “But there’s more,” Twilight continues. “Since your prediction was true, then you must have either myself or Pinkie Pie. And since you correctly predicted that you would not get me …” Twilight pauses, waiting for Yona to reach the logical conclusion: “… then Yona must have Professor Pinkie Pie! YONA GET IT!!!” “WHEEEE … I get to mentor Yona!” Pinkie bounds over to Yona and gives her a big hug, causing Smolder to wince slightly. “Sorry … still not used to all the touchy-feely and such.” “What about when we were in the basement and …” Ocellus stops mid-sentence, upon seeing the dragon stare daggers at her. “Uh … never mind.” Sandbar speaks up: “So, we know that Yona will be mentored by Professor Pinkie Pie. But what about everyone else? Where do we go from here?” “We go to Gallus’s prediction,” Twilight states. Gallus and Smolder both react with surprised looks. “ME??? I didn’t say anything useful at all!” “Yeah, all he said was that I’d be wrong!” “But that makes all the difference,” Twilight states. “Remember, only two predictions were true, and we already know that one of those was Yona’s prediction. So there’s exactly one other true prediction among the remaining five. Now think about Gallus’s prediction for a bit.” The students go quiet as they ponder Twilight’s suggestion. Soon after, Ocellus speaks: “If Gallus’s prediction was true, then Smolder’s prediction would have to be false. And if Gallus’s prediction was false, then Smolder’s prediction would have to be NOT false … it would be true.” “Ooo, ooo,” Silverstream suddenly interjects. “But that means one of them HAD to have the other true prediction!” Sandbar follows: “And that means the rest of us all made false predictions.” Twilight clops her hooves together. “Excellent work, all of you! Yes … that means we know that all the predictions made by Sandbar, Ocellus, and Silverstream are false. Now think about what Ocellus predicted.” Ocellus recollects her prediction. “I said that Silverstream would not get Rarity … and that means Silverstream DID get Rarity.” “And that makes two pairs figured out,” Twilight states. “Can you go further?” Smolder continues the train of thought: “Silverstream’s prediction was false, so Sandbar didn’t get either Rainbow Dash or Fluttershy. And he didn’t get either Pinkie Pie or Rarity, since those two were already assigned. So that leaves either Twilight or Applejack.” “Wait,” Yona interrupts, “but seapony made false prediction, so Sandbar not get Headmare Twilight … Sandbar get Applejack!” she excitedly concludes. “Wow … it’s like unraveling a dress by pulling on a thread!” Silverstream states before quickly backtracking: “Oh … maybe I shouldn’t use that when talking to Professor Rarity.” “Three pairs down, three to go,” Twilight states. “Now back to Sandbar’s prediction.” Sandbar thinks for a minute. “Uh … I said that Ocellus would get Rainbow Dash. That means she didn’t … and since her prediction was false, she didn’t get Twilight either. So Ocellus got Fluttershy as her mentor.” “Very well done,” Twilight again states. “So that just leaves Smolder and Gallus, and myself and Rainbow Dash as mentors.” She turns to Gallus. “Can you figure out which pony you got?” “Uh … no?” Twilight laughs. “Come now, you’re an especially bright griffin. I know you can figure this out.” “I can’t,” he starts with an air of exhaustion. “I mean, I know that either my prediction was true or false. And I know that Smolder’s prediction was either true or false. But how can I know which is which?” “Think back to what Smolder’s prediction was.” Twilight suggests. “Well … she predicted that she would get either you or Rainbow Dash.” Moments later, a revelation comes to him. “But those are the only professors left … so she must have made a true prediction! And that means she got either Twilight or Pinkie Pie … and THAT means Smolder got paired with you!” “And that only leaves Rainbow Dash left to mentor you,” Smolder adds. “And that is the final piece of the solution,” Twilight concludes. “To sum up: I will mentor Smolder, Pinkie Pie will mentor Yona, Rainbow Dash will mentor Gallus, Applejack will mentor Sandbar, Rarity will mentor Silverstream, and Fluttershy will mentor Ocellus.” Brief silence gives way to a chorus of ‘ooo’s and ‘aaah’s from the class. Smolder sighs. “So all this logic and stuff just comes naturally to you?” “It’s something I’ve always been interested in, ever since I was a foal,” Twilight admits. “But we can discuss that in more detail during our one-on-one. And speaking of which,” she states as she beckons the class to rise from their seats, “it’s time for some counseling. Follow me, everycreature.”
Puzzle 2: Who got the 96?“Another ‘history of friendship’ quiz in the books,” Gallus exclaims with relief as he passes his paper up along with his classmates. “You’d think saving Equestria would be enough for graduation purposes.” “From what I’ve seen in my days in Ponyville, we’re not even in the Top Ten around here.” Sandbar’s retort brings about a fresh round of laughter. “It won’t take long to grade these,” Twilight asserts, “so you can just chat amongst yourselves as I get these marks in.” Her horn glows as a quill dips into in an inkwell and she peruses the first of the six papers. The students adjust their desks in a circle and begin pondering how they fared: “Yona think Yona did well. Yona think Yona earn score of 97.” “Why 97?” Sandbar asks. “97 prime, like yaks. Yaks renowned for ways in mathematics.” “I’ll have to take your word on that,” Gallus says, “but I feel like I did better than you did.” Yona’s eyes glare at the griffin. “It’s not a knock on you, Yona. I’m not usually confident on quizzes like this, but I think I did well. Besides, professor Pinkie Pie’s been pushing me to make more predictions.” “Well, I’m not as confident as some of you are,” Smolder states, “but I don’t think I got the lowest score of the six of us.” “Then who do you think did?” Ocellus asks. Smolder replies with a shrug and a trite ‘meh’. “So how did you do?” Sandbar asks Ocellus. The latter opens her mouth to reply, but Silverstream breaks in: “Oh, Ocellus always does well on these. I bet her score is one of the top three among the six of us.” The comment draws a slight blush from the changeling. “OK, that’s four predictions so far,” Sandbar recaps. “So, Yona thinks she got a 97, Gallus thinks he did better than Yona, Smolder thinks her score wasn’t the lowest, and Silverstream thinks Ocellus had one of the top three scores. So I guess that just leaves—” “FINISHED!” Twilight’s exclamation causes the students to suddenly turn to face their headmare. “That was quick,” Smolder asserts. “It’s easy to grade papers when they’re all of such high quality,” Twilight replies with a broad smile. “All of you did incredibly well!” “Really?” Gallus reflexively asks. “Yep. In fact, all of you got scores ranging from 94 to 99. But none of you got the same score.” “All right … A’s for everyone!” Sandbar exclaims. “So tell us,” Ocellus prods, “who got each score?” “Well it’s funny,” Twilight starts. “She sure has a strange notion of funny,” Gallus quietly tells the others, as Twilight continues. “I couldn’t help but overhear your predictions while I was grading. As it turns out, anyone who made a true prediction ended up with an even-numbered score, while anyone who made a false prediction ended up with an odd-numbered score.” Silverstream laughs, causing the others to turn to her. “Oh, she said it was funny, so I thought …” Ocellus turns to Twilight, anticipation in her voice. “So does that mean we can figure out who got what score?” “Oh, not at all. At least not with only that information.” Twilight ponders the situation further. “Although … this is actually quite interesting. IF I told you which of you got the score of 96, then you WOULD have enough information to figure out everyone’s score.” “Whoa … now Yona confused.” “So who did get the 96?” Smolder asks. “I don’t need to tell you that anymore,” Twilight replies. “But, I’ll forego homework assignments for a week if you can figure out who got what score.” “DEAL!” ... Everyone turns to Gallus, surprised by his renewed enthusiasm. “What’re you all staring at, let’s solve this!” Author's Note Admittedly, this puzzle turned out FAR harder than originally intended. Nevertheless, there IS a unique solution, as will be explained in the next 'solution' chapter.
Puzzle 2 solution“You know, Gallus, if you put as much effort into your studies as you do in ways to get out of doing your studies …” Gallus quickly brushes off Ocellus’s observation. “Yeah yeah yeah, whatever, where do we begin?” The changeling sighs, shakes her head, and chuckles to herself. “I’m happy with getting help on this one,” she finally admits. “Well, first we need to figure out who got the score of 96. All we know for certain is that it’s not Yona.” The yak blinks. “Why Yona not get 96?” “You predicted you would get a 97,” Ocellus answers. “Now if you did get the 97, your prediction would be false; that’s a contradiction. So your prediction was false, but that means you got an odd-numbered score. Since it can’t be 97, you must have gotten either the 95 or the 99.” “Wait … Yona might have gotten high score in class? Yona happy!” Sandbar chuckles. “May have gotten the highest score. We don’t know yet.” “Fifty-fifty better than one-in-six!” Yona declares. “Whoa,” Gallus responds with a tinge of surprise, “you really do know math.” His comment causes Yona to beam. “So anyway,” Ocellus says, returning to the task at hoof, “Yona got either 95 or 99; she could not have gotten 96. Now we need to figure out who did … and the key is, knowing who got 96 must lead to a unique solution.” “Sounds good so far,” Smolder states, her thoughts drifting towards the possibility of no homework. “Now I predicted that I would not get the lowest score.” The young dragon ponders her comment for a moment. “Sooo … if I did get the lowest score, that would be a 94 … but that would mean I made a false comment with an even-numbered score, right?” Ocellus nods with an ‘mmm-hmmm’. “Cool … I guess … so that means I didn’t get the lowest score, and I made a true prediction … so I had to get either a 96 or a 98, right?” “That’s right,” Ocellus states. “One more piece of the puzzle.” “Yeah, but I think we’re out of edge pieces,” Gallus adds. Silverstream starts searching the floor around them, causing the griffin to roll his eyes. “Not literally, Silverstream!” The hippogriff gets back to her seat. “Sorry, sorry,” she says with a meek chuckle. “We still have two predictions to examine,” Ocellus continues. “Now Gallus predicted that he would get a higher score than Yona. Now if that prediction was true, then Gallus would had to have scored an even-numbered score … but it couldn’t be 94, since that was the lowest score. So he either scored 96 or 98, and Yona would have scored 95.” Silverstream picks up where Ocellus leaves off: “But if Gallus had a wrong prediction, then he would had to have scored an odd-numbered score, and lower than Yona. But that means he couldn’t have a 99, so he would have either a 95 or 97, and Yona would have the 99.” “EXCELLENT WORK, SILVERSTREAM!” The students turn to Twilight, who blushes a bit for her outburst. “Sorry … I’ll be quiet from now on,” she states before going silent. “So what about that last prediction?” Sandbar asks. “We can’t get much from that, can we?” “Well, no,” Ocellus admits. “If it’s true, then Silverstream got an even score and I got either the 97, 98, or 99. And if it’s false, then she got an odd score and I got either the 94, 95, or 96. So yes, a lot of possibilities.” “So now what?” Smolder asks. “Now we guess … I guess,” Gallus states. “I’m open to starting with anycreature.” “I say we start with Smolder getting the 96,” Ocellus suggests. “If anyone else got the 96, we’ll know she got the 98.” Sandbar speaks up: “All right … if Smolder got the 96, then … it seems like there’s a lot of ways to assign scores.” “If we then assume my prediction was true,” Gallus says, “then I got the 98 and Yona got the 95.” He ponders a bit further. “Silverstream could then have the 94, and either Sandbar or Ocellus could have the 97 or the 99.” “And if griffin wrong,” Yona adds, “then Yona get 99 and griffin get lower score, like 95. And maybe seapony get 94 again, and Ocellus get 98 and Sandbar 97. That work too, right?” “I could have gotten 97 myself, right?” Silverstream adds. “But that’s the whole point, right?” Ocellus concludes. “Knowing that Smolder got the 96 leads to too many possibilities … so Smolder couldn’t have gotten the 96. She must have gotten the 98, and someone else got the 96.” Upon hearing this comment, Twilight’s horn glows as a quiz paper floats from the desk to Smolder’s seat. The dragon picks up the page. “ALRIGHT, a 98!” she proudly declares. “And now we know one of the six scores,” Sandbar adds. “Where do we go from here?” “I’ll give this a go,” Gallus says. “If I got the 96, then that means Yona got the 95. After that … there’s the 94, 97, and 99 left for Sandbar, Silverstream, and Ocellus. Silverstream could have the 94, and if she did, we still don’t know who got the 97 and who got the 99. So … I guess that means I didn’t get the 96.” “And since griffin not get 98 either,” Yona adds, “that mean griffin wrong and Yona get 99!” Another paper levitates from Twilight’s desk towards the group, landing at Yona’s seat. “YES!!!” she triumphantly declares. Gallus turns to the yak. “Yona, you know I meant nothing personal with my prediction, right?” “All good,” Yona replies with a wide smile. With four scores left to determine, Sandbar is next to speak up. “What if I got the 96? That would leave the 94, 95, and 97 for Gallus, Silverstream, and Ocellus. Then Silverstream could have the 97, and Gallus would have the 95 and Ocellus the 94.” “Sounds good to me!” Silverstream declares. “Wait a minute,” Ocellus counters. “Silverstream could have gotten the 94 instead, and I would have had the 97. So again, there’s no way to know for certain.” “So there’s only two possibilities left for the 96,” Smolder adds. “Right,” Ocellus says as she continues. “Now suppose I got the 96. Then that would mean Silverstream made a false prediction, so she would have either the 95 or the 97, just like Gallus. And Sandbar would have the 94 … only two possibilities, but that’s still one too many. So all that’s left is …” Everyone turns to Silverstream. Seconds pass. “… OH! … Oh, that means I got the 96!” The hippogriff thinks for a few more seconds. “So my prediction is true … so Ocellus got the 97 … and since Gallus didn’t get the 94, he got the 95 and Sandbar got the 94.” “Then that’s it, isn’t it?” Gallus asks. “From highest to lowest, that means Yona got the 99, Smolder the 98, Ocellus the 97, Silverstream the 96, me the 95, and Sandbar the 94.” Twilight nods as the remaining papers fly towards the students, each landing at their appropriate spot. Quickly, the group stomp their appendages in triumph. “Never been so happy to have the lowest score in class,” Sandbar admits. “Great job, everyone!” “Indeed, you’ve all exceeded all expectations,” Twilight adds. “I think after that little exercise, you’ve all earned some time off from homework.”
Puzzle 3: Pick a NumberOne-on-one conversations do not come naturally for Gallus. Perhaps this is due to his griffin heritage, or perhaps it’s more of an introvertive nature. Regardless, he often finds comfort in having discussions with his entire group of fellow nonpony students, rather than individual chatter. In light of his recent discovery of Yona’s mathematical prowess, however, he cannot help but initiate a discussion with the young yak, once back in the main foyer of his and his fellow students’ quarters. Thus, he and Yona have a talk, each seated near a table, while most of the others had already retired for the evening to their own rooms; Ocellus remains the exception, reading a tome to herself as the two chat. “So what Ocellus read?” “She mentioned something about a professor Stallyan,” Gallus replies, “and that’s a logic book written by him that Headmare Twilight owns. She said she read parts of it when the two of them took the balloon to parts relatively unknown for that friendship quest they were called for.” Yona turns towards Ocellus and utters a ‘hmm’ in response, the changeling remaining focused on the book. “So,” Gallus continues, “yaks are good at math, huh?” “Yaks best at everything,” Yona proudly responds, “including math. Many yaks renowned for math ability.” “Well I’m not one to brag,” he starts, phased slightly by Yona’s chuckle, “but griffins are also well-known for their abilities in the ways of mathematics.” “That really true?” “Well … it’s not like griffins are known for friendship.” Both he and Yona have a laugh at the comment. “Anyway, I have a simple challenge for you,” Gallus continues. “I’m thinking of a three-digit number, and I want to see how long it takes for you to guess the number.” “There nine hundred numbers like that,” Yona states. “Yona don’t want just guess numbers randomly.” “Let me finish,” Gallus calmly replies as he pulls a bag of gemstones towards them. “You can ask a series of yes-or-no questions to narrow down the possibilities. Smolder was nice enough to let me borrow her snack bag; to keep track of your response, I’m going to use red rubies and green emeralds to signify the answers to your questions. I want to see if you can figure out my number in 5 or fewer gems. So … I hope I haven’t confused you with all that.” “Yona not confused,” the yak insists. “You have number in mind?” “I do … so ask away.” Ocellus keeps an ear towards their conversation as she continues to read from her book. Yona, meanwhile, begins asking questions: “Is number perfect square? Like, number times itself?” “I’m well aware what a perfect square is,” Gallus states as he selects a gemstone from the bag and places it on the table. “OK … is number perfect cube? Griffin know what perfect cube is, right?” she devilishly asks. Gallus sighs and grins. “Yes, Yona, I know what a perfect cube is, too.” He selects another gem from the bag and places it on the table. Yona ponders for a minute. “Does number have same first and last digit?” “Huh … interesting question,” Gallus admits as he chooses a third gem from the bag and sets it alongside the others. “Oooo … Yona close now. Iiiiiiis … number greater than 200?” Gallus takes a fourth gem from the bag and adds it to the collection on the table. “YONA KNOW NUMBER!” “Whoa whoa whoa,” Gallus says with a quiet undertone. “Not so loud, you’ll disturb the others.” “Oh … sorry. Sorry, Ocellus,” Yona calls out across the room. “Not a problem,” Ocellus replies, keeping one ear open to their conversation as she continues perusing her text. Gallus studies the four gems he had placed on the table. “You realize you can’t know the number yet, right? There’s still two possibilities.” “That wrong,” Yona insists. “Griffin wrong. Proof that yaks better at math.” “You’re right about a lot of things, but not this,” Gallus counters … before a thought occurs to him. “Yona, you realize that green means yes, right?” “What???” “I’ve been using the green gems as ‘yes’ responses, and the red gems as ‘no’ responses.” “That make no sense. In yak culture, red always symbol for start of journey or task. Red always mean yes … so green must mean no.” Gallus slaps his head. “Well that explains everything … you took all my ‘no’ answers as ‘yes’ and my ‘yes’ answers as ‘no’. That’s why you think there’s only one solution.” “Well that not sporting of griffin!” Yona grouses. “Look, why don’t we just start over. And I’ll give you a starting hint: my number is greater than 600.” Ocellus promptly blurts out a number, causing the two to turn to her in surprise. ... “What that for?” Yona asks. “Oops … sorry, force of habit,” Ocellus sheepishly admits. “I guess I couldn’t help but listen to your conversation. I thought it an interesting challenge.” Gallus stammers for a response: “But … you didn’t even see any of the gems I placed … you didn’t hear any responses … how in Tartarus could you know the number I was thinking of???” “Well … you said it was greater than 600. That was all I needed to know.” Gallus shakes his head. “You’ll have to do better than that.” Author's Note This one may not be to everyone's taste, as some math knowledge is necessary. That said, there IS enough information to determine Gallus's number … as well as the number Yona thought was the correct answer.
Puzzle 3 solution“Seriously, Ocellus, do changelings have the power to read minds as well as transform into any creature?” “That would be cool,” Yona admits. Ocellus puts down her book and trots over to Gallus and Yona. “It’s nothing so out-of-the-ordinary. I just took the information I had and deduced what your responses must have been. “But you didn’t see any of my responses,” Gallus states incredulously. “I mean, with 4 questions, each one of which could be answered with ‘yes’ or ‘no’, that makes for …” “… 2 to the fourth, or 16 possibilities,” Yona finishes. “… OK, I won’t doubt your math anymore.” Gallus turns from the yak to the changeling. “So yeah, 16 possibilities. How could you know the answers I gave?” Ocellus proceeds with her explanation: “The key is the fact that Yona thought all your responses were the opposite of what you intended. That meant there had to be a series of responses so that both the actual AND negated responses led to only a hoofful of possibilities.” Yona and Gallus both look blankly at Ocellus, who chuckles slightly before continuing. “Take Yona’s first two questions: whether or not the number was a perfect square, and whether or not it was a perfect cube. Now if Yona saw what she believed to be two ‘yes’ answers, she would have immediately thought the number was 729 … that’s the only 3-digit number that’s both a perfect square and a perfect cube.” “Wait … how you so sure of that?” Yona asks. “Because any number that’s both a perfect square and a perfect cube must also be a perfect sixth power. And since 1 to the sixth is 1 and 2 to the sixth is 64, both of those are too small. But 3 to the sixth is 729, and anything bigger is just too big.” Gallus turns to Yona. “No offense, but Ocellus might have both of us beat.” The yak nods in solemn agreement. “OK,” Ocellus continues, “so Yona couldn’t have thought the first two answers were both ‘yes’. On the other hoof, she couldn’t have thought the first two answers were both ‘no’ … there would be far too many possibilities, no matter how the other questions were answered. For example: any of 101, 111, 131, 141, and so on all have identical first and last digits, and all are less than 200. Also, any of 202, 212, 222, 232, and so on all have identical first and last digits, and all of those are greater than 200. And if the first and last digits are not the same, that’s just so many more to pick from. And all that’s to say that Yona must have thought the number was either a perfect square or a perfect cube, but not both.” “Yona getting confused … but Ocellus right; Yona did see different colored gems for first two answers.” “That’s still a lot of possibilities, though,” Gallus counters. “I mean, there’s 21 perfect squares with exactly 3 digits.” “True,” Ocellus admits, “but only three of those have identical first and last digits.” “But how would you know that a ‘yes’ answer to that question went with the perfect square?” “Because if it went with the perfect cube, then either Yona would have ‘known’ the number was 343 … that’s the only perfect cube that satisfies that condition … or she never would have been able to figure the number out with only that last question. So instead, there’s only two possibilities at this point: either Yona thought your answers were ‘yes’-‘no’-‘yes’, or she thought your answers were ‘no’-‘yes’-‘no’. In either case, there’s only 3 possibilities … with the first case, the number could only be 121, 484, or 676; with the second case, the number could only be 125, 216, or 512.” “Incredible,” Yona states in disbelief, “Yona did pick one of those numbers. But … how you know which one?” “Well, in either of those cases only one possibility is less than 200 … so that must have been what you thought.” “But that’s still not enough for you, right?” Gallus asks. “I mean, for all you knew, Yona could have answered either 121 or 125.” “Indeed,” Ocellus replies, “but whichever possibility Yona thought, the other one must have an option greater than 600 … because you said as much.” “… I did, didn’t I?” “Yep … so now I know everything: Yona must have read your answers as ‘no’-‘yes’-‘no’-‘no’ and thought the number was a perfect cube less than 200 … specifically, she must have thought the number was 125.” “That WAS Yona’s guess!” she exclaims in amazement. “But instead,” the changeling continues, “Gallus actually responded with ‘yes’-‘no’-‘yes’-‘yes’, indicating a perfect square greater than 200 with identical first and last digits … either 484 or 676. And since you said the number was greater than 600, it must have been …” “… six hundred seventy-six,” Gallus slowly concludes, still stunned at Ocellus’s reasoning. The sound of a door opening causes all three to turn. Smolder enters the foyer and makes her way towards the table. “Sorry Gallus,” she says as she picks up her bag of gems. “Just had a craving for a late-night snack.” She notices three emeralds and a ruby on the table. “So what’ve you all been up to?” “Trust me,” Gallus replies, “wait until morning before asking that.” Author's Note Originally, I posted that Gallus's number was greater than 500. That would have led to an alternate solution: Yona could have been thinking the number was 121 (perfect square, NOT perfect cube, same end digits, less than 200), with Gallus actually thinking 512 (NOT perfect square, perfect cube, different end digits, greater than 200).
Puzzle 4: Squirrels and Chipmunks“Yer just in time, sugarcube … uh, Ah mean, Headmare Twilight.” Twilight chuckles at Applejack’s backpedaling. “There’s no need to be so formal with me.” “Well, in front of the students is all … Ah thought a bit more formality would be in order.” Applejack turns from Twilight to the class, who all shrug in indifference. “I’ve never been comfortable with anypony addressing me as ‘Princess’; as it is, I’m still adjusting to ‘Headmare’. Formality can be appropriate, but not among friends.” “Uh … we can still call you ‘Headmare’, right?” Twilight laughs at Ocellus’s question. “Whatever makes you most comfortable,” she replies before turning back towards Applejack. “So what do you have for today’s lesson plan?” “So, for today Ah thought it’d be interesting for the students to see first-hoof how confusin’ it can be when competin’ groups try to be uncooperative towards one another. The point, of course, is to show that honesty is always the best policy.” “I’m with you so far,” Twilight accedes. “So how does this work.” “Ah’ve already assigned each of the students here to one of two groups; one group is called the Squirrels, and the other group is called the Chipmunks.” Twilight ponders the names for a moment. “I think all those friendship quests you’ve been sent on with Fluttershy are starting to have an effect.” The two share a laugh as the class quietly observes. “Well, Ah didn’t want to just use ‘A’ and ‘B’ for this; she may have given me suggestions,” Applejack admits. “So anyway, Ah assigned three of ‘em to be Squirrels and three of ‘em to be Chipmunks. Then Ah laid down some simple rules for the students. They can each talk to only one other at a time, and Ah included a rule what causes confusion when ponies … well, any creature gets all underhoofed.” “And what rule would that be?” “Any statement made toward someone from the ‘other’ group has to be a lie, while any statement made towards someone from the same group has to be honest. So if, say, Gallus and Smolder are both Squirrels, they have to be honest when talking to each other … but if Gallus is a Squirrel and Smolder is a Chipmunk …” The other four students burst out in giggles, as Gallus and Smolder sigh in unison. “… uh, anyway, if Gallus and Smolder are in different groups, then any statement made by one to the other has to be a lie.” “I think I got it,” Twilight states. “So I assume everyone knows their group.” “Yes indeed.” Applejack turns from Twilight to the students. “How ‘bout a demonstration for the Headmare, y’all?” “What are we supposed to say?” Ocellus asks. “Ah dunno … maybe keep things simple, like makin’ comments to one another about yer affiliation.” Gallus sighs a second time, swallows his pride, and turns to Ocellus: “I am a Chipmunk.” He pauses for a moment. “Grampa Gruff can never know I spoke those words in that order, okay?” “Ooo, my turn!” Yona exclaims as she, too, addresses Ocellus: “Yona is Squirrel.” Smolder then beckons Yona to face her: “We are different types.” Sandbar goes next, getting Smolder to face him: “You are a Squirrel.” The comment draws rolled eyes from the young dragon. “Oh, I want to go now!” Silverstream insists as she flutters towards Gallus: “You are a Chipmunk,” she states before moving on towards Sandbar: “Either Yona is a Squirrel or Smolder is a Chipmunk.” “Whoa now, love the enthusiasm there Silverstream, but Ocellus hasn’t even had a chance to speak.” Applejack double-checks her list and thinks about the comments made. “Ah think some of you may have forgotten the rule Ah put in place. Two of those statements wouldn’t’ve been made.” “What do you mean?” Ocellus asks. “What Ah mean is that one of those statements is true when it should be false, and one of those statements is false when it should be true … though the other four statements hold up. It’s alright, Ah know it’s a bit confusin’ at first … that’s why Ah always believe in honesty towards all.” Applejack turns to Twilight, noticing the latter lost in thought. A notion quickly comes to her: “Oh shoot, Twilight … Ah never even told you which group each student was in.” “You don’t need to,” Twilight replies with a smile. … “If this is going where I think this is going,” Gallus says, “can I just take my seat and observe this time? I’m still worn out from the last few puzzles.” Author's Note So … which three students are 'Squirrels' and which three are 'Chipmunks'? Also: which two statements should not have been made?
Puzzle 4 solution“Now hold on, Twi … Ah didn’t tell you what student was in each group, you know that some of those six comments made are true and some false, you know that two of those comments are wrong from what they should be … and you still think you got this all figured out?” “Well … yeah.” Applejack takes in Twilight’s simple reply and sighs. “If y’all don’t mind, Ah think Ah’ll just do like Gallus over there and take a seat, and let your headmare do the talkin’ for the rest of the lesson.” She proceeds to her desk and takes a seat, as the others focus in on Twilight’s solution. “So where do you begin with this one, Headmare Twilight?” Silverstream asks. “Well to start, let’s think about Smolder’s comment.” Twilight’s suggestion causes heads to turn Smolder’s way. “What … I just told Yona that we were different types. I thought that was alright.” “But here’s the strange thing,” Twilight explains: “Under the rules Applejack gave, NO ONE could utter that phrase, to ANY creature.” She pauses for the gasps to subside, then gives further details: “Suppose one Squirrel said that to another Squirrel. Then the statement would be a lie, and any statement made between two in the same group is supposed to be true. The same would hold if a Chipmunk said the statement to another Chipmunk.” “But what if a Squirrel told a Chipmunk that?” Sandbar asks. “In that case, the statement would be true … but the rules state that only false statements can be made between two from different groups. So no matter the pair, that statement should never be said.” “Oh,” Smolder utters. “I guess I got a bit confused, like professor Applejack said.” “But there were two statements made in error,” Ocellus reminds the group. “I don’t see how the other statement could be determined … and without knowing that, I don’t know how you could continue.” “The key here,” Twilight continues, “is to focus on two statements … specifically, the two directed at you, Ocellus.” The changeling thinks back to the two comments made to her. “Gallus said that he was a Chipmunk, and Yona said that she was a Squirrel. It seems like either could be true OR false.” “By themselves, either comment could be made … but as you’re about to see, it’s not possible for BOTH comments to be directed towards the same creature under any circumstance.” “Really???” “Let’s start with Gallus’s comment. If true, then he is a Chipmunk, and so is the student he’s addressing. If false, then he’s actually a Squirrel and he’s lying; that means the student he’s talking to is again a Chipmunk. So either way, the recipient of the conversation is a Chipmunk; in this case, that would be you, Ocellus.” “Weird … a comment referring to himself ends up revealing who I am.” “Indeed … and by the same reasoning, Yona’s comment would then force you to be a Squirrel. And that’s where the contradiction lies; you can’t be both a Chipmunk AND a Squirrel.” “Wait,” Yona asks, “did Yona make mistake? Yona not mean to.” “Well either you or Gallus erred, but as of now there’s no way to know who erred. All that’s known is that one of those comments is wrong … and combined with Smolder’s comment, we now know that all the other comments are within the rules. Now we can turn our attention to those comments.” “So, my comment to Smolder was OK?” “It was, Sandbar … and your comment actually forced YOU to be a Squirrel.” Sandbar gapes. “Really? I said that Smolder was a Squirrel.” “Yes, you did. And if your statement was true, that would force you to be a Squirrel so you were in the same group. But if your statement was false, then Smolder would actually be a Chipmunk; since false statements are only made by creatures from different groups, you would again have to be a Squirrel.” “That is incredible!” Silverstream admits. “So if I say that someone else is a Squirrel, that means I’m a Squirrel … and if I say that I’m a Squirrel, that forces the other student to be a Squirrel.” “Exactly; it’s a fun property that exists here.” Twilight catches a slight derisive laugh from Gallus. “Well, I suppose it does depend on one’s idea of ‘fun’.” “So, all that means Sandbar has to be a Squirrel and Silverstream has to be a Chipmunk, right?” Ocellus asks. “It does … and since Silverstream and Sandbar are in different groups, that means her comment to him has to be a lie: it is NOT the case that either Yona is a Squirrel or Smolder is a Chipmunk, so the reality is, Yona is a Chipmunk and Smolder is a Squirrel. And that takes care of four of your affiliations.” “Remarkable,” Sandbar says in awe. “And you’re right about all four of us … but what about Gallus and Ocellus?” “Yeah, couldn’t either of them be the last Squirrel or Chipmunk?” Smolder asks. “There’s still one fact we haven’t used,” Twilight states. “Applejack didn’t just say that two comments were in error. She said that one statement was true that should be false, and one comment was false that should be true. So now go back to your comment.” Smolder recollects what she said. “I said that Yona and I were different types. And now you know that I was a Squirrel and Yona was a Chipmunk … so my statement was true, and should have been false.” “And therefore …” Twilight starts. “… the other wrong statement was false and should have been true,” Ocellus finishes. “And that means the two students involved with one of those statements had to be in the same group.” “But Gallus and Ocellus have to be in different groups,” Silverstream continues. “So it was the other statement that was in error.” Yona looks to the floor for a moment. “Yona not mean to make mistake.” “Well, how about atoning by finishing the solution?” Twilight offers, bringing a smile to the yak’s face. “OK … so, Yona’s comment to Ocellus must be false and we must be in same group … so Ocellus must be Chipmunk.” “And that leaves Gallus to be the last Squirrel,” Smolder adds. “And … that’s it, right?” “That is it,” Twilight concludes. “In summary: Gallus, Smolder, and Sandbar are the Squirrels while Ocellus, Yona, and Silverstream are the Chipmunks. Yona’s comment was false and should have been true, Smolder’s comment was true and should have been false, and the other comments all followed the rules: Gallus lied, Sandbar was truthful, Silverstream lied to Gallus, and she also lied to Sandbar.” … “No matter how often you do that, Twi, Ah’ll never understand how simple you make everything sound,” Applejack states. “Comes from a good chunk of my youth spent working on logic,” Twilight says with a tinge of remorse. “Probably too much … but I’m always happy to impart such knowledge on students such as yourselves.” She glances at the clock in the room. “Oh … class time is almost up! Sorry, Applejack; I didn’t mean to take up all your lecture time.” “Shoot, Twilight, Ah love watchin’ you do yer thing. And hey, less work for me, to be honest,” she adds with a sly smile.” “Less work for me too,” Gallus adds instinctively. His fellow students all turn to him and scowl. “What?” “It’s alright,” Twilight calmly says. “I’m sure Gallus will prove to be up to the next puzzle challenge … whenever that will be.”
Puzzle 5: Buckball GamesOf all the classes to miss … why’d it have to be Professor Dash’s? Gallus muses to himself as he works on a paper for another class while holed up in Headmare Twilight’s office. He stretches his left wing for a moment before wincing and pulling it back. “You probably just overextended your wing,” Twilight suggests as she pores over her latest curriculum updates. “You should be fine in the morning, as long as you don’t exert yourself too much this afternoon.” Gallus sighs. “It was supposed to be a friendly flying competition between me and Smolder … I guess I got a little carried away.” “I would have thought you and the others would know better after seeing how some of your professors can get,” Twilight lectures. “In any case, what’s done is done and now you have time to catch up on your studies.” “But Buckball Day,” Gallus pines. “The one day when classwork doesn’t involve any actual learning.” “Teamwork, good sportscreatureship, being there for your teammates during both wins and losses … seems to me that you can learn a lot from activities like that. And who better to teach?” Gallus ponders Twilight’s comment. “Well, thanks for trying to make it sound boring,” he replies before refocusing on his paper. Twilight chuckles at the comment before setting herself back to her tasks. … A knock on the door catches Twilight’s attention. “Come in!” she calls out. The door gently opens and Ocellus flutters into the office, followed by the rest of her classmates. “You ready to head back to the dorm?” she asks. “Just about,” Gallus replies as he puts his work away. “So how’d your buckball games go?” “Pretty well,” she responds. “We were all on five different teams. No names given, just ‘A’, ‘B’, ‘C’, ‘D’, and ‘E’. There wasn’t enough time for every team to play every other, but all teams played either two or three games, always against different opponents. And of course, there were no ties.” “Sounds like it was fun,” Gallus dejectedly points out. “So who won?” Instinctively, all four remaining students begin calling out different game results. “Whoa, whoa, whoa … I don’t have 20 ears, you know! One at a time.” The four students look to each other, before Silverstream continues: “A beat B, but B beat C. Then again, E beat B. Oh, and E beat D, too.” Sandbar speaks next: “A beat C and A also beat E. Oh, and B beat D.” Yona follows: “B beat D, C beat A, and D beat C.” Smolder finishes up: “Well … A beat E but D beat A. Also, B beat D.” Gallus looks from one student to the next. “Those was some strangely random results you all just gave me.” He turns towards Ocellus. “Don’t you have any results to add?” “No, but I do have this to say: Any result given by somecreature who was on one of the teams mentioned is true, but all the other results stated are false.” Gallus’s jaw drops. “So for example,” Ocellus continues, “If Yona was on team A, then C did beat A … but if she was on team B instead, then C didn’t beat A; C either lost to A or didn’t play A at all.” “I understand what you said,” Gallus declares, “but why would you bother giving me the results this way.” “Well …” Ocellus slowly admits, “… professor Rainbow Dash heard about how much fun Headmare Twilight had solving Applejack’s puzzle with the squirrels and chipmunks, so she wanted to give her a puzzle of her own.” “… You’re kidding.” “She may have asked for my assistance in creating it,” Ocellus continues. Gallus slaps his right wing to his head, then turns to Twilight. “Feel free to explain this one, headmare.” Twilight smiles. “… what?” “Remember what I said after the last puzzle? I think you’ll prove to be up for this challenge … and I would prefer you not being dismissed for the day until you’ve solved it.” Author's Note Can you help Gallus out and determine the results of all the games played?
Puzzle 5 solution Silverstream: A beat B, B beat C, E beat B, E beat D Sandbar: A beat C, A beat E, B beat D Yona: B beat D, C beat A, D beat C Smolder: A beat E, D beat A, B beat D Gallus examines the statements placed on the portable chalkboard. “I guess I shouldn’t be surprised that you have a chalkboard in your office,” he notes. “I like being prepared,” Twilight replies somewhat defensively. “For what, someone giving you a pop quiz?” Ocellus asks. “One never knows,” Twilight states, envisioning the day Princess Celestia makes an impromptu visit. OK, Gallus, you’re smart, you can figure this out. He stares at the predictions for a few minutes … “So … three of you all said that B beat D,” he starts, talking the problem out aloud to himself as the others listen in. “But you couldn’t all be on teams B or D … so that means B didn’t beat D, and it had to be Silverstream and Ocellus who were on teams B and D, in some order.” “Well started,” Twilight says. “OK … so Sandbar, Yona, and Smolder were on teams A, C, and E, in some order. And both Sandbar and Smolder said that A beat E … and at least one of them had to be on either A or E … so that statement has to be true, and both Sandbar and Smolder are on teams A and E. And that means—” “Yona on team C!!!” The group turn to Yona, who sheepishly smiles. “Sorry … Yona excited.” “Yeah,” Gallus says before continuing: “so anyway, knowing that Yona was on team C means that both of her other statements were true: C beat A and D beat C … and the first of those two contradicts Sandbar’s claim that A beat C, so Sandbar couldn’t have been on team A … he was on team E, and Smolder was on team A.” “Nicely done, dude,” Sandbar says with encouragement. His confidence growing, Gallus proceeds: “Now I know that Smolder was on team A, and that means that D beat A.” He pauses and ponders for several seconds. “Now what?” “Would you like a hint?” Ocellus asks. “No … I need to get there on my own,” Gallus insists. He ponders for several more seconds. “Let me recap,” he says aloud to himself: “A beat E, C beat A, D beat C, D beat A … wait a minute … so A played E, C, and D … but no team played every other team … so A didn’t play B at all.” “I think you’re going to do it!” Silverstream exclaims. “So that means Silverstream was wrong by saying that A beat B … Silverstream couldn’t be on team B, so she was on team D and Ocellus was on team B. That’s all the teams!” “Excellently done!” Twilight states. “Now can you finish the results?” “Silverstream was on team D, so E did beat D. And now … yes … I got it now! D played A, C, and E, so D couldn’t have played B either! But since every team played at least two games, B must have played C and E. And since both of Silverstream’s remaining statements must be false, C must have beaten B and B must have beaten E. And that’s all the results: A beat E, B beat E, C beat A, C beat B, D beat A, D beat C, and E beat D.” “You got them all!” Ocellus proclaims. “Way to go!!!” The student all crowd around Gallus to embrace him. He winces slightly at the pressure put on his left wing, causing the others to back away slightly. “It’s OK, everyone,” he reassures the group, “I’ll be all right. Just, maybe not so much crowding right now.” “Puzzle-solving may not be to everycreature’s taste,” Twilight concludes, “but still, you should be proud of your capabilities. Problem-solving skills in general are useful everywhere, to solve all sorts of problems, friendship-based and otherwise.” “Thanks, Headmare Twilight … but if it’s all the same, I hope that next time I’ll be playing rounds of buckball instead of solving a puzzle about them.”
Puzzle 6: Gemstone JumbleIn all their classes, the students receive ample knowledge of the various aspects of friendship. Regarding the personal interests of the professors, however, they got more out of some professors than others … “Professor Rarity,” Smolder speaks up during yet another lesson veering towards fashion. “Does generosity only apply towards dresses?” A tinge of dejection overcomes the unicorn, but she quickly shakes off the feeling. “I admit that my own ventures gravitate towards my profession of choice … but generosity is a vital aspect of friendship regardless of situation. Everything from offering the last slice of pie during a lunch to providing financial assistance to those who cannot afford the finer things in life.” “Is generosity always associated with physical things?” Ocellus asks. Rarity chuckles. “Generosity can go far beyond the mere tangible. One can be generous with their time, volunteering themselves for community service. One can be generous with their emotional state, being there for another when times can get particularly difficult. Why, one can even be generous with one’s rationality.” The sound of Rarity’s final word causes the class to observe their professor with confusion. “What do you mean by being generous with your rationality?” Gallus asks for the group. “Oh … perhaps I should be careful with my choice of words …” Her attempts to backpedal do little to answer Gallus’s question; she sighs and continues: “Obviously, with my line of work, I get all sorts of requests from all sorts of VIPs … Very Important Ponies, of course. And in some cases, those requests can push my sanity beyond tolerance. BUT, one must always work towards the greater good, and not put one’s mental concerns at the forefront.” “Yona confused. Can professor give example?” “Yes, well … I cannot divulge personal information to you, but I suppose I can give you at least a little insight as to what I often have to deal with. Several weeks ago, I got a request from a well-known entertainer.” “Sapphire Shores?” “The Countess?” “An act at Burning Dragon?” “No no no, I simply cannot go into details,” Rarity insists, “but I will say that this particular entertainer, for one particular performance, had five backup performers, and she wished to have each of them made up in a dress consisting primarily of a single type of gemstone. A shiver runs down Gallus’s spine, as he begins to sense the direction the lecture is taking them. “Now again, I cannot reveal details … so let us refer to the backups each by a single letter: A, B, C, D, and E. My client asked that each have a dress made up of one of four gems: either emerald, sapphire, ruby, or diamond. No dress should have more than one gem type, and at least three different gem types had to be used for the dresses, though not necessarily all four gem types.” “Sounds simple so far,” Silverstream states. “Would that it were so simple … but my client had a number of provisions that she simply MUST have; else, she would cancel the entire order.” Don’t ask, don’t ask, don’t ask … “What sort of provisions?” Sandbar’s query causes Gallus to slap his head with a wing and groan to himself. “Is everything all right?” Rarity asks with concern. “Huh? Oh … yeah, everything’s fine,” he replies with as much sincerity that he can muster. “OK then … well to answer your question, Sandbar, my client asked that the gem types used for the backups adhere to strict guidelines that, honestly, still make little sense to me.” “Like what,” Smolder asks. Rarity muses for a moment. “Yes … it’s still fresh enough in my mind … lesse …” “ ‘C must wear sapphire, unless D wears ruby. And D must wear emerald, unless B wears the same gem type as E. And E must wear either sapphire or emerald, with A wearing diamond if E wears sapphire and with A wearing ruby if E wears emerald. Oh, and C should wear ruby, unless B wears either sapphire or emerald, and if B does wear sapphire then C should wear diamond, and if B wears emerald instead then C should wear emerald as well.’ ” … “Now Yona really confused.” “Yeah, it sounds like your client contradicted herself, like, a dozen times,” Smolder adds. “Believe me, it was a lot to process. But the customer is always right, as they say. And after spending a lot of time parsing out her request, I was able to come up with an ensemble that satisfied all her demands.” “Really?” Silverstream asks in surprise. “How?” Rarity opens her mouth to speak, when Gallus suddenly interrupts: “Don’t bother professor; we’ve been through these a lot as of late.” Author's Note So … can you determine which gem type each of the 5 backups wore?
Puzzle 7: Silverstream of ConsciousnessQuizzes are never fun … but for whatever reason, quizzes given by Fluttershy never seem as daunting or fearsome as those for Headmare Twilight or the other professors … “I’m sure you’ve all done wonderfully well on this quiz,” the soft-spoken instructor states as she takes the last of the students’ papers. “We have some time left before class is over, so I’d like to introduce you to a most unusual piece of flora.” “It’s not dragonsneeze trees, right?” Smolder cautiously asks. “Oh, nothing like that … but I must ask that you not touch.” The students turn to each other with mixed feelings of curiosity and trepidation as Fluttershy walks over to a closet and carefully clutches a potted plant sporting blue foliage, gingerly bringing it to her desk. “OK, everycreature; come on up and look, but again, do not touch.” The group of six approach the front desk and crowd around the plant. “So what’s so unusual about this?” Gallus asks. “This is ‘poison joke’,” Fluttershy begins to explain. “Don’t you mean ‘poison oak’?” Sandbar interrupts. “No … ‘poison joke’. And it has strange properties, causing reactions that would make Discord proud.” The students exchange more strange glances towards each other. “Yona confused. Plant causes different reactions for different creatures?” “It does. One time, when all your professors once stumbled into a field of poison joke, we all came down with various afflictions. Headmare Twilight lost her magical ability, Pinkie Pie lost her ability to speak, and Rarity came down with an awful case of frizzy mane.” “And what happened to you?” Ocellus asks. Fluttershy chuckles. “My voice became very deep,” she explains, attempting to naturally recreate a baritone sound. “At first I got very depressed … but with the help of Zecora, I and everypony else got better, and we’ve since been able to laugh at the whole incident.” “Uh … professor? …” Fluttershy turns to address Silverstream, who by now had grown rather quiet. “Goodness, you look nervous. Is everything all right?” “Uh, yeah, everything is fine … so … how long does it take for symptoms to appear?” Fluttershy thinks about the question for a moment. “About a day, I guess? We all got a good night’s sleep before coming down with any ailments. Why do you ask?” “Well yesterday I may have taken a stroll around the area, and I may have come across a patch of this ‘poison joke’ … and I may have rolled around in it because I enjoy the feel of plants.” Instinctively, the other students back away quickly from both Silverstream and the desk. “Oh dear … wait here, everycreature, while I get Twilight,” Fluttershy requests before bolting from the room. The other five students examine the hippogriff in awkward silence. “How do you think this ‘poison joke’ affects non-ponies?” Smolder asks Ocellus. “I couldn’t even imagine,” the changeling replies. “How seapony feel?” Yona asks. “I feel fine, I guess. At least I don’t feel any differently but then I usually feel like this like the other day I was telling myself I-feel-great-but-I-could-always-feel-better-if-I-just—” Silverstream abruptly clasps a talon to her mouth, catching the others by surprise as her eyes dart this way and that. “Silverstream, what’s wrong?” Sandbar asks. Carefully, Silverstream removes her talon from her mouth. She opens her mouth: “I-don’t-understand-all-of-a-sudden-I-can’t-help-but-talk-and-go-on-and-on-and-on-but-I-want-to-be-quiet-but-I-just-have-this-impulse-to-keep-going—” She quickly reclasps her mouth with her talon. “Whoa … and I thought dragonsneeze tree were strange,” Smolder admits. “Don’t worry,” Ocellus reassures, “Twilight and the others will be here to cure you soon enough.” Moments later, Twilight and Fluttershy rush into the room. Twilight trots over to Silverstream, noting the talon over her mouth. “Silverstream, how are you feeling.” She uncovers her mouth: “I-feel-fine-except-I-can’t-stop-talking-which-is-weird-since-I-like-having-conversations-but-this-feels-like-too-much-so—” and immediately re-covers her mouth. Twilight nods and turns to Fluttershy. “Can you recreate Zecora’s cure?” Fluttershy nods. “Go ahead and start a bath; let us know when you’re ready.” Fluttershy nods again and heads off. Twilight turns back to Silverstream. “You’ll be fine, it’s just a strange sensation you’re feeling now. One thing we’ve learned is that ‘poison joke’ is easily cured.” Silverstream nods. Twilight looks towards the desk, seeing the quiz recently taken by the students. “Maybe we can take your mind off your illness for a bit. So, how do you think you did on the quiz you just took.” Silverstream again opens her mouth: “Gallus-was-higher-than-Sandbar-but-Sandbar-was-higher-than-Ocellus-while-Ocellus-was-higher-than-Yona-but-then-Yona-was-2-higher-than-Smolder-while-Smolder-was-3-higher-than-Sandbar-and-Gallus-was-4-higher-than-Yona-but-Ocellus-was-5-higher-than-Gallus—” Silverstream clasps her mouth once again, as Twilight and the others take in the long list of declarations. Minutes later, Fluttershy reenters the room. “I think everything is ready for Silverstream.” “That’s wonderful. Let’s go, everycreature …” The last of the daylight having left Equestria, the students settle down in the dorm, grateful that Silverstream is now rid of the bizarre disease. “It was weird,” she relates to the others. “It was like I felt compelled to keep talking, even ahead of breathing. I never really appreciated pauses in conversation before now.” “What important is seapony all better,” Yona declares. The others nod in agreement. A knock on the door causes the group to turn to the entrance. “Come in,” Gallus calls out; moments later, the door opens and Twilight enters the foyer. “I just wanted to check in on all of you, and especially Silverstream.” “I’m feeling better, thank you. And thanks to Fluttershy for that curing bath; it felt invigorating!” Twilight smiles. “You can thank her face-to-face when you see her tomorrow. Oh, and she finished grading your quizzes in case you’re interested …” She chuckles to herself. “What’s so funny?” Smolder asks. “Well, it’s just that … all those statements Silverstream made while infected with the poison joke.” “You mean the ones concerning our scores on Fluttershy’s quiz?” Ocellus adds. “Yeah, it’s just that … well, she made 7 statements regarding how you placed. And every comment where the students mentioned were either both above or both below Silverstream was true, while every comment where one student was above and one student was below Silverstream was false.” Twilight’s horn glows as a quill jots down the statements on a sheet of paper for reference. “Well, I have to go. We’ll see you all tomorrow, and again, I’m glad to see Silverstream isn’t the worse for wear.” With that, the alicorn departs. “Why headmare not tell us order?” Yona asks. “I think she did,” Sandbar replies. Gallus shakes his head. “Unbelievable … now the plants are giving us puzzles!” Author's Note In what order did the students place, 1st through 6th? (Assume no ties.)
Puzzle 8: Chaotic ReasoningThe last of their classes completed, the students head down the hallway leading to Starlight’s office. “A non-eventful day for a change,” Ocellus points out. “Just a quick meeting with the counselor, and the rest of the day is ours.” “It’s weird,” Silverstream says, “I got so used to dealing with solving puzzles in class, now it feels strange not to experience one in a class.” “Just as well,” Smolder replies. “My brain could use a cooling down after the last couple days.” “Yeah, it’s nice to just have a normal school day for a change,” Gallus adds as they reach Starlight’s office. He opens the door … as water pours from the office, quickly enveloping the hooves, claws, and talons of the students. “Why do I tempt fate like that?” he asks as he and the other peek into Starlight’s office. They are surprised to see a downpour taking place within the confines of the office, as Starlight stands atop her desk while barking orders towards a familiar former administrator: “FOR THE LAST TIME, DISCORD, STOP THE RAIN!!!” Discord snaps his fingers; immediately, the clouds that had been enveloping the ceiling disappear as the precipitation ceases. “I still don’t see why you’re so irate … you said you had to water the flowers, I thought I’d save you some time,” he replies. “And now that I’ve done you a favor, our friendship dictates that you reciprocate in kind.” Starlight shakes off as much moisture from her coat and mane as possible. “There’s no position available at the present time, Discord.” “Oh, be reasonable,” he insists, drawing a laugh from the unicorn. “And just what does that mean?” “Discord, you are one of the most unreasonable creatures I’ve ever met! And I’m not saying that to be mean; it’s just your nature.” “Why, I am aghast at your insinuation, Starlight. I can be plenty reasonable, much more than you, of course!” Starlight rolls her eyes at the notion. “Uh, Counselor Starlight,” Silverstream meekly asks, drawing the attention of the two in the office. “Is this a bad time, because we can come back later.” An idea pops in Discord’s head. “Actually, my little nonponies, this is a perfect time to demonstrate my superior powers of reasoning.” “Didn’t you say ‘reasonable’?” Sandbar asks. “Six of one … speaking of six, how about each of you six think of a number, and I’ll try to figure out what numbers you’ve thought up.” Smolder delivers a flat ‘No’ to Discord. “We’ve been doing all sorts of puzzles like that all week. We need a break!” “Fine,” Discord responds in a huff. “I guess I’ll create my own puzzle, one that Starlight won’t be able to figure out. And to relieve you of the burden of thinking up numbers, I’ll assign numbers to you. I’ll even direct you as to what to say.” “Nasty trickster can’t put words in Yona’s mouth,” the yak defiantly states. Discord snorts, then snaps his fingers. Moments later, Yona feels a queasiness from her stomach, causing her to groan. “What trickster do to … Yo … na … AAAUGH!” Before finishing her sentence, Yona spits out a scrolled-up parchment onto the floor. Similar groaning sounds cause her to turn to her fellow students, all of which have experienced the same sensation, and all of which now stand before similar parchments. “Blecch,” Silverstream moans. “How does Spike do this on a regular basis.” Cautiously, she picks up her scroll from the ground and unfurls it; the others do likewise. “What does this even mean?” Sandbar asks. “Just read them, all of you,” Discord requests. “And you may want to take notes, Starlight.” “And if we refuse?” Gallus asks. Discord groans. “You want motivation? Fine, here’s my proposal: if your counselor can solve my puzzle, I’ll rescind my request for a more permanent position here.” Starlight rolls her eyes. “Whatever … anything that will get you out of my office.” Her horn glows as a parchment and quill fly from a far, dry corner of the office. “Excellent,” Discord says. “Now there are rules to follow, so pay attention: I won’t be repeating myself. I have assigned each of these six a number from 1 to 60. And because I like things on the odd side, all their numbers are odd, as well as different. All you have to do is figure out their numbers.” Starlight turns to the students. “What do the scrolls say?” Each student, in turn, reads off his or her scroll: Smolder goes first: “No one has a number that is a multiple of 9. I have the largest number.” Gallus follows: “My number is a perfect cube. Yona’s number is a perfect square. Silverstream’s number contains a ‘3’.” Ocellus goes next: “Yona’s number contains a ‘4’. Smolder's number contains a ‘9’. No two of us have numbers within 7 of each other.” Sandbar continues: “My number contains a ‘1’. Silverstream’s number is 21. Gallus’s number is between 16 and 24.” Silverstream follows: “My number is 10 less than Gallus’s number. Sandbar’s number is 27.” Yona goes last: “Ocellus truthful. Sandbar lies. Both Gallus’s and Silverstream’s numbers prime.” “Oh, and one other thing: any statement made by a creature with a number greater than 30 is true, unless the number is prime, in which case those statements are false. In addition: any statement made by a creature with a number less than 30 is false, unless the number is prime, in which case those statements are true. Got it? Good luck … I suspect you’ll need luck more than reasoning!” Discord gives a final laugh before *POP*-ing out of the office. Author's Note Probably my most contrived storyline, but I really like the puzzle I created. I hope you, the reader, enjoy solving the puzzle as much as I enjoyed creating it.
Puzzle 8 solution Smolder: “No one has a number that is a multiple of 9. I have the largest number.” Gallus: “My number is a perfect cube. Yona’s number is a perfect square. Silverstream’s number contains a ‘3’.” Ocellus: “Yona’s number contains a ‘4’. Smolder's number contains a ‘9’. No two of us have numbers within 7 of each other.” Sandbar: “My number contains a ‘1’. Silverstream’s number is 21. Gallus’s number is between 16 and 24.” Silverstream “My number is 10 less than Gallus’s number. Sandbar’s number is 27.” Yona: “Ocellus truthful. Sandbar lies. Both Gallus’s and Silverstream’s numbers prime.” Over 30 TRUE … unless prime, in which case FALSE. Under 30 FALSE … unless prime, in which case TRUE. All numbers different, odd, and under 60. Starlight peruses the parchment for minutes. “Twilight taught me a lot of things,” she admits to the students, “but we never got around to logic puzzles. I don’t know where to begin.” “Leave everything to us,” Gallus reassures. Starlight turns to Gallus and blankly blinks. “Long story, we’ll tell you about it later,” he adds before turning to the other students. “Everyone focus on the statements ‘you’ said; divide-and-conquer seems like the best way to start … and I think I can lead us off.” The others refocus their thoughts to their individual statements as Gallus begins the thought process: “I said my number was a perfect cube. Now the only perfect cubes less than 60 are 1, 8, and 27. It couldn’t be 8, because all our numbers are odd. And if it were 1 or 27, my statement would have to be false.” “Wait,” Sandbar asks, “isn’t 1 a prime number?” “1 neither prime nor composite,” Yona explains. “1 special, like Yona,” she adds with a smile. “OK,” Gallus continues, “so the upshot is I could never truthfully state that my number is a perfect cube. That means all my statements are false, so Yona’s number isn’t a perfect square and Silverstream’s number doesn’t contain a ‘3’.” “Wow … I’m impressed,” Starlight states. “You say you’ve been doing this a lot?” “More than I’d like to admit,” he concedes. “Anyone got anything else?” “I … I think I do,” Sandbar says. “Based on what you just said.” “What do you got?” Ocellus asks. “Well … I said that Silverstream’s number was 21 and Gallus’s number was between 16 and 24. If both of those were true, then his number would have to be 17, 19, or 23 … but all of those are prime, which would mean Gallus’s statements would be true. So I must have lied, too. And my number doesn’t contain a ‘1’.” “Oh, oh,” Yona interrupts, “that mean Yona truthful when Yona say Sandbar lied. So Yona truthful, and that mean Ocellus truthful and both Gallus and Silverstream have prime numbers!” “And that means my statements are true, too,” Ocellus adds. “So Yona’s number contains a ‘4’, Smolder’s number contains a ‘9’, and all our numbers are 8 or more apart from each other.” “Remarkable,” Starlight says in awe. “So you already know the veracity of four of your sets of statements.” “We’re very well-versed … city,” Silverstream says. “But do we know any actual numbers yet?” Yona ponders Ocellus’s statement regarding her own number: “Ocellus truthful, so Yona have number with ‘4’ in it … but 4 can’t be last digit, because all numbers odd. So Yona have 40-something.” “But not a prime number,” Smolder adds, “or else you’d be lying. So … that means it’s not 41, 43, or 47.” “It’s not 49, either,” Gallus adds, “because that would mean Yona’s number is a perfect square, making one of my statements true. So that only leaves—” “YONA HAVE 45!!!” the young yak enthusiastically concludes. “Awesome,” Sandbar says. “One down, five to go. So … now what?” Smolder comments on her own statements: “45 is a multiple of 9 … so I guess I’m a liar, too.” “I’d say you’re in good company,” Gallus responds with a grin. Smolder remains unfazed. “What, you don’t get? … whatever. As you were saying?” “Yeah, well … since I’m a liar, I don’t have the largest number. Now, my number has a 9 in it, so the choices are 9, 19, 29, 39, 49, and 59. And it sure isn’t 59, since that’s the largest number possible.” “It’s also not 39 or 49,” Ocellus adds, “since you’d be telling the truth if you have those numbers.” “Same with 19 and 29,” Sandbar adds as well. “So I guess I got 9,” Smolder concludes. “There’s our second number. Now what?” The group studies the parchment for several minutes. “Now it gets difficult,” Gallus concludes. “Let’s go back to your number,” Ocellus suggests. “What could it be?” “A lot of things, right?” “Not really. I mean, we know it’s a prime number, and that mean it’s greater than 30. It can’t be within 7 of 45, so … it can only be 31, 37, 53, or 59.” “Ooo, now I know something!” Silverstream excitedly asserts. “If my statements were true, then my number would be 10 less than Gallus’s number. That would make my number either 21, 27, 43, or 49.” “But Silverstream have prime number,” Yona adds. “So number would be 43.” “And that would make Silverstream a liar,” Sandbar adds as well, “so Silverstream can’t be truthful, and her number isn’t 10 less than Gallus’s number.” “But her number is still prime,” Smolder follows, “so it’s greater than 30 just like Gallus’s. And, her number doesn’t contain a 3. So what does that leave?” Gallus runs through possibilities: “Greater than 30, doesn’t have a 3 in it, so greater than 40 … not within 7 of 45, so at least 53 … but that contains a 3, so …” “Oh, I have 59!” Silverstream finishes. “And that means my number isn’t 59 or 53,” Gallus adds, “so it has to be either 31 or 37.” “And everyone else’s number has to be less than 30,” Ocellus states. “Let’s go back to my number,” Sandbar suggests. “It has to be less than 30, so it’s can’t be prime. It’s not 1, it can’t be within 7 of 9, it can’t be 17 or 19, or 21 or 23. It has to be 25 or 27 … but It’s not 27 either, because of Silverstream’s statement. I have to have 25!” “And that means I can’t have 31,” Gallus notes, “so I have 37.” “So what’s left for my number? Can’t be within 7 of either 9 or 25, can’t be 1 … my number can only be 17.” “That all numbers,” Yona says with finality: “Smolder have 9, Ocellus 17, Sandbar 25, Gallus 37, Yona 45, and Silverstream 59.” … Silence pervades for several seconds. “Did we … just solve that puzzle?” Silverstream asks. “YOU MOST CERTAINLY DID, YOU MEDDLING KIDS!!!” The students all turn to the entrance and eye Discord nonchalantly standing there, eyeing the group right back. “Technically I win the challenge, since your counselor did nothing more than watch and maybe learn. Still … the way you six demonstrated finding reasoning amidst the chaos I put into that puzzle speaks volumes:” He snaps his fingers, and a number of tomes fly off the bookshelves, ‘flapping their covers’ as they speak accolades about the students: “Well done, chaps.” “Jolly good show.” “Much better than some unicorns!” Starlight’s eyes glare as Discord snaps his fingers a second time. The books immediately return to their places along the bookshelves. “When indoors, it’s always proper to have volume control,” he points out. “In any event, I’m late for work, so I’ll let you be … for now.” “WHAT???” Starlight shouts. “Since when do you have a job?” “Since there seems to be little in the way of educational job openings, I thought I’d offer my services elsewhere to pass the time.” “So … exactly what do you do?” Ocellus asks. “Air-traffic control. I’ll be monitoring the hot-air balloons for the foreseeable future. Anyway, well done to the six of you … and considering yourself lucky,” he adds to Starlight before disappearing with a final *POP*. The puzzle now behind them, the students take in one last look around the water-logged office. “You want us to get a mop or something?” Silverstream asks. “You’ve done so much for me already,” Starlight insists. “I am thoroughly impressed what you were able to accomplish this afternoon.” “Because we solve puzzle?” Yona asks. “Because you got rid of Discord,” Starlight corrects. “Hopefully, now we can return to a sense of normalcy around here.” “Maybe,” Smolder says, “but if I’m taking a break near the fountain outside, I’ll still watch for falling balloons.” Author's Note Bonus points if you figured out my 'Shout-Out' at the end.
Puzzle 9 solution“I thought you weren’t interested in puzzles,” Smolder says to Gallus. “I wasn’t … but we’ve been getting so many of these lately, I can’t help but grow at least some interest, right?” “So the fact that I can’t solve this is enough to let you solve this?” Ocellus asks. “It’s not just that you can’t solve the whole puzzle … it’s that you can’t figure out anyone else’s number. That’s what gives me your number; I already know both Smolder’s and Sandbar’s.” “Really?” Sandbar asks. “How?” “Smolder said her number was three less than Sandbar’s. That forces their numbers to be either 1 and 4, 2 and 5, or 3 and 6 … but they can’t be 2 and 5, because Yona already said her number was 2. So that leaves either 1 and 4 or 3 and 6.” “OK, so how you know Ocellus number?” Yona asks. “If Ocellus had either 1 or 4, she would know that Smolder and Sandbar had 3 and 6. And if Ocellus had either 3 or 6, she would know that Smolder and Sandbar had 1 and 4. In any of those cases, she would know more than just her own number and Yona’s. The only way she could be completely stuck is if she has 5.” Ocellus shows her number to the others. “You’re right … I do have 5 … and I don’t feel so stupid, now.” “ ‘Stupid’ is the last word I would ever use to describe you,” Gallus flatly states. “Yeah, you’re brilliant,” Silverstream adds. Ocellus smiles at the comments. Yona speaks up: “So … since griffin say he have higher number, he have 6. And that mean dragon have 1 and pony have 4.” “And that leaves 3 for Silverstream,” Ocellus adds. “And that’s all there is to it,” Gallus concludes: “Smolder has 1, Yona has 2, Silverstream has 3, Sandbar has 4, Ocellus has 5, and I have 6 … and considering my baking prowess, it’s just as well I go last.” The group shares one final chuckle before heading to their next class.
Puzzle 10: QuibbleFic Part I (Bears and Wolves)Closing her eyes, Silverstream leans back and falls into the waiting hooves of Sandbar. “Wow, so that’s what a trust fall is, huh? That was incredible!” “I’m still uneasy about the whole thing,” Smolder states despite Gallus successfully catching her. “It’s all part of loyalty, I guess,” Ocellus concedes, Yona having caught her moments earlier. “So what now?” she asks Rainbow Dash, noting her professor perusing a collection of hoofwritten notes. Rainbow looks up with a start. “Wha? … oh yeah, go ahead and change places and do it again.” Ocellus turns towards Yona, then back to her professor. “Um … I don’t wish to cause trouble, but …” “What are you looking at?” Gallus interrupts. “You haven’t paid any attention to us since we got here.” Rainbow groans and shoves the papers aside. “I’m sorry, everyone. I’m just trying to do somepony a favor, but it’s proving harder than I thought.” “Are you trying to do someone’s taxes?” Silverstream asks. “I’ve heard those are super-challenging.” “No, it’s nothing like that,” Rainbow says before sighing. “I’m reading fanfiction for a friend.” “What fan-fiction?” Yona asks. “Lemme back up … how many of you are familiar with Daring Do?” Sandbar nods as the others shrug in ignorance. “Well,” she starts, careful not to reveal her favorite author’s true identity, “Daring Do is this pegasus adventurer who goes on quests for rare artifacts and treasures. Her stories are really incredibly awesome,” she adds with a touch of over-enthusiasm. “Sounds interesting … for a pony, anyway,” Smolder concedes. Rainbow continues: “Now many of us … I mean, many other fans are so impressed with these stories that they create their own stories related to the original.” Gallus scoffs. “Really? Ponies can’t make enough of their own lives, so they have to live the life of made-up characters?” Smolder and Yona chuckle at the comment, as Rainbow bites her tongue. “In any case, I have this friend that I met at a Daring Do convention, named Quibble Pants. The other day, I got a message from him: he’s been writing his own Daring Do stories, and he wants my opinion.” “Let me guess,” Gallus says, “his writing is terrible and you don’t know how to break it to him.” “That’s an awful thing to say,” Ocellus chides, before pondering for a moment and turning towards Rainbow. “That isn’t what the problem is, is it?” “His writing is OK, I guess … but he and I have differing thoughts as to what makes for the best stories. I love the action-adventure aspects, but he’s all about the puzzle-solving that was more … uh … there was more of it in the earlier stories.” “You mean it was more prevalent?” Ocellus asks. “Yeah, that … saaay …” Rainbow continues, an idea popping into her head. “Say what?” Yona asks. “I’ve been hearing a lot from the other professors about how well you six solve puzzles … like, thinking puzzles and whatnot … maybe you can help me out with these.” “Seems like our reputation is spreading,” Smolder states. “So this ‘Quibble’ guy, he made puzzles?” “Yeah, kinda like a lot of the earlier stories, but way harder,” Rainbow states. “Like, he’s got a whole chapter where Daring Do is imprisoned by some new villain that he dubs a logitaur.” “A what?” Sandbar asks. “Basically, a minotaur who likes logic puzzles the same way Quibble does. So without going into the whole plot, the logitaur has Daring Do trapped in a room with 3 doors; she has to choose the correct door to make her way to freedom.” “What if she chooses the wrong door?” Silverstream asks. “Then she gets mauled by either a bear or a wolf.” The others exchange funny looks with each other. “I skipped some stuff,” Rainbow admits. “Basically, the logitaur keeps a number of bears and wolves, whose job is to prevent trespassers from escaping.” “Wonderful,” Gallus deadpans. “So what does any of this have to do with puzzles?” “Here, have a look,” she says as she passes a sheet of paper across her desk. The students gather round to inspect the puzzle: Door 1: “This room contains a bear.” Door 2: “This room contains a wolf.” Door 3: “This room leads to freedom.” “According to the story,” Rainbow explains, “the logitaur placed a bear in one room and a wolf in another room; the third room leads towards the final exit. If Daring Do chooses the wrong door, really bad things will happen to her. Each door has writing on it to hint at the correct way forward.” “So why not just take Door 3?” Sandbar asks. “Because the logitaur put in rules regarding the messages: Any room containing a bear has a true statement, and any room containing a wolf has a false statement. The room leading onward could have either a true or a false statement on it.” The group ponders the rules set out for a minute. “Seems like a convoluted way to get the reader to solve a puzzle,” Smolder states. “But still, an interesting puzzle,” Ocellus adds. “Fairly easy though, this one.” “There’s more,” Rainbow states. “After Daring Do solves the first puzzle, she moves on to another room that contains 5 doors. Of those, two contain bears and two contain wolves, with one room leading onward. Here’s what the doors to those rooms say,” she continues as she pushes a second page towards the group: Door 1: “Rooms 2 and 3 both have bears.” Door 2: “Rooms 4 and 5 both have wolves.” Door 3: “Rooms 2 and 4 contain animals of different types.” Door 4: “Rooms 2 and 5 contain animals of the same type.” Door 5: “This room leads to freedom.” “That one look harder,” Yona states. “And Quibble created a third puzzle,” Rainbow states. “After Daring Do solves the second puzzle, she moves on and eventually finds herself in a third room, this time containing 7 doors. Three bears, three wolves, and one room that leads onward.” She pushes a third page across her desk: Door 1: “No two adjacent rooms contain animals of the same type.” Door 2: “An even-numbered room has the exit.” Door 3: “The room with the exit is adjacent to a room with a wolf.” Door 4: “Either this room has a wolf or Room 5 has a bear.” Door 5: “Room 2 has a wolf and Room 6 has a bear.” Door 6: “Room 3 has a wolf.” Door 7: “This room leads to freedom.” “And after that,” Rainbow starts. “Don’t tell me this guy put in a fourth puzzle!” Gallus says with some exasperation. “Now you know how I feel … but no, he has something completely different after this. But still, he expects me to solve all three of these puzzles. Can you believe that?” Yona speaks: “So when professor say class can help out, professor mean …” “Can you solve these?” Author's Note Moving on for the moment from E. R. Emmet to Raymond Smullyan for the next couple puzzles. Those who have read this far and have not already done so may be interested in another of my Fics: The Chalice of Rhinyorgrun. That Fic contains similar puzzles to what I have here, but uses colors rather than words to signify instructions. If you do decide to peruse, let me know what you think.
Puzzle 10 solutionThe group take another look at the three pages containing Quibble’s door puzzles. “We can probably solve these,” Gallus states, “but what does solving puzzles have to do with loyalty?” “Oh, well … it has everything to do with loyalty,” Rainbow explains. “You see … one key aspect of loyalty is being there for your friends when they need you the most. And, uh … I need you, right now.” “So … you’re not just our professor, you’re our friend?” Smolder asks. “We’re all friends,” Rainbow earnestly asserts. “So … all we do now is puzzles?” Yona asks. “Sure looks like it,” Sandbar replies. “May as well get to it.” The group focuses on the first puzzle: Door 1: “This room contains a bear.” Door 2: “This room contains a wolf.” Door 3: “This room leads to freedom.” “This one’s pretty easy,” Ocellus states. “The exit must be Door 2.” “Why is that?” Rainbow asks. “If Room 2 had a bear, the statement would be true, meaning the room had a wolf, which is doesn’t. And if Room 2 had a wolf, the statement would be false, meaning the room didn’t have a wolf, but it does.” Rainbow’s head begins to spin. “Could you go over that one more time?” “If statement true, room have wolf, but wolf room lies, so statement false, which mean no bear either, so room lead to freedom,” Yona sums up. “Oh … I think I get it. But what about the other rooms?” Silverstream takes a shot: “Well, Door 3 must be false now, so that’s where the wolf is. And that leaves Door 1 to lead to the bear. Easy-peasy.” “Huh … not bad,” Rainbow admits. “What about the next puzzle?” The students shift their focus to the second puzzle: Door 1: “Rooms 2 and 3 both have bears.” Door 2: “Rooms 4 and 5 both have wolves.” Door 3: “Rooms 2 and 4 contain animals of different types.” Door 4: “Rooms 2 and 5 contain animals of the same type.” Door 5: “This room has the exit.” “So now what?” Rainbow asks. “To start,” Smolder says, “Room 5 can’t have a bear. If it did, the statement would be true and it would lead to the exit, not contain a bear.” “Meanwhile,” Ocellus continues, “Room 1 can’t have a bear, either. If it did, then Doors 2 and 3 would both lead to bears as well, and there’d be at least three bears … but we know there’s only two bears to go with two wolves and one door to freedom.” “So that means Door 1 is false,” Gallus says, “and either Room 2 or Room 3 doesn’t have a bear. And that means Room 4 has a bear.” “Uh, Gallus?” Ocellus says. Gallus pays no heed and continues: “And since Room 4 has a bear, Rooms 2 and 5 have animals of different types … but Room 5 doesn’t have a bear, so it has a wolf and Room 2 has a bear.” “Gallus?” “And since Room 2 has a bear, Rooms 4 and 5 both have wolves. So Room 3 has the other bear, and Door 1 leads to freedom. Nothing to it,” he concludes. “GALLUS!” The griffon turns to Ocellus. “What?” “Door 1 CAN’T be false! You said Room 4 had a bear, but then you said Room 4 had a wolf. It can’t be both!” Gallus blinks … and goes over his reasoning a second time. “Hmmm … if Door 1 was false, then Door 4 would have a bear, so Rooms 2 and 5 would different animals, but Room 5 doesn’t have a bear, so Room 2 has a bear, and then Rooms 4 and 5 have … so Room 4 would have a wolf … so there’s no solution!” “No,” Ocellus insists. “There IS a solution.” “How? Now I know Door 1 can’t be false so it has to be true, but there can’t be a bear in Room 1.” “Right … so therefore …” The realization dawns on the griffon. “Ohhh, Door 1 leads to the exit!” He continues to ponder the possibility: “Lesse … Door 1 is true means Rooms 2 and 3 have bears, leaving Rooms 4 and 5 to have wolves. Do all the statements work? … Huh, I guess they do.” “Whoa … harder than it looked,” Smolder observes. “What about the last puzzle?” The six ponder the final puzzle: Door 1: “No two adjacent rooms contain animals of the same type.” Door 2: “An even-numbered room has the exit.” Door 3: “The room with the exit is adjacent to a room with a wolf.” Door 4: “Either this room has a wolf or Room 5 has a bear.” Door 5: “Room 2 has a wolf and Room 6 has a bear.” Door 6: “Room 3 has a wolf.” Door 7: “This room has the exit.” “Anyone wanna start?” Gallus asks the group. “Room 7 not contain bear,” Yona points out. “Just like last couple puzzles.” “OK,” Gallus continues, “What else?” … A minute of silence passes. … “What about Room 4?” Sandbar finally says. “What about it?” Smolder asks. “It can’t have a wolf, right? Because if it did, the statement on the door would be true.” “Good job!” Ocellus states. “I must be slipping.” “Or we’re getting better,” Silverstream adds. The others, including Ocellus, chuckle at the comment. “All right,” Smolder says, “so Room 4 can’t have a wolf. Can it be the exit?” “Going right for the finish, huh?” Sandbar says. “But then the statement could be true or false.” “So let’s start with ‘true’,” Ocellus suggests. “that would force Room 5 to have a bear, and that would mean Room 2 has a wolf and Room 6 has a bear.” “But Door 2 true,” Yona states. “So Room 2 not have wolf.” “That’s right,” Sandbar concurs. “So if Room 4 was the exit, the statement would have to be false.” “Which means Room 5 doesn’t have a bear,” Ocellus adds. “And since it would have to have an animal, it would have a wolf.” “Room 7 also have wolf,” Yona adds, “since it not exit.” “And based on the statement on Door 6,” Gallus continues, “either that room or Room 3 has the final wolf.” “Meaning Rooms 1 and 2 both have bears,” Smolder states, before pausing. “… but Rooms 1 and 2 are adjacent, so Door 1 would be false, and that can’t be.” “And all of that goes to show that Door 4 doesn’t lead to the exit!” Ocellus concludes. Rainbow eyes the students quizzically. “Wait … how long have you been doing these?” Yona ponders her professor’s question. “Maybe one week?” Rainbow blinks, then shakes her head. “That’s … that’s ...” “So after all that,” Silverstream states, “what does that mean?” “It means,” Gallus replies, “that Room 4 has a bear. Which means the statement on that door is true, and that means Room 5 has a bear as well.” “And that mean Room 6 have bear and Room 2 have wolf,” Yona adds. “No more bears left.” “But we have two wolves, too: Room 2 and 3,” Sandbar says. “All that’s left are Rooms 1 and 7.” “What haven’t we used,” Smolder asks aloud. Several seconds pass. “What about Door 3?” Silverstream finally asks. Ocellus responds: “Door 3 has to be false … which means the room with the exit isn’t adjacent to a room with a wolf. So the exit can’t be Room 1, since Room 2 has a wolf. But it could be Room 7, since Room 6 has a bear.” “And that means it has to be Room 7,” Gallus concludes. “That’s funny,” Silverstream adds: “The room that says it’s the exit really is the exit this time!” “Probably intentional, considering the previous puzzles,” Ocellus suggests. “ ‘Fool me once’, and all that?” “So that’s that,” Smolder states. “Hope that helps, professor … uh, professor?” The group all turn to face Rainbow, noting the spiral formation in her eyes. “Is professor pony OK?” Yona asks. “… why yes, I would love some eggs,” Rainbow replies in a half-daze. Sandbar turns his attention to the clock on the wall. “Whoa, class is almost done for the day.” “Looks like whatever that other puzzle she wanted us to look at is gonna have to wait,” Ocellus states. “You guys wanna hang by the fountain for a bit?” Gallus suggests. The group collectively nod in approval, and quietly file out, leaving Rainbow Dash to stare blankly at the back wall. “… no more eggs, I’m good …” Author's Note Gallus's reasoning in the 2nd puzzle was my own. I knew a bear couldn't be in Room 1, but I forgot that the statement could still be true. Fortunately, the location of the exit was the same either way ... … and then it was pointed out that my original solution was wrong! I rewrote the section containing the error. I originally planned on doing the other 'QuibbleFic' puzzle immediately after, but I'm giving myself time to properly develop that one.
Puzzle 11: Random Inference“Just tell me, already!” “Maybe Gallus tell Yona first!” “You first!” “No, you first!” Unable to focus on her reading, Ocellus groans and turns her attention towards her two vitriolic classmates. “Why are either of you making a big deal about this?” “It’s a matter of personal pride,” Gallus explains. “Look, I can accept that Yona scored higher than me on whatever you call that ‘random quiz’ thing Professor Pinkie Pie gave us, but I’d still like to know by how much.” “Then just tell each other what you got,” Ocellus suggests with a tinge of exasperation. “Private information is private,” Yona states, “and should be earned rather than given away.” Ocellus shakes her head and returns to perusing her book, keeping an ear towards the conversation. “How about this,” Gallus suggests: “we each ask a single question about the other’s score, then try to deduce what that score is. That sound fair to you? Up for the challenge?” he concludes with a grin. “Fine,” Yona says. “And Yona ask first.” “Go right ahead.” Yona ponders for a moment before asking: “Is Gallus score multiple of six?” Ocellus perceives a slight pause, assuming the griffon either nodded or shook his head. “My turn,” Gallus counters: “Is your score a triangular number? Oh … do I need to explain what that is?” “Yaks know what triangle numbers are!” Yona flatly states. “OK then … well, is your number triangular?” Ocellus perceives another pause. A smile comes across both Gallus’s and Yona's faces. They announce simultaneously: “I know your number.” / “Yona know Gallus number!” “Well here,” Gallus says as he pushes a piece of scrap paper towards the yak. “You write down your guess, and I’ll write down my guess.” “Yona not guess,” she replies in kind. She and Gallus write down their guesses and simultaneously show each other. Moments later, Ocellus catches a pair of cries of “HEY!!!” She puts down her book and turns to the others. “What now?” “YOU LIED TO ME!” “GALLUS LIE TO YONA!” “You two really deserve each other,” Ocellus states. Gallus and Yona look at each for several seconds, before bursting into laughter. “This really did spiral out of control, didn’t it?” Gallus admits. “Professor Applejack not approve,” Yona states. “All right,” Gallus adds. “I’ll just tell you first, and then you tell me.” “That quiz was out of 20 points, right?” Ocellus asks. Yona responds with a simple ‘uh huh’. “Then it’s settled,” Ocellus says … before stating both their scores. Gallus stumbles for a response: “How … how could you possibly know our scores???” “Yeah, and how you know Yona score?” “Well to be honest, I only knew for certain that one of those scores was correct … but apparently, your scores differ by more than one, and that’s all I need to know for certain.” Author's Note So, what score did Gallus and Yona each receive? Also, what score did each think the other got? Apologies for the delay; I was a bit under the weather, and was also in pure 'Netflix binge' mode. Also, I admit this puzzle is fairly simple … but I've struggled in the past with solving puzzles that require inference, and wanted to make my first such creation simple just to be sure I didn't err. My next puzzle should have a bit more meat on it.
Puzzle 11 solutionAuthor's Note The original version of the puzzle contained incorrect information, as mentioned in the 'Comments' section for that chapter. Those who were stuck the first time may wish to re-read that chapter before continuing with the solution below. Puzzle 11 solution “If those were just guesses,” Gallus states, “I need to introduce you to the Griffonstone Lottery.” “Or maybe trip to Las Pegasus,” Yona adds. “They weren’t pure guesses,” Ocellus insists. “I knew that at least one of those numbers had to be; the luck was that your scores differed by more than one, which forced you to have those scores.” “But how could you know?” Gallus reiterates. “Your head was buried in that tome the whole time; you never saw us nod or shake our heads.” “Yeah, so how you tell what we answer?” Yona follows. “It’s all about inference,” Ocellus says as she begins her explanation. “The key is that each of you thought you knew the other’s score. So, start with Yona’s question to Gallus. Yona, you asked whether or not Gallus’s score was a multiple of six.” Yona nods. “Well suppose Gallus said ‘no’.” “Gallus not say anything,” Yona corrects. “Gallus just nod head.” Ocellus sighs. “You know what I … never mind. Suppose Gallus shook his head to mean ‘no’. Then how could you possibly know his score? I mean, your score could be 8, or 10, or 13, and you wouldn’t be able to figure much out about Gallus’s score. From your view, he could have a score of 1 or 2 or 3, right?” Yona ponders Ocellus’s statement. “Yona guess that make sense.” “Now it could be possible for you to know Gallus’s score … but only if your own score was 2, in which case you would know his score was 1. Anything higher, and you couldn’t know if his score was 1, 2, or possibly something else. And if your score was 2, Gallus would have a score of 1 and he would NEVER know your score after his one question.” “OK, OK, you got me,” Gallus admits. “Yes, I nodded to Yona’s question: I did claim that my score was a multiple of six.” Ocellus smiles. “All right, so now I figured out what Yona’s score could be.” Yona interrupts: “How? Yona not say whether number triangular or not.” “True, but again, you claimed to know Gallus’s score. So suppose your score was 13 or higher. How could you know whether Gallus got a score of either 6 or 12?” Yona struggles to respond. “Exactly … you wouldn’t know. And that means your score has to be between 7 and 12, and you thought Gallus’s score was 6.” Yona stares at Ocellus for a moment, then flips over a scrap of paper near her, revealing a guess of ‘6’ for Gallus’s score. “Yona impressed.” “Me too,” Gallus concedes. “But you narrowed it down further, didn’t you.” “Yes, because now I can focus on your question to Yona. You asked if her number was triangular; if she shook her head to say ‘no’, then you would have no way of knowing whether her score was 20 or 19, or possibly other values, unless your own score was 19 … and In that case, Yona would have a score of 20, which we now know cannot be.” “Yeah, Yona did nod her head, just like I did with her question to me.” “So now you’re thinking ‘Yona’s score is a triangular number.’ … what could that number be? The triangular numbers less than or equal to twenty are 1, 3, 6, 10, and 15.” “You know triangle numbers?” Yona asks in surprise. “I do … I know a lot about numbers,” Ocellus admits with a tinge of embarrassment. “I suppose I really shouldn’t be surprised,” Gallus states. “OK,” Ocellus continues, “so how can you think to know her score? If your own score was 9 or less, you couldn’t know whether her score was either 10 or 15 … so your score has to be between 10 and 14, and you must have thought Yona’s score was 15.” Gallus picks up the scrap containing his guess and shows it to Ocellus, confirming her belief. “Right again, you little detective.” Ocellus smiles at the comment. “So to sum up: Yona must have a score between 7 and 12, thinking Gallus scored 6, and Gallus must have a score between 10 and 14, thinking Yona scored 15. But Gallus’s score is less than Yona’s, so the only possible combinations are 10 and 11, 10 and 12, or 11 and 12. No matter what, either Gallus scored 10 or Yona scored 12, so that’s what I guessed … and, well … I got lucky with both, I guess.” ... “Wow … it’s like you’re in our heads,” Gallus states. “Yona prefer head privacy … but that was incredible.” “Thank you both,” Ocellus says. “I’ll just, uh, get back to my book, then.” As she turns to return to the couch, Gallus catches her with a closing question: “So are all changelings as smart as you with logic?” Ocellus chuckles. “Funny you should ask … Headmare Twilight pulled me aside after classes today to tell me to expect a visitor tomorrow. That’s kinda the reason why I’m rereading this,” she adds as she indicates the tome on the couch.” “Ooo, Ocellus have family visit? Can we meet with Ocellus family?” “Actually, Headmare Twilight says he’ll be a special guest lecturer for us tomorrow.” “Really?” Gallus asks. “Who’s coming to speak to us?” “My uncle Coccyx.”
Welcome Back, Coccyx“Good morning, everycreature,” Twilight addresses the group of six. “I have a special announcement for you today.” “Special announcement?” Silverstream repeats with great anticipation. “Ooh, are we taking another field trip? Are we finally going to learn basic plumbing?” Confusion spreads across Twilight’s face, but she quickly shakes off the feeling and chuckles. “No plumbing lessons today, Silverstream.” “Aww,” she replies with a tinge of sadness. “Instead, you’ll have a special guest lecturer today!” Twilight cheerfully continues. Sandbar’s eyes prick up, as the others gaze on in mild interest. Slightly put off, Twilight addresses Gallus: “I thought there would be more enthusiasm from all of you.” “Sorry, Headmare, but we kinda already know who’s coming … Ocellus told me and Yona last night.” “Really,” Twilight sternly responds as she walks over to Ocellus. “You know, it’s not very friendship-like to spoil a surprise for others.” “I, oh, um … I’m sorry,” Ocellus nervously replies. Twilight stares down Ocellus for several seconds … then bursts into laughter. “I’m kidding; it’s all right if you told your friends already!” The news causes the young changeling to breathe a sigh of relief. Twilight returns to Gallus. “OK then, why don’t you tell the class who’s visiting today?” “Ocellus said her uncle Coccyx is visiting … but that’s all I know about it.” “I see,” Twilight continues with a wry smile. “So you don’t even know what this ‘Coccyx’ individual looks like, do you?” “No, she didn’t say. But he’s a changeling, so he could take on any appearance.” “He could, couldn’t he …” Gallus takes a second look at Twilight, as an idea dawns on him. “Wait … you’re Coccyx, aren’t you!” His declaration draws interest from the others. “Whoa,” Sandbar admits, “that’s impressive. You fooled me, anyway.” “You fooled all of us,” Smolder adds. “Nicely done.” ‘ Twilight’ peruses the students. “So, just between us … how do you feel about your real headmare?” “Headmare Twilight is awesome!” Silverstream quickly exclaims. “She and the others have taught us all so much, and we’ve even saved Equestria because of her lessons!” “Well, just between you and us,” Gallus starts … before catching himself. “Yes?” ‘Twilight’ prods. “Waaaaait a minute … what if you really are Headmare Twilight? Are you just trying to figure out what all of us really think about you?” “What? Don’t be silly,” ‘Twilight’ states with a nervous chuckle. “I mean, as if I really needed to … I mean, as if your headmare would really …” “Is everything all right?” The class turns to the door as Twilight … another Twilight … trots into the room. The second Twilight takes a look at the first Twilight, then Ocellus, then back to the first Twilight. “What’s going on?” she asks the group. Ocellus looks from one Twilight to the other. “I really want to give my uncle a big hug … but which one do I hug?” “I can’t be certain,” Smolder starts, “but I’d say the first Twilight is the real Twilight.” “What makes you say that?” the two Twilights ask in unison. “Well for one, you got pretty jittery when Gallus started questioning which one you were.” The first Twilight blushes at the comments. “And second, you knew Silverstream’s name.” The first Twilight turns to the second. She nods her head, and the second Twilight morphs before everyone’s eyes into a yellow changeling with bulbous green eyes and fluorescent tan wings. “Uncle!” Ocellus cries out as she approaches and hugs her fellow changeling. “It’s great to see you again,” Coccyx says as he looks around the room. “So these are your friends. A most eclectic group if I’ve ever seen one.” “So what was with the whole ‘pretending to be someone else’ bit?” Gallus asks Twilight. “Just trying to hone my sense of humor,” she sheepishly replies. Gallus utters a flat ‘uh huh’ in response as Twilight composes herself. “In any case, I’m sure Coccyx has much to relate to you, and I have more planning to do at the present time. Enjoy!” She turns and heads out the door, leaving the elder changeling with the class. Author's Note The next couple puzzles revolve around the 'twin towns' of Utopia and Paradise, as well as the 'third town' of Elysium. It will not be absolutely necessary to read my earlier Fics to solve these upcoming puzzles, but for those who are interested: Twilight's Logic Puzzle Adventure Twilight's New Logic Puzzle Adventure
Puzzle 12: Silent Pegasi“So Ocellus,” Coccyx starts, “care to introduce me to your fellow classmates.” “Oh … well,” Ocellus replies, “these are Sandbar, Smolder, Silverstream, Yona, and Gallus.” Each student nods upon hearing his or her name. “And … those are my classmates … and my friends.” “It’s a pleasure to meet all of you,” Coccyx states. “Do any of you have any questions about me?” “From what Ocellus has told us,” Gallus says, “you’ve been spending time in some pony village that served as inspiration for some logic puzzles. That seems … odd.” Coccyx laughs. “Yes, the twin towns of Utopia and Paradise … now the triple towns, including Elysium. It’s not as foreboding for travelers as it used to be, at least from what I’ve heard of their history. However, they still very much embrace what made their towns special.” “Oh? What would that be?” Silverstream asks. “It would take too long to delve into what I know of their history, so I’ll keep it simple. Utopia and Paradise were founded by unicorns and pegasi, and over the years a quirk developed regarding the inhabitants of those towns: in Utopia, all the unicorns tell the truth and all the pegasi lie; in Paradise, it’s the opposite: all the unicorns lie and all the pegasi tell the truth.” “That’s … really weird,” Sandbar admits. “I’m only giving a summary, but I’ve been told that the twin towns were the basis of a number of logic puzzles. As a simple example: suppose you came across two unicorns, either of which could come from either town. One unicorn states that they both come from Paradise. Could you tell which town each come from?” The group ponder the question for a few minutes. Soon after, Yona responds: “First unicorn say both from Paradise … but you say all unicorns from Paradise lie, so it can’t be that both unicorns from Paradise.” “On the other hoof,” Sandbar adds, “the first unicorn lied, which means she is from Paradise.” “But since the statement was false,” Silverstream continues, “they can’t both be from Paradise … so the second unicorn must be from Utopia!” “Very impressive,” Coccyx admits to the class. “And a very quick response, no less,” “We’ve had some practice as of late,” Gallus states. “But what about earth ponies?” Smolder asks. “Ah, that’s where more recent history comes into play. Initially, earth ponies were not even part of either town … they were left behind and forced to form a third town, away from the others; that town, they dubbed ‘Elysium’.” “Wait,” Yona says, “so ponies not always friends?” “I think that was the basis of Hearthswarming,” Ocellus replies. “Long ago, there was animosity among the three pony types.” Sandbar nods in agreement. “And from what I know,” Coccyx continues, “that animosity existed in some form when these towns were formed. Now, I can tell you that things aren’t that way anymore. All three towns now sport all pony types … but in a nod to the old traditions that made their towns unique, they still adopt their truthfulness regarding place of residence to the old ways.” Gallus thinks for a moment. “So what that means is …” “In terms of where each come from, the only truth-tellers are Utopian unicorns, Paradisian pegasi, and Elysian earth ponies. The unicorns from either Paradise or Elysium lie, as do the pegasi from Utopia or Elysium, and the earth ponies from either Utopia or Paradise.” “That’s … even weirder,” Silverstream states. “I know … it’s weird to me, too. But it does make for interesting puzzles; for example, it would render the previous puzzle as having multiple solutions: now the first pony could be from Elysium and the second pony could be from any of the three towns. Or for that matter, the first pony could still be from Paradise, but now the second pony could be from Elysium.” The class silently takes in the explanation. “Here’s another, simpler example: if a unicorn from one of the three town stated that he was from Paradise, what could you conclude?” After a brief pause, Smolder offers a quick response: “The unicorn can’t be from Paradise, since then he’d be telling the truth. But he can’t be from Utopia either, since all Utopian unicorns are truthful. So that means he must be from Elysium.” “Very good,” Coccyx states. “Maybe you guys need a bigger challenge,” he adds with a sly grin. “What sort of challenge?” Sandbar asks. “Recently, I and a number of fellow changelings have acclimated ourselves to the customs of the triple towns. The elder statesponies of the towns insist that we changelings understand and abide by these customs, and as such have created a number of challenges to keep us on our hooftips. One such challenge was presented to me just this past week; I’m curious if you are able to solve this puzzle.” Gallus snorts. “Friendship is logic, I guess. Go ahead.” Coccyx chuckles. “OK then. As a test, I was introduced to six ponies: two unicorns, two pegasi, and two earth ponies. I was told in advance that of each pair, one would tell the truth and one would lie, in accordance with the rules stated beforehoof. For simplicity’s sake, I’ll refer to the unicorns as A and B, the pegasi as C and D, and the earth ponies as E and F. They made the following statements: A said “None of the others have the same hometown as me.” B said “F and I have the same hometown.” E said “C is from Utopia.” F said “D and E have different hometowns.” … “What about the pegasi?” Ocellus asks. “For some reason neither one said anything,” Coccyx explains. ‘Pega-shy’ was more like it, I guess.” “Well it’s not like we don’t know a shy pegasus,” Gallus admits. “So did that invalidate the test?” Coccyx smiles. “On the contrary, there’s enough information to figure out all six ponies’ hometowns.” Author's Note So … what are the hometowns of each of A, B, C, D, E, and F?
Puzzle 13: Button, Button, Which One's 'Gray Button'?“Fake wings?” “Fake horns?” Coccyx nods his head at both phrases. “All in the name of pony type anonymity. Quite a fascinating backstory, actually; by not revealing their true nature while adhering to a higher moral code, the Alicorn Society was able to bridge the gap between unicorns and pegasi … and later, earth ponies.” “But wouldn’t that make puzzle solving, like, waaay harder?” Silverstream asks. Coccyx chuckles. “Indeed … though before the inclusion of earth ponies and the town of Elysium, it actually led to some fascinating logical truths.” Hearing a chorus of ‘Oh?’s, he continues: “Here’s a simple example: suppose you met one of these ‘alicorns’, and I give you the knowledge that she is not an earth pony and is not from Elysium. The alicorn states that she is, in fact, a unicorn. What could you conclude?” The class goes silent for several seconds. Then, Sandbar speaks up: “If she’s telling the truth, then she must be from Utopia. If she’s lying, then she’s not a unicorn, so she must be a pegasus … and she can’t be from Paradise, so she must be from Utopia.” “Exactly,” Coccyx concludes. “An alicorn that claims to be a unicorn must be from Utopia, but could be either pony type, unicorn or pegasus. And at the same time, an alicorn that claims to be from Utopia must, necessarily, be a unicorn from either Utopia or Paradise. In short: stating that she is a particular pony type reveals her hometown, and stating her hometown reveals her pony type. I really think there’s an elegance to the symmetry, don’t you?” Ocellus, Sandbar, Silverstream, and Yona nod in agreement, while Smolder and Gallus each shrug off the perceived beauty with a ‘meh’; the latter two cause Coccyx to laugh again. “I suppose such things are not for every creature,” he admits. “But … symmetry gone with Elysium and earth ponies?” Yona asks. “Indeed … without the knowledge I gave you, the previous example would have multiple possibilities. Now if she were telling the truth and was a unicorn, then of course she would still be from Utopia; but now, lying doesn’t mean she’s from Utopia: she could be a pegasus from either Utopia or Elysium, or she could be an earth pony from Utopia or Paradise. So now there’s five possibilities, and further information would be needed to reduce this number.” “I’m afraid to ask how much more complicated the puzzles can get with these ‘alicorns’,” Smolder states. “You don’t have to,” Coccyx replies. “Recently, I was given a particularly daunting challenge that revolved around a quintuplet of alicorns.” A sly smile spreads across his face. “Let’s see how you fare on this one.” The class murmurs to itself in excitement, save for Gallus. “It beats the usual history lessons,” he eventually states. “I’m game.” “Wonderful … now before we begin, I do want to remind you of an important distinction: if an alicorn, for example, states that she is a Utopian pegasus, that is different from the alicorn stating that she is from Utopia, and then stating that she is a pegasus in a separate statement.” “What’s the difference?” Silverstream asks. “In the former case, you know she’s lying but she could be any of the following: Utopian earth pony, Paradisian unicorn, Paradisian earth pony, Elysian unicorn, Elysian pegasus. In the latter case, she still must be lying, but now she could only be either a Paradisian unicorn, Paradisian earth pony, or Elysian unicorn.” “So, in one case she could be a Utopian earth pony, but not the other?” “Exactly. In the first case, all we know is that she is not the specific specified combination; she could be either half of the combination. In the second case, she cannot be either half.” “Got it … I think,” Silverstream replies with a tinge of confusion. “Excellent. Well then … I was introduced to five gray alicorns. I was informed that from this group of five, every pony type and every hometown was included at least once. I was also informed that no two of them had the same combination of pony type and hometown, so there couldn’t be two Utopian pegasi, for example. One of these five, I was told, went by the name ‘Gray Button’; my task was to determine the pony type and hometown of all five, as well as identify which one was, in fact, Gray Button.” Ocellus begins calculating: “3 possibly pony types and 3 possible home towns for each alicorn, so 9 times 9 times 9 times 9 times 9 is … 59,049 possible combinations. Not accounting for which one is Gray Button.” “But alicorns all different,” Yona corrects, “so actually 9 times 8 times 7 times 6 times 5, or 15,120.” “Then times 5 for Gray Button,” Gallus adds, “so actually 75,600.” “Uh, guys?” Sandbar asks. “Can we slow down with the ‘math’ part?” “Oh … sorry, Sandbar,” Ocellus says in apology. “I see your math skills are high as well,” Coccyx admits. “But I assure you, there’s only 1 possibility once the statements are taken into account. I’ll continue to use ‘A, B, C, D, E’ for simplicity. These were the statements made: A: “I am a unicorn. I come from Elysium. Gray Button comes from Paradise.” B: “I am NOT an earth pony. I come from Paradise. Three of us are the same pony type.” C: “I am an earth pony. I do NOT come from Elysium. At most one of us comes from Utopia.” D: “I am NOT a unicorn. I do NOT come from Paradise. Gray Button is a pegasus.” E: “I am a pegasus. I do NOT come from Utopia. Three of us come from the same town.” “And from those statements, along with the information I provided, you have enough to figure out the pony type and hometown of all five alicorns, and state which one is Gray Button.” “So,” Coccyx concludes, “Can you figure this one out?”
Puzzle 13 solution A: “I am a unicorn. I come from Elysium. Gray Button comes from Paradise.” B: “I am NOT an earth pony. I come from Paradise. Three of us are the same pony type.” C: “I am an earth pony. I do NOT come from Elysium. At most one of us comes from Utopia.” D: “I am NOT a unicorn. I do NOT come from Paradise. Gray Button is a pegasus.” E: “I am a pegasus. I do NOT come from Utopia. Three of us come from the same town.” Having written the statements on the chalkboard, along with the information that there at least one of each pony type and hometown among the five, with no two being the same for both … Coccyx waits for several minutes as the class ponder the information. “Yona confused,” the yak admits as she looks around the room, “but Yona not alone in confusion.” “You got that right,” Gallus adds. “Where do we even begin?” “Where did you begin,” Ocellus asks her uncle. “Well you can’t identify the pony type or hometown from any of the five at the onset,” Coccyx explains, “but you can at least narrow down the possibilities. Try starting there.” “Hmmm.” Ocellus focuses on the first of the five alicorns. “Well ‘A’ must be a liar; if not, she would be an Elysian unicorn, and that contradicts what we know of them. And that means that ‘A’ must be either a pegasus or earth pony, and from either Utopia or Paradise … but not a Paradisian pegasus, since they’re truthful. But that still leaves three possibilities.” “That’s fine,” Coccyx assures. “File away that information and move on.” “Ooh, let me go next,” Silverstream calls out.” “Go ahead,” Ocellus replies. “OK, so about ‘B’ … if she’s telling the truth, then she’s from Paradise so she’d be a pegasus. If she’s lying, then she is an earth pony but not from Paradise … and not from Elysium either, ‘cause that would make her truthful. So she must be either a Paradisian pegasus or a Utopian earth pony.” Smolder follows: “What about ‘C’? If she were telling the truth, then she’d be an earth pony who’s from either Utopia or Paradise … but in either case, she’d be a liar. So she must be a liar, and she’s not an earth pony but is from Elysium. She must be either a unicorn or pegasus.” “Yona look at ‘D’. Uh … ‘D’ either truthful or liar. If ‘D’ truthful, then ‘D’ not unicorn and not from Paradise … so ‘D’ must be earth pony from Elysium. And if ‘D’ liar, then ‘D’ unicorn from Paradise.” Sandbar picks up the final alicorn: “ ‘E’ could be truthful or lying, too. If she were truthful, then she would be a pegasus, and would have to be from Paradise. And if she were lying, then she would be from Utopia and not a pegasus … and not a unicorn, either. So she would be an earth pony in that case.” “Wait a minute,” Gallus interrupts. “Didn’t we conclude that ‘B’ was either a Paradisian pegasus or a Utopian earth pony?” Coccyx chuckles. “You did … and well done, all of you. You’ve done the dirty work that needed to be done. Now you just need to put the pieces together.” “But how,” Ocellus asks, the others looking on with similar confused looks. “Maybe this will help.” Coccyx picks up a piece of chalk and draws a 3x3 figure onto the board, adding letters above and to the left: -- upe U: - - - P: - - - E: - - - “The lowercase letters represent the pony types: unicorn, pegasus, earth pony. The uppercase letters represent the towns: Utopia, Paradise, Elysium. And if you recall, no two alicorns can share both pony type and hometown. Now … who can go where?” “Well,” Ocellus starts, “we know that ‘B’ and ‘E’ could each be either a Paradisian pegasus or a Utopian earth pony. And since they can’t be the same, that means that one of them has to be a Paradisian pegasus and the other one has to be a Utopian earth pony. But I can’t tell which is which.” “It doesn’t matter,” Coccyx says, “you now have two spots filled, at least in some order. Now what about the others?” Yona follows: “ ‘A’ must be Utopian pegasus or Paradisian earth pony or … wait, ‘A’ not Utopian earth pony, if ‘B’ or ‘E’ has to be.” “What about ‘C’?” Smolder adds. “She has to be Elysian, and has to be either a unicorn or a pegasus.” “And ‘D’ has to be either a Paradisian unicorn or an Elysian earth pony,” Sandbar adds. “Oh, let me add this to the board,” Silverstream says as she flutters to the board. Picking up the chalk, she fills in the possibilities: -- u p e U: - A * P: D * A E: C C D “I used a star for ‘B’ and ‘E’,” she clarifies as she returns to her seat. “So where do we go from here?” Gallus studies the letters. “No Utopian unicorns, I see. Wait, we know that ‘A’ and ‘C’ are liars, right?” “Good catch,” Ocellus says. “I think we should wait on ‘A’s comment until later, but what about ‘C’? She said that at most one of them came from Utopia, so in fact, at least two of them do.” “And there aren’t any Utopian unicorns,” Smolder adds, “so there has to be both a Utopian pegasus and a Utopian earth pony. Well the ‘earth pony’ slot is either ‘B’ or ‘E’; we already knew that.” “But now we know that ‘A’ has to be a Utopian pegasus,” Sandbar states. “That’s the only way we can get at least two Utopian ponies.” “We got one!” Silverstream triumphantly asserts. “Now … now what? “What about ‘B’ and ‘E’ comments?” Yona asks. “One truthful and one liar, right?” “You’re right,” Ocellus replies, studying each of their third comments. “One of them says that three of them are the same pony type, and the other says that three of them have the same hometown. So one of those facts has to be true. Which one?” “What can we have three of?” Smolder asks. “We don’t have any Utopian unicorns or Paradisian earth ponies,” Gallus states. “That rules out four of the six categories. There has to be either three pegasi or three Elysians.” “But there can’t be three Elysians,” Ocellus realizes. “ ‘C’ is the only one would could be an Elysian unicorn, and she’s also the only one who could be an Elysian pegasus. She can’t be both.” “That just leave three pegasi,” Yona concludes, “so ‘B’ truthful and ‘E’ lie. That make ‘B’ Paradisian pegasus and ‘E’ Utopian earth pony.” Sandbar speaks up: “If three of them are pegasi, then ‘C’ has to be an Elysian pegasus.” “And since at least one of the five has to be a unicorn,” Smolder adds, “ ‘D’ must be a Paradisian unicorn.” “BRAVO!” Coccyx calls out. “You got the pony types and hometowns of all five alicorns! All that’s left is to figure out which one is Gray Button.” “There’s only two comments to work with,” Gallus says, “and both of them are lies. Gray Button isn’t from Paradise, and she isn’t a pegasus. And the only one who fits both of those statements is … ‘E’. ‘E’ must be Gray Button.” “Wonderful,” Coccyx gushes. “Top notch; top notch! I’m starting to think there’s no puzzle the six of you can’t solve.” Other than why we’re solving so many of these, Gallus thinks to himself. . . . “Uncle Coccyx?” “Yes, Ocellus?” “I really enjoy these puzzles … but none of them seem to have anything to do with you?” Coccyx chuckles. “I’m sure you have concerns since you and Twilight left, but I assure you, relations between the townsfolk and the visiting changelings are very much cordial. It’s been made easier since they’ve made allowances for our presence.” “What do you mean by ‘allowances’?” Silverstream asks. “When we first got there, some of us may have made comments that caused … friction. But we’ve come to some agreements that those in the Alicorn Society have signed off on, and those agreements have made our presence more than tolerable; they now embrace our presence.” “Whoa,” Yona says with surprise. “Maybe yaks take lessons from changelings.” “Dragons, too,” Smolder adds. “And Griffons,” Gallus concludes. “So how did you get them to embrace you so much?” “Simple, really … our presence allowed for even greater complexity of puzzles. Let me explain …”
Puzzle 14: Balancing Act“ ‘Logic bomb’? So, these ponies would black out over a sentence?” Coccyx nods as Smolder processes the answer to her question. “Nothing personal, but I couldn’t see any dragon passing out over a bunch of words.” “A by-product of their way of reasoning,” Coccyx assures. “From the beginning, Utopia was created solely for unicorns and Paradise for pegasi. Even after unicorns began settling in Paradise, their pride and sense of loyalty persuaded them to maintain an allegiance to the town of their ancestors’ origin. Hence, no unicorn would ever admit to coming from Paradise, and vice versa. From there, notions of truth and falsity spread organically, up to the point when Twilight first made contact.” Yona asks for clarification: “So, if unicorn claim to be from Paradise, that caused fainting?” “Before Elysium came into the picture, yes,” Coccyx replies. “At that time, all unicorns claimed to come from Utopia and all pegasi claimed to come from Paradise. For that matter, once one factors in the faux-alicorns, all Utopian claimed to be unicorns and all Paradisians claimed to be pegasi.” “But Elysium changed all that, right?” Sandbar asks. “In many ways … now a unicorn could claim to be from Paradise, if in fact she was from Elysium. But no unicorn would claim to be from either Paradise or Elysium.” Coccyx’s last statement draws quizzical looks from the class. “You just said a unicorn could claim to be from Paradise if she was from Elysium,” Gallus says. “And I assume a unicorn could also claim to be from Elysium, if she was actually from Paradise.” “Yes, and yes,” Coccyx replies. “But it’s not possible for a unicorn to claim to be from either Paradise or Elysium? That makes no sense!” “I realize it sounds weird, but it’s true. Suppose a unicorn did say ‘I come from either Paradise or Elysium.’ Assuming that unicorn came from one of the three towns, which one could it be?” Gallus mulls over the question. “… if the unicorn was from Utopia, then she would be truthful, but her statement would be a lie. So the unicorn would actually have to be from Paradise or Elysium, making her statement true … but unicorns from those towns always lie … huh … that’s weird.” “And believe me, I found out the hard way,” Coccyx continues. “While posing as a pegasus, I commented to a townspony that I wasn’t from Paradise; that pony dropped like a rock.” He smiles as he continues. “But they don’t have that problem anymore.” “Why not?” Silverstream asks. “Because of changelings like myself,” he proudly proclaims. “Now that the townsponies are aware of our presence, they can rationalize that such a comment could only come from a changeling. Rather than fear our presence, the knowledge of our being there prevents similar occurrences from happening.” “Wow,” Ocellus admits, “it’s a logic-based symbiotic relationship.” “In a sense, yes. We fill the gaps that would normally come from certain statements. Really, we complete the towns.” “Like imaginary numbers!” Yona asserts. Sandbar, Smolder, and Silverstream turn to Yona with curious looks. “It math thing,” she continues. Ocellus re-addresses her uncle: “So, the changelings are still in good standing with the Alicorn Society?” “Indeed … in fact, new tests have been created that incorporate our presence. Recently, one particularly difficult challenge was created for new recruits. I would like to see how you as a class respond to it.” “Harder than the last one?” Silverstream asks. “Well … you be the judge,” Coccyx says before explaining the challenge: “You are introduced to four alicorns. You are informed that one of them is a changeling with no particular hometown. The other three alicorns consist of one unicorn, one pegasus, and one earth pony; in addition, each one comes from a different one of the triple towns. Each alicorn will make one or two statements, with the usual rules of truth and falsity applying: only a Utopian unicorn, Paradisian pegasus, or Elysian earth pony will make truthful statements, while ponies of the other combinations will lie.” “As for the changeling, he or she will either be truthful or lie … whichever is less prevalent among the others. So if two or three of the others are truthful, the changeling will lie; if two or three of the others lie, the changeling will be truthful. Think of the changeling as attempting to balance out the veracity of the group.” “I will refer to the four alicorns as A, B, C, and D, as before. They make the following statements:” A: C is a pegasus. B: A is from Utopia. C: D is a changeling. A is a unicorn. D: Only one of us is a liar. B is from Paradise. Ocellus ponders the statements. “That doesn’t seem like enough to go on.” “It’s not,” Coccyx replies. “You are also given two additional facts:” A recruit who took the test before you was then told which alicorn was from Elysium, but that recruit still did not have enough information to solve the puzzle. Another recruit who took the test before you, who knew nothing of the first recruit, was then told which alicorn was the earth pony; that recruit also has insufficient information to solve the puzzle. “And now YOU have enough information to solve the puzzle! Which alicorn is the changeling, and what are the pony types and hometowns of each of the others?”
Puzzle 14 solution A: C is a pegasus. B: A is from Utopia. C: D is a changeling. A is a unicorn. D: Only one of us is a liar. B is from Paradise. A recruit who took the test before you was then told which alicorn was from Elysium, but that recruit still did not have enough information to solve the puzzle. Another recruit who took the test before you, who knew nothing of the first recruit, was then told which alicorn was the earth pony; that recruit also has insufficient information to solve the puzzle. “Where to begin, where to start, what to do, what to think, …” Silverstream rambles to herself for several seconds, before Ocellus turns to her. “Maybe we should start with ‘D’s first statement. What if it’s true?” Gallus responds: “If it is, then 3 of them are truthful and 1 is a liar. Could all 3 truthful alicorns be ponies?” “That would force them to be a Utopian unicorn, Paradisian pegasus, and Elysian earth pony,” Sandbar replies. “Also, the changeling would be a liar.” “That not work,” Yona counters. “That make ‘A’ and ‘C’ both liars.” “Why so?” Yona answers Ocellus’s query: “ ‘C’ say ‘D’ changeling, so if ‘D’ truthful, then ‘C’ must be liar. And ‘D’ say ‘B’ from Paradise, so ‘B’ must be pegasus, make ‘A’ liar.” “She’s right,” Smolder states. “So the 3 truthful ones can’t all be ponies. It would have to be 2 truthful ponies and a truthful changeling.” “That can’t work either,” Silverstream says. “The changeling isn’t ‘balancing’ in that case.” Ocellus adds: “Besides, given what we know there can’t be exactly 2 truthful ponies; the third would have to be truthful, assuming there’s one of each pony type and one from each hometown.” Gallus concludes: “All of which means ‘D’ is lying, as well as at least one other. And I guess that means ‘B’ isn’t from Paradise … what now?” “What about ‘C’?” Smolder asks. “Could ‘C’ be truthful?” “If so,” Ocellus answers, “that makes ‘D’ a lying changeling. And that would mean either 2 or 3 truthful ponies. It still can’t be exactly 2, and if it were 3, ‘D’ would be truthful; we already ruled that out.” “So ‘C’ is a liar, too,” Smolder concludes. “And ‘A’ isn’t a unicorn. Anything else?” “Well at least one of the others is truthful,” Sandbar states. “Could both of them be?” “Can’t be,” Yona responds. “If both truthful, then ‘A’ truthful Utopian, would have to be unicorn. That not work if ‘C’ liar.” “All right,” Ocellus says. “So far we know that exactly one of them is truthful, and it’s either ‘A’ or ‘B’.” … “We might be stuck now,” Gallus laments. “You give yourself too little credit,” Coccyx insists. “Well done, all of you! You’ve come farther than most. But now I suggest you start figuring out possible solutions, and then use the additional information I provided.” “Possible solutions, eh?” Gallus ponders the suggestion. “OK then, we know that either A or B is the only truthful one, and whichever one she is is also the changeling. So I guess we just ‘pick a path’.” “ ‘A’ or ‘B’?” Ocellus asks the group. “May as well start with ‘A’,” Smolder recommends. “If ‘A’ is the changeling, then ‘C’ is the pegasus … but could be from either Utopia or Elysium.” “If ‘C’ is from Utopia,” Sandbar adds, “then ‘B’ has to be from Elysium; she can’t be from Paradise, since ‘D’ lied.” “That make ‘D’ from Paradise,” Yona continues. “And if ‘C’ Utopian pegasus, then … ‘B’ must be Elysian unicorn and ‘D’ Paradisian earth pony.” “Maybe we should put this on the board,” Silverstream suggests as she flutters over to the front of the class and picks up a piece of chalk. “So, Case 1: ‘A’ changeling, ‘B’ Elysian unicorn, ‘C’ Utopian pegasus, ‘D’ Paradisian earth pony” “OK … but what if ‘C’ was from Elysium instead?” Ocellus asks. “ ‘B’ still can’t be from Paradise,” Gallus points out, “so she has to be from Utopia. And ‘D’ would have to be from Paradise.” “And ‘B’ can’t be the unicorn then,” Smolder adds, “so she’s the earth pony and ‘D’ is the unicorn.” “Got it,” Silverstream states as she writes down the next possibility: “Case 2: ‘A’ changeling, ‘B’ Utopian earth pony, ‘C’ Elysian pegasus, ‘D’ Paradisian unicorn” “That’s all the possibilities with ‘A’ as the changeling,” Sandbar notes. “What about ‘B’?” “If ‘B’ changeling,” Yona starts, “ ‘A’ from Utopia but still not unicorn. Either pegasus or earth pony.” “If we assume ‘pegasus’,” Ocellus says, “then ‘C’ and ‘D’ have to be a Paradisian earth pony and an Elysian unicorn, in some order.” She ponders the original statements for several seconds. “But there’s no way of knowing which is which.” “Looks like Silverstream’s got double-duty, then,” Gallus states. “Can you add those cases?” “I’m on it,” Silverstream assures: “Case 3: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Paradisian earth pony, ‘D’ Elysian unicorn … and Case 4: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Elysian unicorn, ‘D’ Paradisian earth pony” “Only one path left to take,” Ocellus states, “If ‘A’ is an earth pony.” “ ‘C’ can’t be a pegasus in this case,” Smolder adds, “so she’s the unicorn and ‘D’ is the pegasus.” “Then ‘D’ not from Paradise,” Yona concludes, “so ‘D’ from Elysium and ‘C’ from Paradise.” “Last one,” Silverstream echoes as she writes the fifth possibility down: “Case 5: ‘A’ Utopian earth pony, ‘B’ changeling, ‘C’ Paradisian unicorn, ‘D’ Elysian pegasus” Silverstream flutters back to her seat, and the students take in the five possibilities: Case 1: ‘A’ changeling, ‘B’ Elysian unicorn, ‘C’ Utopian pegasus, ‘D’ Paradisian earth pony Case 2: ‘A’ changeling, ‘B’ Utopian earth pony, ‘C’ Elysian pegasus, ‘D’ Paradisian unicorn Case 3: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Paradisian earth pony, ‘D’ Elysian unicorn Case 4: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Elysian unicorn, ‘D’ Paradisian earth pony Case 5: ‘A’ Utopian earth pony, ‘B’ changeling, ‘C’ Paradisian unicorn, ‘D’ Elysian pegasus “We’ve used everything,” Smolder states. “No we haven’t,” Gallus corrects, the realization dawning on him. “We were told that two recruits before us were given additional information, and that information wasn’t enough in either case.” “So what?” Yona asks. “So consider which alicorn could be from Elysium,” Gallus responds. “In order, the cases state that the Elysian is either B, C, D, C, or D. But it can’t be ‘B’ … or that first recruit would have had enough information!” “But if that recruit was told that either ‘C’ or ‘D’ was the Elysian,” Ocellus adds, “that wouldn’t be enough. We can rule out Case 1!” “That still leaves 4 cases,” Sandbar points out. “But now focus on who the earth pony could be,” Smolder states, the final piece coming into place for her as well. “Going in order again, the earth pony is either D, B, C, D, or A … the only one that doesn’t immediately give away the answer is if ‘D’ is the earth pony.” “But didn’t we rule out Case 1?” Yona asks. “But that’s just it,” Ocellus answers: “We know it’s not Case 1, but the second recruit doesn’t know that! If Case 4 is the actual solution, then the first recruit being told that ‘C’ was the Elysian would still allow the possibility that it’s Case 2 … and the second recruit being told that ‘D’ was the earth pony would still allow the possibility that it’s Case 1.” “So … we’re done, right?” Silverstream asks. “It’s Case 4, isn’t it?” “It is!” Coccyx admits: “Indeed, the final solution is that ‘A’ is a Utopian pegasus, ‘B’ is a changeling, ‘C’ is an Elysian unicorn, and ‘D’ is a Paradisian earth pony. EXCEPTIONAL WORK, EVERYCREATURE!” “Sounds like I missed some good work.” Coccyx and the others all turn to the doorway, observing Twilight as she observes them. “What are you doing here?” Ocellus asks. “More like ‘what are you all still doing here?’ … you’ve been here for an hour and a half.” The group turn to the clock on the wall, noting that indeed, ninety minutes had elapsed since Coccyx entered the room. “Wow,” Gallus says, “I don’t think we’ve ever had a class go by so fast.” “I guess time is relative,” Twilight replies, “but I’m afraid you do have other classes to get to.” She turns to Coccyx: “Thank you again for taking time out of your busy schedule to visit our school.” “The pleasure was all mine,” Coccyx insists. “Your students work exceedingly well together … something I suppose I should expect from true friends.” Silverstream, Yona, Sandbar, and Ocellus smile and blush, as Smolder and Gallus simply sigh and smile. “Well, I should be heading back,” Coccyx tells the class. “It was great seeing you again, Ocellus … and it was a real pleasure to meet your classmates.” “Take care, uncle Coccyx … hope everything continues to go well at the twin … er, the triple towns.” Coccyx chuckles. “Hey, maybe a field trip is in order at some point.” He again turns to Twilight. “What do you say?” “There’s a lot on our schedule,” Twilight sadly replies … “but we’ll see. Maybe next semester.” “Well then … until next time ...” Giving one final bow, Coccyx turns and flies out of the room. Twilight scans the chalkboard. “Wow … sorry to have missed whatever puzzle this came from.” “Don’t sweat it,” Gallus says, “these days, the next one’s probably just around the—” *OOF* “… corner?” The class gather around Twilight and get her back on her hooves.” “Oh dear, I am so sorry,” Rarity pleads. “I was just looking for … oh, there you all are!” “Our apologies, Miss Rarity,” Sandbar says. “We kinda went over our time with the last class.” “Oh, that’s quite all right,” Rarity insists. “But come; there’s fashion to be done!” Satisfied that Twilight is no worse for wear, Rarity heads back out the doorway, with Twilight and the students following in tow …
Puzzle 15: Bejeweled and BefuddledI can’t believe what I’m about to say, but can we get back to solving puzzles? Gallus knew better than to voice his thoughts to his professor. In fact, he has become rather burned out at the whole notion, even as adept as he and the others had become. But more to the point: fashion never really inspired him the way it did others, such as his professor. And given a choice, he’d rather toil through another puzzle than— “Gallus?” The griffin looks up with a start, now aware of Rarity staring back at him. “Uh … could you repeat the question?” “How much impact would you say fashion has had on griffin culture?” “Fashion? Uh …” Gallus’s eyes dart around the room as he searches for an acceptable response. “Well? … er …” He sighs before concluding: “Not much.” Rarity frowns at the response. “Surely there must be some emphasis made.” “Griffinstone isn’t much for finery,” he sheepishly replies. “Well perhaps we can change that. Come on up and let me take your measurements: we’re going to design an outfit specifically for griffins!” Gallus gulps inaudibly as he gets up and walks towards the front of the class. He catches a glimpse of various photos on the teacher’s desk. “So what are these?” he asks, pointing out the pictures. Rarity sighs. “A failed attempt at pleasing the unpleasable, mixed with a tinge of revenge, no doubt.” Her response draws the attention of the rest of the class. Gallus takes a second look at the photos. “Really? All I see are scarves … six of them, each with some strange markings … are those exes and ohs?” Rarity’s horn illuminates as the photos separate and spread themselves atop her desk. The students get up and gather around the pictures, each of which is of a scarf trimmed with five gems of various colors. “Who are all these for?” Smolder asks. “Chroma-chaotica, the avant-garde fashion designer,” Rarity replies. “Every year, the stallion gets in his head *THE* color combination that is ‘in’, and every year designers from all of Equestria follow his lead. This year, I had hoped to get a jump on the crowd.” Ocellus takes a closer look at one of the photos. The scarf displays a series of five gemstones: diamond, ruby, another diamond, onyx, and a third diamond. In the bottom-right corner, she discerns an ‘x’ and an ‘o’. “Miss Rarity … what do these symbols at the bottom mean?” “I had my trusted Manehattan assistant Coco take six samples to Chroma’s representatives; these photos bear those samples. For each sample, she received feedback as to what truly worked, what worked in a minor sense, and what didn’t work at all.” “Is that what ‘x’ and ‘o’ for?” Yona asks. “The ‘x’ signifies that Chroma fully agrees with the color and placement of exactly one of those gems,” Rarity explains. “So either he thinks a ruby should be in the second spot, an onyx should be in the fourth spot, or one of the diamonds is correctly placed.” “And the ‘o’?” Sandbar asks. “The ‘o’ means Chroma likes the idea of one of the gems, but not its placement. So maybe he believes the ruby should come first, instead. Or maybe last. Or maybe one of the diamonds should be fourth. Unfortunately, there’s no way to know for sure.” “But you have six of these photos,” Silverstream notes. “That’s enough information, isn’t it?” As the hippogriff speaks, Ocellus picks up a piece of chalk and notes each photo’s gem ensemble and associated marks on the board: D R D O D – x o D R S R R – o o o O S S E O – x x R O O S T – x o o S E T R E – o o T S O T D – x o Gallus scans the board. “I guess ‘D’ stands for diamond. What about the other letters?” “ ‘R’ is for ruby, ‘O’ is for onyx, ‘S’ is for sapphire, ‘E’ is for emerald, and ‘T’ is for topaz,” Ocellus replies. “Really pretty colors: white, red, black, blue, green, and yellow.” “So … this like puzzle,” Yona points out. “That means we can solve.” “If only it were that simple,” Rarity sullenly responds. “I’m not particularly adept at figuring things like this out, as you have seen.” “We are,” Smolder asserts. “Yes, the whole school has witnessed your capabilities, but I’m afraid there’s an added issue, for which you can thank Suri.” “Suri?” Ocellus asks. “Suri Polomare … and old rival of mine. Shortly after receiving these photos, I received a letter from Coco that Suri caught wind of my plans. She was able to gain access to the photos and changed the comments on one of them, and I don’t know which one.” “Why just one?” Gallus asks. “Why not just alter all of them?” “It’s her way of toying with me, like baiting a cat with a piece of string. In any event, I daresay the situation is now beyond even your abilities. Only five of these six photos can be trusted as having useful information, and there’s no way to figure out which photo is the wrong one.” “Some might consider that a challenge,” Gallus retorts. “No, no, no … I appreciate all you’ve shown myself and the others, but even if it were possible to solve this conundrum, I’m afraid it would take too much time out of our already abbreviated day. But, if you wish to stay late and inspect the photos on your own time, they’re of little use to me now. Now then, let’s discuss proper griffin regalia …” As early evening comes over Ponyville, the students reflect upon another day’s work … “Miss Rarity can really work fast under a deadline.” Silverstream notes. Ocellus turns to Gallus. “So what do you think Griffinstone will make of your new modeling duties?” she asks with a chuckle. He replies with a laugh. “I don’t want to be the one to disillusion our resident fashion guru, but her first trip to Griffinstone would probably be her last. It’s not exactly Manehattan where I’m from.” “Maybe she’ll get the idea to create outfits for Yakyakistan next,” Sandbar says as he turns to Yona. “Yona not think other yaks interested in attire … but then, Yona not think other yaks interested in others at all, so first for everything?” “Ooh, maybe we’ll design outfits for dragons next,” Silverstream notes. “By the way … shouldn’t Smolder be back by now?” Ocellus muses for a bit. “She said she wanted a few minutes to take a longer look at those photos. Honestly, I’m surprised she wanted to; she never showed a big interest in puzzle-solving before.” “Even if she did,” Gallus adds, “there’s no way she’d still be at it. I just assume she caught up with Spike afterwards. Not like any of us have fellow students of our own species around here.” Sandbar opens his mouth to speak, when the door to their foyer bursts open. The group turn to face Smolder, her eyes wide open and twitching with retinas bulging. “Smolder,” Yona exclaims. “We just talk about you!” “Where’ve you been,” Ocellus adds. “It’s been hours since we last saw you.” “You were with Spike, right?” Gallus asks. “I mean, it’s not like you spent all that time on Miss Rarity’s photos … did you? …” Smolder replies stiltedly: “Four hours … thirty-seven minutes … twenty-five seconds … but I did it.” “You did what?” Sandbar asks. “I figured … it … out …” Author's Note So basically, a solitaire 'Mastermind' puzzle with the added twist that one of the six clues is bogus and needs to be ignored. I'll admit, this proved harder than I imagined … and I suspect it may not get solved without further hints. I do have a logical solution, but it's long and makes assumptions that lead to contradictions. If anyone comes up with a simpler explanation (and NOT just a guess-and-check solution), let me know. If anyone wants a starting point, send me a private message and I'll reveal my first few steps.
Puzzle 15 solution D R D O D – x o D R S R R – o o o O S S E O – x x R O O S T – x o o S E T R E – o o T S O T D – x o [one clue wrong] The students note the six clues scrawled into the upper-left portion of the chalkboard; the remainder of the board filled with an alphabet soup’s worth of letters, squiggles, arrows, and other assorted ‘short-claw’ notation. Atop the desk are papers upon papers of additional scribblings. Their attention shifts from the desk and chalkboard to Smolder herself, still wide-eyed and twitchy as she explains her motives to the others. “Dragons competitive, must be best, so I must be best at solving, and pretty fashion, I like pretty fashion, so I put two and two together to get four, or maybe two more for six, so six clues, but I figured out which one wasn’t really a clue, and I figured out the whole thing. Yes, oh yes I did!” Gallus reaches out a talon to console Smolder. “It’s okay … we came here to see all you’ve done, and we’re very, uh, impressed, with the work you put in tonight. Maybe come back to the dorm and get some rest, and tomorrow you’ll be—” “DON’T TOUCH!” Smolder snaps, as Gallus quickly recoils. “Sorry, sorry, out of line there, but really, it’s all so simple now, I just have to explain it, while it’s all still fresh in my mind and spread out here. Allow me this, and everything will be revealed.” The others turn to each other, then back to Smolder. “Go ahead,” Ocellus calmly beckons. “Good, good. So anyway I said to myself, I said ‘You have to start somewhere’ and so I did, I started by making a simple assumption that some clues were right and see where that led, and what better ones than the ones that give the most information, so I assumed that clues 3 and 4 were correct, because one places two gems exactly and the other places three gems with one correct, so I started there, you see?” “Yona see,” the yak replies apprehensively. “Great, I knew I could count on you seeing this, so anyway if clue 3 is correct then exactly two of those five gems are correct and in the right place but which ones … yes, which ones, indeed … so I have to just go through all of those possibilities and see where each led. So it could be that both onyxes are right, or both sapphires, or one of each, or maybe the emerald with exactly one of the others, so I went through all of those while assuming clue 4 was also correct. Got it?” “Uh, yeeeah, we got it,” Silverstream carefully replies. “Good, so start with both onyxes which would then be in spots 1 and 5, while there also wouldn’t be any sapphires or emeralds, but then by clue 2 or clue 5 there would have to be a ruby, since they’re not both false, but if there’s a ruby and two onyxes then by clue 4 there’s no topaz, and that means that clue 5 is invalid. But that forces clue 2 to be valid, and that means there has to be a diamond since there’s no sapphire and three rubies can’t all be in incorrect places, but that means there’s a ruby, an onyx, and a diamond in the solution and that contradicts clue 1 so there can’t be two onyxes. See, we’re on our way!” “Yes we are,” Gallus states. Right to a padded cell … “So it can’t be two onyxes, but what about two sapphires that would have to be in spots 2 and 3, while there then wouldn’t be any onyxes or emeralds? Glad you asked: by clue 4, there would then have to be a ruby and a topaz, which means clue 5 is invalid, but then clue 1 is valid and there’s a diamond somewhere and that contradicts clue 2 so there can’t be two sapphires, either. Now what about having exactly one onyx and one sapphire. Here’s where it gets complicated!” The group blinks in unison. “This is where it gets complicated?” Sandbar asks incredulously. “Oh my, yes, because now all I know is that there’s no emerald, as well as no second onyx or second emerald. But since either clue 1 or clue 2 has to be valid, there has to be either a diamond or a ruby in the solution. So we break it down into cases: Case 1 and Case 2. Now for Case 1, assume there’s a diamond in the solution, so no ruby. Well then of course clue 2 is invalid, but then so is clue 6 since now there can’t be a topaz but clue 4 implies now that there is a topaz, so we can clearly disregard Case 1.” “Are you sure you don’t want to take a break?” Ocellus soothingly asks. Smolders pays no mind as she continues. “But what about Case 2? Well now there’s no diamond and still no topaz, to go with no emerald, no second onyx, and no second sapphire. All that’s left is to have three rubies to go with the one onyx and the one sapphire, so clue 2 has to be invalid. All I had to do now was check all the combinations that existed.” “You mean permutations, right?” Gallus corrects. “Yes of course I did but thank you anyway so now there’s RRROS and RRRSO and a bunch of other ones but that would take forever right? Of course it would but then I remember clues 3 and 4 which combined force a ruby in spot 1 and then by clue 1 I know the onyx can’t be in spot 4 so another ruby must be in spot 2 and that means the sapphire is in spot 3 and the onyx is in spot 5 leaving the last ruby for spot 4, so RRSRO.” Silence pervades for several seconds. Gallus opens his mouth to speak, when— “I KNOW, RIGHT? That CAN’T be right since now clue 6 is false too, so all that proves that there can’t be exactly one onyx and one sapphire! So now there has to be an emerald, and that makes things a lot easier.” Everyone nods and murmurs agreement. Satisfied, Smolder continues on. “So now that there’s an emerald, can there also be an onyx? If so, there’s no sapphire and if clue 2 were valid there would have to be two rubies and a diamond but that goes against clue 4 so clue 2 has to be the invalid one now, but meanwhile by clue 4 either the ruby is in spot 1 or the topaz is in spot 5 and in either case an onyx has to be in the other spot while an emerald has to be in spot 4. Now the first of these would force a second ruby into spot 2 because of clue 1 but then by clue 6 the only spot left that could be correct is spot 3 which would be a second onyx which we know can’t be, so that rules out the ‘ruby in spot 1’ case, but what about the ‘topaz in spot 5’ case? Well now clue 6 forces either a sapphire into spot 2 or a second onyx into spot 3, but we know there’s no sapphire or second onyx. And just like that, there can’t be an onyx and an emerald.” “What about a sapphire and an emerald? With no onyx, clue 4 forces there to be a ruby and a topaz, so clue 5 has to be invalid, and if clue 1 is valid, there has to be a diamond too, and that goes against clue 6, too, so there can’t be a sapphire and an emerald. So you see what all this means, right?” … “IT MEANS CLUES 3 AND 4 CAN’T BOTH BE VALID!!!” … “How long did it take you to get this far?” Sandbar asks. “Oh about three hours, but now everything gets a lot simpler, because now I know that clues 1, 2, 5, and 6 are all valid, and in particular clue 2 is valid so there’s at least one or two rubies in the solution. Could there be only one ruby? Then there would also be a diamond and a sapphire and by the other ‘valid’ clues there wouldn’t be any emerald, onyx, or topaz, so clue 4 would be invalid and clue 3 would have to be valid with both sapphires correct, but that is counter to clue 2, so there has to be two rubies in the solution, but not three since they couldn’t all be incorrectly placed.” “Yona worried for Smolder sanity.” “Why? I’m perfectly fine, great even, because now I know that there’s two rubies AND I know they’re in spots 1 and 3. And since clue 1 has one gem correctly placed, it’s either the onyx in spot 4 or the diamond in spot 5. And now I can prove that clue 3 is the false one: if not, then there would have to be a sapphire in spot 2 and either an emerald in spot 4 or an onyx in spot 5, so either RSROO or RSRED. The first case can be ruled out by clue 3 itself: it has a sapphire and two onyxes, which is one gem too many; the second case can be ruled out because it goes against clue 5. And that means clue 3 is the false clue!” “It’s all downhill from here. From clues 5, there’s can’t be a sapphire and a topaz to go with the ruby, so from clue 4 there has to be an onyx, and now from clue 1 there can’t be a diamond, so by clue 2 there has to be a sapphire. And since the ruby in clue 1 is in the wrong place, the onyx must be in spot 4, and now by clue 6 the sapphire must be in spot 2. All that’s left is spot 5, and it can’t be a diamond, emerald, onyx, ruby, or topaz; it must be a sapphire. And that is the final solution:” “R-S-R-O-S” Smolder takes a number of deep breaths upon completing her reasoning. “And now I can go to sleep.” She then collapses to the ground. “Wha … where am I?” The young dragon’s eyes flicker open. She lifts herself up from her bed and looks around, finding herself in her sleeping quarters, a pale moonlight giving slight illumination to the room. “How did I end up …” She pauses upon seeing her fellow classmates asleep on the floor, atop a number of blankets and cushions. “Oh …” she says to herself, before turning back in. “Dragon feeling normal?” Yona asks the following morning. “Yeah … thanks again, everyone, for putting up with … whatever that was last night.” “No worries,” Sandbar states. “But I’m still curious … why?” “I dunno for sure … I guess with all the puzzle-solving we’ve been doing, and particularly what I’ve seen from the rest of you, I felt like I hadn’t really proven my own abilities. I kept reminding myself of Dragonlord Ember, and how she became ruler through intelligence and savvy rather than brute force. But she’s still competitive, like all dragons … and I wanted to prove to others, and to myself I guess, that I was capable of solving something like this.” “But why this puzzle?” Ocellus asks. “Well … you don’t all know this, but there’s a part of me that really … likes fashion and girly stuff. I wanted to impress professor Rarity with this, really blow her away.” “You blew all of us away,” Gallus states. “Even if I thought I could solve this one, I never would have had the patience to see it through to a conclusion.” “Yeah … thanks, everyone. But had I known, I probably wouldn’t have chosen this puzzle,” she says with a laugh. “Hopefully today will be less strenuous.” “I think professor Rainbow Dash has more buckball planned for today,” Silverstream says. “Lot of running and flying around, getting the heart pumping.” Yona chuckles. “Yep. Good break from strenuous activity!” Author's Note Mea culpa: this puzzle got away from me. I wanted to do a 'Mastermind' puzzle with the twist of a single false clue, but I never thought it'd be so difficult to solve (short of a guess-and-check method that wouldn't disprove an alternate solution). Next puzzle will be more along the lines of earlier ones, I promise.
Puzzle 16: More Buckball GamesHaving missed the previous ‘buckball day’, Gallus was secretly thrilled to be able to participate in the next one. Not that anyone else had to see this enthusiasm. “Looking forward to professor Dash’s class today?” Sandbar asks as the group make their way down the hallway. “Oh? What’s so special about today?” “Oh please,” Smolder retorts. “Don’t act like you don’t know what today is; we flat-out mentioned it this morning!” “Oh, right …” Gallus chuckles as they continue making their way. “Yeah, I want to make up for lost time. Show you all my mad skills as a defender.” “Griffin get real mad after Yona score on him!” The group laughs at the young yak’s counter. “Talk is cheap, Yona. You’ll have to prove it in competition.” “Bring it on!” Ocellus turns to Silverstream. “You think some creatures are getting the wrong lessons from professors Rainbow Dash and Applejack?” “Eh … I’m still waiting on that plumbing lesson we’re owed. Who do you think would teach us that?” Sandbar looks at Silverstream with a wry smile. “Maybe headmare Twilight needs to open a Friendship Vo-Tech school for you.” The group reaches the gymnasium, but they’re surprised to see Applejack waiting for them instead of Rainbow Dash, her focus on a large piece of paper held in her forehooves. “Come on in, y’all,” she calls out to the group, her eyes still glued to the paper. “Professor Applejack,” Ocellus calls out, “where’s Rainbow Dash? She usually runs the buckball games.” “We’re still playing buckball today, right … RIGHT???” Applejack quickly alleviates Gallus’s concerns. “Yeh, ye’re all still playin’ buckball, just like the last group. Ah agreed to take over for Rainbow Dash for this period.” “Did she take another trip to Las Pegasus?” Sandbar asks. Applejack chuckles. “Not this time, Sandbar. She muttered somethin’ about another chapter by some friend of hers that she met at the Daring Do convention. Said she needed some time to figure out things …” She pauses for a moment. “Actually, Ah think she might be askin’ fer yer help about this.” Gallus rolls his eyes, now aware of what’s to come later in the day. “Just to be clear,” he says, “we are playing buckball today, right?” “Yes yes, of course …” Applejack dismisses as she turns to her attention back to the paper. “Uh, professor?” Silverstream asks, “What’s that in your hoof?” “Huh … OH, sorry, everycreature. Ah’m just tryin’ to make sense of these here results.” “Results?” Ocellus asks. “The previous group of students what come down here for some buckball. Rainbow Dash was here for that group, and Ah got here just before they all headed out. It was a big enough class that she split ‘em up into four teams and had ‘em play each other. RD told me every team played at least once and no two teams played each other twice. When they were all done, she had a member of each team summarize the results and asked me to record all the game results right before she took off. Unfortunately …” “May I?” Silverstream asks as she reaches for the paper. Applejack obliges, and the students peruse the paper. ….GP...W L T...GF GA A....3.....1 0 2.....4 4 B....2.....0 1 1.....2 4 C....2.....1 1 0.....2 1 D “I take it ‘GP’ means ‘Games Played’,” Ocellus says, “with Wins, Losses, and Ties … but what do ‘GF’ and ‘GA’ stand for?” “They stand for ‘Goals For’ and ‘Goals Against’,” Applejack replies. “She gave me summary totals, but she didn’t give me actual scores. Shoot, I don’t even know who played who?” “Where are D’s results,” Sandbar asks. “Ah guess all the members of that team left before Rainbow did. So I got nothin’ regardin’ their scores, and that ain’t even the worst of it.” “Oh?” Yona asks. “What worst of it.” “Last thing Rainbow said before leavin’: she said one team was havin’ a prank at her expense, and all the figures in that team’s row are all off by 1. Guess that prank is on me now. Dangit, Rainbow!” “All right, so just so we’re all clear on all this,” Smolder states, “These results are missing anything regarding D’s games, one of the three rows is completely wrong with every number off by 1, and professor Rainbow Dash expects you to still recreate the results of every buckball game.” “Eeyup,” Applejack replies, mimicking her brother’s verbal tick. “Professor?” Silverstream asks, “Don’t buckball games usually go until some team scores 5 goals?” “Usually, but due to time constraints, Rainbow only had teams play until someone got to 3 goals. Further, she had a time limit in place, so some games ended before any team got to 3.” “Okay,” Silverstream says as she studies the paper a bit longer. “… aaand that’s it.” Applejack blinks. “What’s ‘it’?” “I figured out all your scores,” the hippogriff cheerfully replies. Author's Note This one's a bit easier than usual, but after that last one, I didn't want to go crazy. Simple question: which teams played each other, and what were the scores in each game?
Puzzle 16 solution ….GP...W L T...GF GA A....3.....1 0 2.....4 4 B....2.....0 1 1.....2 4 C....2.....1 1 0.....2 1 D Applejack takes the sheet of summary results back and peruses it. “Seriously? You gone figured out the whole thing based on this? Even with the missing row and mistake row?” “Uh huh,” Silverstream replies. The other students turn to each other and exchange odd looks. Finally, Gallus speaks: “I don’t mind sitting this one out; anyway, I came to play buckball, not go on about scores.” “And I still need a break after last night,” Smolder adds. “Anyone else mind if Silverstream solos this one?” The remaining students shrug, and Silverstream beams at the chance of demonstrating her knowledge. Applejack sighs. “Guess I shouldn’t be surprised anymore. Still, where do you even begin with this one?” “With row ‘A’,” the hippogriff replies. “That one has to be the ‘prank’ one.” “Why’s that?” “Because it’s not possible for ‘A’ to have the same number of goals for as goals against if they have at least one win and no losses; they would have to have more goals for. So all those numbers are off by one, and that means ‘A’ played two games, with no wins, one loss, and one tie.” “Well Ah’ll be …” “And more! Since ‘A’ has a loss and no win, they have to have more goals against than goals for … so they must have three goals for and five goals against.” “Huh … hold on while Ah jot this down.” Applejack picks up a pencil from the ground and fills in the correct information: ….GP...W L T...GF GA A....2.....0 1 1.....3 5 B....2.....0 1 1.....2 4 C....2.....1 1 0.....2 1 D “So Ah take it that means ‘D’ played two games as well?” “It has to be an even number, and it can’t be zero or four, so yes: ‘D’ played two games. And since the other teams combined for one win and three losses, ‘D’ has to win both its games.” “This is really helpful,” Applejack admits as she jots down the further information. “But that still don’t give us the final scores.” “Well for that, look at ‘A’s goal totals. They had one tie and one loss, so their loss had to be by two goals. Their two games had to have scores of either 0-0 and 3-5, 1-1 and 2-4, 2-2 and 1-3, or 3-3 and 0-2 … but the first two of those can be ruled out since no team scored more than three goals, and the last case is out since a team that got to three goals won immediately. So ‘A’s games had to be scores of 2-2 and 1-3. And since only ‘B’ tied as well, ‘A’ must have played ‘B’ to a 2-2 tie.” Applejack scribbles notes as fast as Silverstream recites her solution. “Now ‘C’ didn’t score three goals in total, so they didn’t beat ‘A’; ‘D’ must have, so ‘A’ lost to ‘D’ by a 1-3 score and ‘A’ didn’t play ‘C’ at all. That means ‘B’ didn’t play ‘D’, and their other game must have been a loss to ‘C’ by a 0-2 score. That leaves ‘C’ with one goal against them, so ‘C’ lost to ‘D’ by a 0-1 score, and that’s all the games played.” “Was anyone timing that?” Gallus asks. Sandbar looks up at a clock hanging on the gymnasium wall. “That was, what, 3 minutes?” “Sounds like a record,” Ocellus points out. “Good job!” “Ah’ll say … that was remarkable.” Silverstream beams. “Seriously, if there’s anything Ah can do to repay you …” “Maybe professor give Silverstream one-on-one lesson on plumbing,” Yona suggests. The students laugh, while Applejack ponders the request. “Sounds strange … but Ah do know mah way around a toolbox. Shoot, Ah helped out the spa ponies not too long ago. Ah’ll talk to Twilight about settin’ somethin’ up.” “Really? That’d be super-awesome!!!” Silverstream does aerial somersaults in celebration. “Hey now, save some of that energy for buckball. Alright, everycreature, split up an’ lets get yer bodies a workout for a change …”
Puzzle 17: QuibbleFic Part II (The Cursèd Code) Having mastered the puzzles of the bears and wolves, Daring Do found herself in the final chamber before escape and total freedom. However, it was the most heinous of traps, which nopony before her had ever solved: the Chamber of the Cursèd Code. Sandbar looks up from the hoofwritten outline and turns to Rainbow Dash. “Quibble Pants wrote all this?” “All sixty-four pages,” Rainbow laments. “I can appreciate the effort, but there’s a lot better work out there?” “Like yours?” Gallus asks “Exactly … I mean, yeah, there’re so many others that … you know … are better handled to …” Not wanting to continue watching her professor writhe in mental agony, Ocellus refocuses back to the story, reading aloud for those unable to crowd around the desk: Daring observes the room, her body shivering at the shattered bones of so many previous adventurers strewn about the room. Shaking off the macabre images of those unfortunate souls, she catches sight of a large bin containing … children’s blocks? ‘A strange medium’, she thinks to herself when musing about the inner-workings of the mind of the logitaur. ‘What to do with these’ … but the notes she had acquired had already prepared her for this final challenge. In a recess of the far wall, she makes out some sort of contraption, with a shelf capable of holding a number of the small blocks, but an ominous nozzle from some sort of weapon pointing directly at where one would have to stand to place said blocks. ‘Looks like the others also had only one attempt at this’, she ponders as she pulls the notes from her bag. “If X represents any sequence of blocks, then LXR will lead to X. Further, if X leads to Y, then note the following: KX leads to YL, QX leads to YR, DX leads to Y ‘doubled’, and CX leads to Y ‘curtailed’ … find the sequence that leads to itself.” ‘Find the sequence that leads to itself’ … she repeats the phrase over and over, pondering how to crack this infernal code … Ocellus looks up. “That’s it? Where’s the rest of it?” “Quibble didn’t send me the rest … all he cares about right now is that this final puzzle he created has a unique solution.” The students turn to each other quizzically. “What would he care about that for?” Smolder asks. “Probably still bitter about the last time he came across a puzzle like this.” “You mean, like in another book or something?” “Uh, yeeeah, something like that,” Rainbow replies to Sandbar, careful not to mention the true ‘something’. “He’s something of a picky perfectionist. So anyway, can you guys help?” The students take another look at the page containing the puzzle’s description. “Yona confused.” “I have some experience with this, so I can demonstrate.” “Really?” Gallus asks. “You read the same puzzle books as this friend of yours?” “Uh, again, something like that,” Rainbow carefully replies before going to the board and fitting a hoof-glove with a piece of chalk cradled within, allowing her to speak as she writes. “So ‘X’ and ‘Y’ each mean any sequence of letters. If we take the word PONY, for example, the first line means that LPONYR would return PONY. It’s like a machine, where LPONYR is the thing you put in and PONY is the thing you get out.” “So it’s basically a function, where LPONYR is the input and PONY is the output,” Ocellus states. “Yeah, sure, if that helps. So like, that works for anything that starts with ‘L’ and ends with ‘R’. So LYAKR would return YAK, LDRAGONR would return DRAGON, and so on.” “OK, Yona less confused.” “What about the other lines,” Silverstream asks. “The other lines all begin with knowing some in-and-out pair, like LPONYR and PONY. Putting a ‘K’ before the former returns the latter with an ‘L’ at the end, so KLPONYR would return PONYL. Also, KLYAKR would return YAKL, and … you get the idea.” “And the same for ‘Q’, right?” Ocellus asks. “So, QLPONYR would return PONYR?” “Hey, you catch on quick … guess I shouldn’t be surprised about that by now, should I?” “What about the other lines?” Gallus asks. “ ‘Doubled’ means repeated, so DLPONYR would return PONYPONY. And ‘curtailed’ means cutting off the end, so CLPONYR would return PON.” “And you can combine those later lines, right?” Ocellus continues. “So, QKLPONYR would return PONYLR, and DQKLPONYR would return PONYLRPONYLR, and CDQKLPONYR would return PONYLRPONYL, and …” She catches herself before going any further. “So what’s the point of all this?” Smolder asks. “ ‘Find the sequence that leads to itself’ … Daring Do has to find a sequence that returns the same sequence you put in.” “So, the output matches the input: it’s a ‘fixed point’, right?” Ocellus asks. “Exactly!” Rainbow exclaims, hiding her own confusion regarding those terms. “Professor?” Gallus asks, “Who would want to pick up an adventure novel just to solve puzzles?” Rainbow shrugs. “It takes all kinds, I guess. So can you guys help?” The group ponder the puzzle for some time … Author's Note Truth be known, this puzzle is a re-skinning of Quibble's fix of the final puzzle in the 'Chalice' fic. As no one had posted a solution to that one, I felt it OK to re-use it here. I had a similar puzzle in mind … but by pure coincidence, my wife found the Smullyan book that had gone missing ('Puzzles of Scheherazade'), and the puzzle I created was a pure re-skinning of the puzzle in that book. I was inspired by that puzzle (and similar ones from the 'Lady or the Tiger' book), but I didn't want to just re-use that exact puzzle.
Puzzle 17 solutionThe group ponder the puzzle for some time … “If X represents any sequence of blocks, then LXR will lead to X. Further, if X leads to Y, then note the following: KX leads to YL, QX leads to YR, DX leads to Y ‘doubled’, and CX leads to Y ‘curtailed’ … find the sequence that leads to itself.” … and after some time had passed, little has been accomplished. “Yona hope letters don’t appear in nightmares after this.” Sandbar ponders the yak’s comment. “Not sure what a nightmare involving letters would look like.” “With Headmare Twilight, the letter ‘F’ would probably keep her awake.” The group chuckles at Smolder’s insight, Rainbow Dash included. “So, this one’s really got you stumped, huh?” Rainbow asks the students. Ocellus continues to peruse the descriptions on the board. “Since LXR leads to X, I feel like we need some sort of X that leads to LXR. But after that, I’m stumped.” “It looks like everything begins with LXR,” Gallus adds, “so maybe we need to focus on what comes before LXR.” He picks up a piece of chalk and writes on the board: LYR --> Y ………… XLYR --> ??? “So ‘X’ is basically any sequence of letters that can act on a given string,” he explains. “Since the only other letters that do anything are K, Q, D, and C, ‘X’ probably has to be some sort of combination of those letters. But what do we want to get from them?” “Well we want to get back what we start with,” Smolder adds, coming to the board to replace the question marks: LYR --> Y ………… XLYR --> XLYR “Wait a minute,” Silverstream says as she approaches the board. “ ‘Y’ is anything right … but it’s a fixed anything. Like, we can put it anywhere on the right, so what if …” She erases one of the X’s and replaces with another Y: LYR --> Y ………… XLYR --> YLYR Yona studies the new line. “Soooo … if X is Y, then XLYR is YLYR, and we get solution, right?” “You’re right,” Sandbar adds. “So now it’s like a construction puzzle. How do we build YLYR from just Y?” “There’s two Y’s,” Ocellus notes, “so we’ll need a D somewhere. But not right away, because there’s an L between them.” “So add the L first,” Smolder adds. “That turns Y into YL, and if we then double, we get YLYL.” “And now we cut last L off,” Yona adds. “And replace with R!” “It looks like we have all the tools we need,” Gallus concludes. “Let’s see … add the L, then double, then cut off the last L and add the R. In order, those correspond to K, D, C, and Q … but they have to be written from right to left, right?” “Wait, left or right?” Silverstream asks. Gallus puts a palm to his face. “Let me rephrase that last part: they have to be written from right to left, correct?” “Oh … yeah, right … I mean, correct. That sounds right … I mean …” “We get it!” Smolder exclaims in slight exasperation. “So from right to left, that’s QCDK. That’s our X, right?” “And that means we should use QCDKLQCDKR,” Sandbar concludes. “Only one way to find out,” Gallus states as he writes down a check on the board: LQCDKR --> QCDK KLQCDKR --> QCDKL DKLQCDKR --> QCDKLQCDKL CDKLQCDKR --> QCDKLQCDK_ QCDKLQCDKR --> QCDKLQCDKR ... Rainbow Dash inspects the work shown on the board. “Wow … it looks like you did it! You guys are as good at this as Quibble was in that chamber.” The students turn to their professor. “What chamber?” Sandbar asks. “The marble chamber underneath the trap door in the Spectrum Catacombs in the library of the Crystal Empire where we …” Too late, Rainbow Dash catches herself before saying any more, aware that she had said too much already. “Uh, saaay, look at the time. Guess we’re done our lessons for today!” “You mean the friendship lessons that we seem to have stopped having ever since we started getting these puzzles?” “Exactly,” Rainbow answers Gallus. “Gotta run!” Quickly, she turns and flies out, leaving a rainbow-colored trail in her midst. The students look towards each other. Soon after, Smolder sighs and turns to the door. “Where Smolder going?” Yona asks. “To track down Rainbow Dash and tell her that Ember already shared with me her story regarding the Chalice.” Smolder proceeds out the door, leaving the others with still more questions. “If this about Daring Do,” Yona adds, “yaks already know she exist. Daring spotted outside of Yakyakistan in past.” “Griffinstone, too,” Gallus adds. “Not sure what the big deal is.” Silverstream and Sandbar look at the two with wonder in their eyes. “Wait,” the latter says, “Daring Do is real? Like, a real pony?” “Uh oh … maybe Yona say too much.” Gallus sighs. “Just … maybe, keep it to yourself for now.” He turns to Ocellus. “So have the changelings ever been paid a visit?” Ocellus pays no mind, her focus on the chalkboard. “Ocellus? You OK?” Her concentration broken, Ocellus turns to Gallus. “Huh? Oh, sorry, just that …” “I know that look,” Gallus says. “What did you find?” “It’s just … there’s multiple solutions to this puzzle.” “Really?” Silverstream asks, essentially speaking for the group. “Well, yeah. I mean, you could cut off and replace the last letter as often as you like, so the letters C and Q could be repeated in X. That means a string like QCQCDKLQCQCDKR would also lead to itself. Or QCKCDKLQCKCDKR, which adds an extra L before cutting it off for another R. You could even add and cut off a letter before doubling, so QCDKCKLQCDKCKR and QCDKCQLQCDKCQR work too.” “We should probably tell Professor Rainbow Dash to tell her friend about that,” Sandbar says. “At least if only one solution is intended.” “Story could ask for shortest solution,” Yona adds. “That would fix, right?” “Good thing there’s no actual machine to punish a different answer, then,” Silverstream states. “So far as we know,” Gallus adds. “Anyway, may as well head to our next class.” “You think someone else is gonna ask us to solve something?” Sandbar asks. Gallus turns to face his pony friend. “At this point, I’d be surprised not to have anyone ask us to solve something.”
Puzzle 18: It's 'The Mind'Seconds of dead silence pass by, with both Sandbar and Yona nervously looking at each other. Finally, Yona plays a card face-up on the table. It shows the number 97. “Aaah, nuts,” Sandbar exclaims, revealing a 95. The others groan in exasperation. “Yona knew Yona shouldn’t play,” the yak bemoans. “I had a high number,” the pony replies in defense. “I thought it’d be higher than yours. Smolder groans again. “Why did Headmare Twilight recommend this game to us. It’s hopeless!” “I think it’ll just take some more plays to calibrate our minds,” Ocellus answers. “I actually like this game.” Silverstream picks up the cardboard box that houses the cards. “ ‘The Mind’,” she says. “Weird name for a card game.” “As in, ‘Does anyone mind playing random cards to the table?’ ” Gallus says. “Where do ponies come up with these ideas?” “I like the idea,” Sandbar says. “It’s simple but thought-provoking: cards numbered from 1 to 100, everyone gets one card, and everyone tries to play their cards in order.” “Our headmare likes games that mess with our heads, doesn’t she,” Smolder states. “Yaks not like headgames,” Yona pouts. “Well I’ll try another round,” Silverstream says as she shuffles the cards anew. She deals one card facedown to everyone. “Yona sit this round out, if OK with others.” “Meh,” Gallus replies. “I’ll just take your card.” The five students each look at their own cards, and all wait for another to play the first. Seconds pass … “Look like everycreature got small cards,” Yona comments. “This is ridiculous,” Smolder states as she reaches for everyone’s cards, save the two in Gallus’s talons. She inspects the four cards now in her claws. “Yeah, we all had small numbers; this would’ve taken a while to start.” “So much for that round,” Sandbar says with a slight sigh. Smolder continues to inspect the cards. “Actually, this is interesting. Ocellus, your card was the sum of these other three cards: mine, Sandbar’s, and Silverstream’s. You’re a smart changeling; that give you enough to figure out our three cards?” Ocellus takes back her own card. “Too many possibilities; I have no idea.” Smolder takes a second look. “Well these three cards multiply to 360. That should do it, right?” “Maybe,” Ocellus replies as she ponders the new information … before shaking her head. “No … there’s still multiple possibilities. Gallus ponders the back-and-forth conversation between Smolder and Ocellus. “Huh … you know, this is interesting.” “What is?” Silverstream asks. “If I said what either of my cards was, Ocellus would have enough information to know all three of your cards. Maybe that gives you enough information.” “Hmmm …” Ocellus muses for a minute, before … “YONA KNOW ALL YOUR CARDS!!!” Author's Note So … what are Ocellus's card, Gallus's two cards, and the other three cards. For more information on 'The Mind', go here: https://boardgamegeek.com/boardgame/244992/mind
Puzzle 18 solutionGallus and the others turn to Yona, with Gallus in particular observing the young yak through skeptical eyes. “Really,” he deadpans. “Uh huh.” “You know all six cards, despite not seeing any of them.” “Yaks good at math.” “You get all these cards right, and I’ll spend next Yikslirbirtfest with you in Yakyakistan.” Yona’s eyes enlarge at the possibility. “Ah ah ah … I want proof that you know these, not some lucky guesses.” “Griffon on!” Yona declares. Having pondered her own card during the conversation, Ocellus looks up and turns to Gallus. “You may want to bring a scarf.” “And no help!” Gallus adds. “All right … what’s your reasoning.” “OK, well … first, nocreature have ‘1’, or somecreature would have played immediately.” The fivesome turn to each other and nod. “Can’t argue that,” Smolder comments. “So now, three cards multiply to 360, so for example, cards could be 2, 3, and 60, but then Ocellus card 65 and then she know three cards since no other group of three cards add to 65 and multiply to 360. Instead, Ocellus must have card where two groups of three cards exist that add to Ocellus card and multiply to 360.” “She’s right,” Ocellus adds, careful not to reveal her card. “I found two possibilities … but is my card the only card that leads to multiple possibilities?” “No … Yona do math in head and figure out three possible sums: 23, 25, and 29. Hang on, Yona write down.” The young yak grabs a pencil in her mouth and jots down several numbers: 4+9+10=23 and 5+6+12=23 3+10+12=25 and 4+6+15=25 2+12+15=29 and 3+6+20=29 Yona spits out the pencil. “So now Yona know Ocellus have either 23, 25, or 29; all other sets of three numbers that multiply to 360 have unique total when added.” Gallus blinks in astonishment. “You … figured all that out. In your head.” Yona shrugs. “Like Yona say, yaks good at math.” “You sure are,” Smolder admits. “But now what?” “Now Yona remember Gallus comment, that Ocellus know everything if she know either card of Gallus. But that can only be case if Gallus cards are 6 and 12.” “Why is that?” Sandbar asks. “Because 6 and 12 are only numbers that appear in all three rows. So if Ocellus know Gallus has 4, and Ocellus have 23 or 25, then Ocellus know other cards, but not if Ocellus have 29. Or if Ocellus know Gallus has 3; then Ocellus know if she have 25 or 29, but not if she have 23.” “Huh … Guess I’ll need that scarf after all,” Gallus admits, revealing his two cards to the others: 6 and 12. “And since only one combination have neither 6 nor 12, Ocellus must have 23 and other cards must be 4, 9, and 10.” Ocellus reveals her card, the 23, and the other three show their cards, confirming the solution Yona pointed out of 4, 9, and 10. “You never cease to amaze me,” Smolder admits. “Wait a minute,” Silverstream says, an idea popping into her head as she turns to Gallus. “The only way for this puzzle to make sense is for you to know all this when you made your comment … you did all that math in your head, too?” Gallus shrugs, his eyes darting around the room. “Well … yeah. I’m good at math; I just don’t like to brag about it. Remember when Trixie gave that guest lecture about her first visit to Ponyville?” Sandbar turns to Ocellus. “And when you commented that Gallus might need a scarf … you figured this all out, too, didn’t you?” Ocellus blushes. “I just like puzzles,” she meekly replies. Smolder’s gaze turns from Gallus to Ocellus to Yona. “I suppose when we all graduate, the three of you could turn around and open a school for mathematical logic.” “Don’t go selling yourself short,” Gallus replies. “I think we’ve all shown a lot over the past several weeks.” “Mostly the rest of you,” Sandbar says with a tinge of regret. “I feel like I’ve been more of a follower than a leader during most of these.” Ocellus ponders Sandbar’s comment. “If you feel that strongly about it, there’s still some other ordering tests that the rest of us took that you haven’t heard about.” Her words elicit a groan from the others. “What?” “You want to dredge up those meaningless tests?” Smolder asks. “Those things were completely pointless.” “But not useless,” Ocellus corrects. “I think I could create an interesting puzzle based on some of those.” Author's Note Ocellus alludes to this previous one-off logic puzzle, which spawned this collection: A Question of Rankings
Puzzle 19: Double Vision, Part 1Daylight shines through the window, a natural alarm clock for Sandbar as he rises from his bed and goes through his daily before-school routine. Entering the kitchen area, Sandbar mumbles greetings to his fellow non-pony students. “Mornin’, Silverstream. Mornin’, Yona, Mornin’, Gallus. Mornin’, Silverstream. Mornin’ …” His eyelids snap open as a realization hits him. He turns around to see a giggling Silverstream, then back to his front to eye the hippogriff chuckling at him. “What gives?” he suddenly bellows, as all around him his friends break out into laughter … soon causing Sandbar himself to join in. “It’s a challenge,” the Silverstream in front of him states. “Can you figure out which Silverstream is me, and which one is … oh wait … shoot!.” This Silverstream smacks her head with a claw as the other one flutters over to her. “Don’t worry about it,” the other hippogriff states before emanating a flash; an instant later, Ocellus hovers next to her friend. “I meant to say ‘which one of us is Silverstream’,” the ‘real’ Silverstream notes. “Don’t sweat it, that wasn’t the test anyway,” Ocellus replies. “Test? What test?” Sandbar asks. “You said you wanted a chance to prove your mental acuity, right?” Gallus states. “Wait, you’re saying Sandbar cute? Don’t we all know that?” Yona asks, causing the griffon to suppress a laugh. “He means Ocellus created a challenge for Sandbar … well, several challenges, I guess,” Smolder corrects. Sandbar turns to Ocellus. “What about those ‘ordering tests’ you mentioned last night?” Ocellus blushes. “I just couldn’t think of anything to come up with that we hadn’t already done. And believe me, I thought about those tests for hours … but it occurred to me that I could create a whole new test for you, based on a test that Thorax gave our professors.” “Wait,” Silverstream interrupts, “our professors were given tests?” “Thorax told me about a ‘Spot the Changeling’ challenge he did, where he would pick one of the six ponies and take her place; the others had to figure out which of them was Thorax in disguise.” “Couldn’t they just ask each other questions that only the professors would know?” Gallus dryly asks. “They put conditions in to prevent that. In any case, I thought we could do that, but on a smaller and more logic-based scale.” Ocellus flutters out to the foyer, returning with the deck of ‘Mind’ cards from the previous night. She quickly searches the deck, pulling out four cards: the 1, 2, 3, and 4. “Here’s how this will work,” Ocellus starts. “You’ll close your eyes, and when you open them you’ll see two Silverstreams, one of which will be me. Each of us will make a number of statements, and then you have to figure out which of us is Silverstream and which of us is me, Ocellus.” “Seems simple,” Sandbar says, “but what’s with the cards?” “This is where the ‘logic’ part comes in. Each of us will have one of these cards in front of us. Any statement made by somecreature with an even-numbered card in front of her will be true, and any statement made by somecreature with an odd-numbered card in front of her will be false. In other words, having the 2 or 4 means true statements and having the 1 or 3 means false statements. And you have to figure out which card we each have as well.” “Whoa … that sounds a lot harder,” Sandbar notes. “We’ll do five rounds … let’s see how you do. Ready?” Sandbar closes his eyes. “Ready.” Ocellus turns to Silverstream. “Are you ready?” “I hope so,” the hippogriff says somewhat apprehensively. “I did try to remember what you told me to say.” Ocellus smiles. “You’ll do fine,” she assures her friend. The sole pony’s eyes now shut, Ocellus changes her appearance to mimic that of her hippogriff friends. Each takes a card, and the rounds begin: [In each round, one Silverstream appears on Sandbar’s left (L) and one appears on his right (R).] [Round 1] L: “I am Silverstream. I have the 1.” R: “I am Ocellus. I have the 2.” [Round 2] L: “I am Silverstream. I do not have the 3.” R: “I am Ocellus. I do not have the 4.” [Round 3] L: “I am Silverstream. Both of us have even-numbered cards.” R: “I am Silverstream. My card’s number is the smaller of the two.” [Round 4] L: “Either I am Silverstream or I have the 1.” R: “Ocellus has the 1. Neither of us has the 4.” [Round 5] L: “Either Silverstream has the 1 or Ocellus has the 3. Neither of us has the 2.” R: “Either Silverstream has the 3 or Ocellus has the 4. Neither of us has the 1.” Author's Note Sorry for the delay; I had an idea for a more traditional Fic and spent time on that before coming back to these. The puzzle is original, but the idea is not: it's a combination of the 'Tweedledum & Tweedledee' chapter of "Alice in Puzzleland" by the late, great, Raymond Smullyan and the Fic "Spot That Changeling" by Inspector Brown. A link to the latter is provided, in case you missed it: Spot That Changeling
Puzzle 19 solutionThe sole pony’s eyes now shut, Ocellus changes her appearance to mimic that of her hippogriff friends. Each takes a card, and the rounds begin: [In each round, one Silverstream appears on Sandbar’s left (L) and one appears on his right (R).] [Round 1] L: “I am Silverstream. I have the 1.” R: “I am Ocellus. I have the 2.” Sandbar ponders the two sets of statements. “Oh wait,” he quickly says, pointing a hoof towards the one on his left. “You have to by lying; nocreature would claim to have the 1. So you’re Ocellus, and you can’t have the 1, so you have the 3.” He turns to the one on his right. “And that means you’re Silverstream, and you’re lying too so you have the 1.” One flash of light later, the ‘Silverstream’ on his left reverts to her natural, changeling form. “Good job,” she remarks. “But a pretty easy round, right?” Gallus states. “They’ll get harder,” Ocellus promises. “Close your eyes again.” Sandbar obliges, and the second round starts: [Round 2] L: “I am Silverstream. I do not have the 3.” R: “I am Ocellus. I do not have the 4.” Sandbar considers the two sets of statements, but quickly replies: “This is like the last one, but in reverse, right?” He turns to the one on his right. “Nocreature would lie and claim to not have the 4, just like nocreature would truthfully claim to have the 1. So you’re telling the truth, and you’re Ocellus with the 2.” He turns to the one on his left. “And that makes you Silverstream, and you’re truthful as well so you have the 4.” Another flash of light reveals Ocellus as the ‘Silverstream’ on his right. “Right again,” she comments. “Still pretty simple,” Gallus reiterates, drawing a glare from Sandbar. “Whaaat?” “You could be a little more supportive, you know.” The dragon points out. “Ready for Round 3?” Ocellus asks. “Can I just check the note cards you gave me one more time?” Silverstream asks. “These later rounds get harder to remember.” “Go ahead,” the changeling replies. “Ooo, ooo, Yona want to play too!” She turns to Sandbar. “Unless you want to keep doing these yourself. Yona understand if you do.” “I don’t mind,” Sandbar responds as he shuts his eyes. Smiling, Yona does likewise; moments later, the third round begins: [Round 3] L: “I am Silverstream. Both of us have even-numbered cards.” R: “I am Silverstream. My card’s number is the smaller of the two.” Sandbar turns to Yona. “What do you think?” “Well, Silverstreams can’t both be Silverstreams, right? So one liar.” “Sounds good to me … and that means the one on the left is lying when she says they both have even-numbered cards. So the one on the right is the real Silverstream.” “But what numbers?” Yona asks. “Oh wait, right-hoof Silverstream truthful, so her card 2 or 4 and left-hoof Silverstream has 1 or 3.” “But the one on the right has the smaller number,” Sandbar reasons. “So Ocellus is on the left with the 3 and Silverstream is on the right with the 2.” Ocellus [on the left] reveals her true self once again. “Not so easy this time, was it?” she comments, her eyes focused on Gallus. “I’ll admit that there was more to that one,” the griffon says. He turns to Smolder: “You wanna try to solve the next one?” “Sure, why not?” the dragon replies. She and the others all shut their eyes as the fourth round commences: [Round 4] L: “Either I am Silverstream or I have the 1.” R: “Ocellus has the 1. Neither of us has the 4.” Yona looks at Gallus. “OK, smart guy. You know answer?” “Uh …” he stammers, before admitting: “All right, this one has got some kernels on the cob.” “What do you think,” Smolder asks of Sandbar. The young pony thinks about the statements made by the two Silverstreams. “Well, suppose the Silverstream on the left was lying. That would mean she wasn’t Silverstream and she didn’t have the 1; she would have to be Ocellus, and she would need to have the 3. And that would like the other Silverstream’s first statement false, so they’d both be liars … but then her second statement would be true, and that can’t be!” “I’m impressed,” Gallus says with conviction. Yona picks up where Sandbar leaves off: “That mean Silverstream on left truthful, so she can’t have 1; it must be real Silverstream, and one on right Ocellus.” “And since nocreature would claim to have the 1,” Smolder adds, “Ocellus is lying and must have the 3. And that means one of them does have the 4; it must be Silverstream.” Ocellus [on the right] changes back to her usual form. “Well done!” Silverstream turns to Ocellus. “We still have one more round, right.” “We do … and it’s the hardest round. All of you ready?” “I hope so,” Gallus says, hoping to redeem himself in the eyes of the others. The group of four shut their eyes and the final round plays itself out: [Round 5] L: “Either Silverstream has the 1 or Ocellus has the 3. Neither of us has the 2.” R: “Either Silverstream has the 3 or Ocellus has the 4. Neither of us has the 1.” “Whoa,” Sandbar quietly says. “Where do we begin?” “We gotta begin somewhere,” Gallus starts. “Let’s just say the one on the left is telling the truth. What would that lead to?” Smolder continues: “Well if the Silverstream on the left is telling the truth, then that one has to have the 4 since neither would have the 2. And the Silverstream on the right would have to have either the 1 or 3, making that one a liar.” “And since Silverstream on right liar, that one have 1 and not 3,” Yona follows. Sandbar puts the information together: “So the one on the left is truthful with the 4 and the one on the right is lying with the 1.” He ponders for a minute … “That doesn’t work!” The other three turn to Sandbar, awaiting an explanation. “If Silverstream is on the left, then her first statement is false: she can’t have the 1 and Ocellus couldn’t have the 3 since she would have the 1. But on the other hoof, if Ocellus is on the left, then Silverstream’s first statement would be true since Ocellus would have the 4.” “OK,” Gallus says, “that’s proof enough for me that you’re as adept at this as the rest of us. “So what all that mean?” Yona asks. “We do all this for nothing?” “Not for nothing,” Smolder corrects: “Now we know the one on the left is lying. And that means the one on the right has the 2.” “And that means the one on the left has the 3, since neither can have the 1,” Gallus adds. Sandbar concludes the process: “Silverstream can’t be on the right, ‘cause that would make her first statement false. Silverstream must be on the left with the 3, and Ocellus might be on the right with the 2.” At long last, the ‘Silverstream’ on the right morphs back into Ocellus. “Great job! she exclaims. “I really didn’t think you were going to get that.” “I had good help,” Sandbar replies somewhat bashfully. “Sandbar stop selling self short,” Yona pleads. “Sandbar smart, like all Yona’s friends.” “And Yona herself,” Gallus adds. “That looked like fun for you guys,” Silverstream states. “Can I play along too?” Ocellus ponders the hippogriff’s request. “I do have some more challenges, but I thought those might be a bit too much. Besides, if you’re playing, then who am I copying?” Smolder raises a claw …
Puzzle 20 solution“OK,” Ocellus begins, “for these five rounds, you’ll each see two Smolders, one of which will be me. Each of us will have a card from 1 to 4, and of course, we won’t have the same card. You’ll have to figure out which of us is really Smolder, as well as which card each of us has.” The foursome look at each other quizzically, before Silverstream addresses the changeling: “Uh … how is this any different from what we just did?” “One small change, but it makes a big difference: this time, if I have the 1 or 3 then my statements will be false, and if I have the 2 or 4 then my statements will be true.” Ocellus’s comment is met be more strange looks. “That exactly how first five rounds went,” Yona points out. “Oh … yeah, that part’s the same.” Ocellus begins to blush as she continues: “What’s different is—” “What’s different,” Smolder interrupts, “is that if I have the 1 or 3 then my statements will be true, and if I have the 2 or 4 then my statements will be false. It’s the opposite of Ocellus.” “That … really doesn’t sound all that much more challenging,” Gallus states. “Hey, I’m game either way,” Sandbar replies. “Go ahead and start.” The foursome collectively close their eyes, and the rounds begin: [In each round, one Smolder appears on the group’s left (L) and one appears on their right (R).] [Round 1] L: “I am Smolder. I have the 1.” R: “I am Smolder. I have the 2.” It takes only a few seconds for Silverstream to respond: “Well … one is lying and one is telling the truth. I’m sure of that.” “Shouldn’t be hard to figure out which is which,” Gallus adds. “Which one said something that couldn’t be said?” Sandbar ponders for a moment … “The Smolder on the right said she had the 2 … but the real Smolder wouldn’t truthfully say she had the 2, so that has to be Ocellus. And, she has to have the 1 or 3.” “But then Smolder on left truthful,” Yona adds, “so Smolder on left have 1. That only leave 3 for Ocellus on right.” In a flash, the Smolder on the right changes back into Ocellus. “Not bad. Let’s see how you do on the next one.” Eyes close, next round begins … [Round 2] L: “I am Smolder. Ocellus has the 3.” R: “I am Ocellus. Smolder does not have the 4.” Silverstream again begins the thought process: “They each to be somecreature different, so they’re either both truthful or both lying this time.” “But if both truthful, Ocellus have 3,” Yona points out. “That not right.” “So both are lying,” Gallus concludes. “Ocellus is on the left and Smolder is on the right. Now as to their numbers …” “They both lied,” Sandbar states, “so Smolder does have the 4. And since Ocellus can’t have the 3, she has the 1.” Another flash, and the Smolder on the left reverts to her natural form. “It feels weird to say ‘Ocellus has’,” she admits. “Meh,” Smolder says. “I know dragons who like to refer to themselves in the third-creature. No big deal.” “Just big egos,” Gallus says to himself, before quickly backtracking upon noticing Smolder’s death-glare. “Sorry, sorry … old habits … my apologies!” “Wellllll, you may not be too far off with certain dragons.” “We’re all cool, right?” Sandbar asks. “It’s fine,” Smolder says. “Let’s just get back to the rounds …” [Round 3] L: “I have the 1. The other card is not the 2.” R: “I have the 3.” “Wait,” Yona says, “neither one claimed to be Smolder or Ocellus. This round actually have solution?” “There has to be a solution,” Gallus states, “… and it begins with the fact that Ocellus is lying.” “How do you know that?” Silverstream asks. “Ocellus would never truthfully state she has an odd number, either 1 or 3.” “Good point,” Sandbar says. “Does that mean Smolder is telling the truth?” Yona ponders for a moment … “No! Smolder could still be liar; all we know is Ocellus definitely liar.” “All right,” Silverstream continues, “so which one is Smolder? Could she be on the left?” Gallus thinks the possibility over. “Suppose she was on the left and telling the truth. Then she does have the 1 … and if Ocellus is lying, she would have to have the 3.” “But that would make her statement true,” Sandbar adds. “That doesn’t work.” “So now say she’s lying,” Gallus continues. “Then the ‘other’ card is a 2, which would mean Ocellus is telling the truth … that doesn’t work, either.” “So that mean Smolder not creature on left,” Yona concludes. “Ocellus on left, and both statements false; that mean other cards is 2.” Silverstream excitedly wraps up the solution: “That means Smolder is lying and has the 2, and Ocellus being on the left means she has the 3!” The Smolder on the left changes back into Ocellus as the one on the right comments: “Wow, you guys really are getting good at this.” “Two more rounds,” Ocellus points out as they begin anew … [Round 4] L: “The sum of our numbers is 5. My number is the smaller of the two.” R: “One of our numbers is double the other. My friend’s card is not the 1.” Silence pervades for a full minute. Finally … “The one on the left is lying,” Gallus states. “How do you figure that?” Sandbar asks. “If she was truthful, their cards would have to be one even and one odd, and that would force both of them to be telling the truth. But that’s impossible, since the only ways to have one card be double the other is with the 1 and 2, or the 2 and 4 … and neither of those add up to 5.” “Whoa,” Silverstream says, “that is some high-level thinking there!” “So where we go from here?” Yona asks. “What if both of them were lying?” Sandbar asks. “Then they’d still have one even and one odd card, right?” Gallus nods. “But,” Sandbar continues, “we already know their cards don’t add up to 5, so the cards would have to be either 1 and 2, or 3 and 4, right?” “Sounds good to me,” Gallus replies. “Wait,” Silverstream says, “1 and 2 are out, right? ‘Cause, that would make the ‘double’ statement true.” “That leave 3 and 4,” Yona adds, “but that make other statement true, about friend card not being 1.” “So all that means the statements from the Smolder on the right can’t be false; they must be true,” Gallus concludes. “That means the numbers are either both even or both odd,” Sandbar states. “They can’t be both odd, since one is double the other, so they must be 2 and 4.” “And since the Smolder on the left lied,” Silverstream adds, “her number is the bigger of the two. And that means—” “Smolder on left with 4 and Ocellus on right with 2,” Yona concludes. Ocellus (on the right) reveals herself. “That was amazing! I really thought I had you stumped on that one.” “That just leaves one final round,” Smolder says. “Let’s see if you can go five-for-five …” [Round 5] L: “Either I am Smolder or I have the 3.” R: “I am Smolder and I do not have the 4. Neither of us has the 2.” “I don’t like conjunctions,” Silverstream admits. “They make my head hurt with these puzzles. “We gotta start somewhere,” Gallus states. “What if the one on the left is the real Smolder?” “That mean one on left truthful,” Yona replies. “Also, one on right lying if that one actually Ocellus.” “And if she’s lying,” Silverstream adds, “then one of them has to have the 2.” “But that’s not possible,” Sandbar insists. “Either Smolder would have the 2, and that would force her to lie, or Ocellus would have the 2, and that would force her to be truthful. Neither of those works with these assumptions.” “So much for figure out which one is Ocellus,” Gallus says. “The one on the left is Ocellus.” Rather than wait for the final solution, Ocellus (on the left) changes back. “Wow … I really thought this was the hardest one.” “They still need to figure out the cards,” Smolder points out. “But if Ocellus truthful, she would have 2 or 4,” Yona says. “That not work here, so Ocellus lying and have 1 or 3 … but not 3, since that make statement true. So Ocellus have 1.” “Meanwhile,” Silverstream adds, “the only way Smolder could by lying is if she has the 2 or 4. If she had the 2, her first statement would be true, and if she had the 4, her second statement would be true. So Smolder must be telling the truth, and her card has to be the 1 or 3 … but not the 1, since Ocellus has it, so she has the 3.” Ocellus and Smolder each reveal their cards, confirming the final piece of reasoning. “That was actually a lot of fun!” Silverstream states. “So what’s next?” Ocellus shrugs. “There’s not much more I can do with this format? I mean, I could make it so one of us is truthful with 1 and 4 and the other is truthful with 1 and 3, or some other combination, but really, it’s all pretty much the same.” “Maybe we need two pairs, like two Sandbars and two Galluses,” Yona says. The others chuckle at the thought. “Maybe we should just have two of all of us,” Gallus says. “That would mess with someone’s mind.” “But if we did that, who’d be left to actually solve the puzzle?” Sandbar asks. Ocellus takes all the comments in as her mind races …
Puzzle 6 solutionRarity gives Gallus an odd look. “I beg your pardon?” Gallus pauses for a few seconds before responding. “Professor, how long did it take for you to figure out an appropriate ‘ensemble’ for your client?” “Oh … I don’t remember, really … maybe, a couple hours?” “I see,” Gallus replies. He turns to his fellow students. “Ocellus, did you write down the requests made by Rarity’s client?” Ocellus looks up from her sheet of notebook paper, gives a sheepish smile, and nods. “Professor, may I come to the board?” “Well … uh … yes, that would be OK.” Gallus takes the paper from Ocellus’s desk and casually writes down a number of sentences on the board: No dress should have more than one gem type, and at least three different gem types had to be used for the dresses, though not necessarily all four gem types. C must wear sapphire, unless D wears ruby. D must wear emerald, unless B wears the same gem type as E. E must wear either sapphire or emerald, with A wearing diamond if E wears sapphire and with A wearing ruby if E wears emerald. C should wear ruby, unless B wears either sapphire or emerald, and if B does wear sapphire then C should wear diamond, and if B wears emerald instead then C should wear emerald as well. “That’s all the information, correct?” Gallus dryly asks his fellow students. They nod in unison. “Ocellus, care to begin?” Ocellus responds quickly: “The last request indicates that C should wear either ruby, diamond, or emerald. That means that C should not wear sapphire. But then there’s the request that C must wear sapphire unless D wears ruby … since C cannot wear sapphire, that means that D must wear ruby.” “Wow … I’m very impressed with your acumen, Ocellus. And yes, D did indeed wear ruby. But then—” “Sandbar,” Gallus continues, “what do you see?” Sandbar studies the requests for another minute. “Ocellus said that D wore ruby. But that one requests says that D should wear emerald, unless B and E wear the same gem type. And that means that B and E have to wear the same gem type.” Gallus turns to Yona. “Your turn.” “What? Oh, yeah, Yona ready. Uh … B and E wear same gem, but E wear either sapphire or emerald, so that mean B also wear either sapphire or emerald. And if B wear sapphire, then C wear diamond, and if B wear emerald, then C wear emerald.” “Smolder?” The young dragon sighs. “If B and E both wear emerald, then C wears emerald as well. And if E wears emerald, then A wears ruby. But D already wears ruby, so everyone would be wearing either emerald or ruby, and that can’t be since at least three gem types have to be used.” “Silverstream? Care to finish?” The hippogriff enthusiastically continues: “That means B and E must both wear sapphire instead, and since B wears sapphire that means C wears diamond, and since E wears sapphire that means A wears diamond.” “So in summary,” Gallus concludes as he strolls back to his seat, “A wears diamond, B wears sapphire, C wears diamond, D wears ruby, and E wears sapphire.” Rarity gapes in astonishment before sputtering with a response. “What? … but … I mean … that’s all correct, but … I spent hours on … and you just …” “We’ve had a lot of practice as of late,” Gallus casually admits. “Now I can’t believe what I’m about to say, but can we get back to discussing fashion?”
Puzzle 7 solution (1) Gallus was higher than Sandbar (2) Sandbar was higher than Ocellus (3) Ocellus was higher than Yona (4) Yona was 2 higher than Smolder (5) Smolder was 3 higher than Sandbar (6) Gallus was 4 higher than Yona (7) Ocellus was 5 higher than Gallus “Did I really blurt all these statements out?” Silverstream asks as she peruses Twilight’s notes. “In about 5 seconds, yeah,” Smolder replies. “So, Ocellus, care to explain?” Ocellus opens her mouth to answer, but Gallus puts a wing on her shoulder. “How ‘bout someone else take the lead for a change?” He turns to Smolder. “Why don’t you begin this one?” Smolder groans. “What does it matter? For that matter, why don’t we just ignore this and hear our order from Fluttershy tomorrow?” “If you haven’t noticed, we’ve been getting involved in a lot of these puzzles as of late, almost as if the universe wants us to keep our minds honed. I’m just saying we shouldn’t be overly reliant on a single creature to handle all the mental heavy lifting.” The others give a quizzical look towards the griffin … before he and the others burst into collective laughter. “Couldn’t keep a straight face with that one,” Gallus mirthfully admits. “OK, OK, Ocellus, care to start us off.” “Actually, I do want to begin,” Smolder insists. “I mean, you never know, right?” she adds with a grin. After a further round of chuckles dies down, Smolder inspects the list of sentences. “It was nice of Headmare Twilight to number the sentences for reference. So, they’re only true if both students mentioned are either above or below Silverstream at the same time, right?” “That sounds right,” Ocellus says. “OK then … where to begin …” Smolder inspects the statements for a few minutes before coming to a realization: “Wait … if sentence 7 was true and Ocellus was 5 higher than Gallus, then Silverstream would have to be between them. And that can’t be … so that statement is false.” “Nice,” Sandbar states. “Can I give it a go?” “Go ahead,” the young dragon replies as Sandbar peruses the statements. “So … I see Smolder’s name in sentences 4 and 5, so if both were true, then Yona would be 5 higher than me … but that can’t be either, so at least one of those statements is false. But that’s all I got. What now? Anyone else want to jump in?” “Yona give shot,” the yak replies. “… false statement mean students on opposite side of seapony. That mean Ocellus and Gallus on opposite sides, and Smolder opposite either Yona or Sandbar. So at least two above and two below seapony … Silverstream either third or fourth!” “That’s brilliant!” Ocellus states. “Honestly, I was a little stuck there. OK, since Silverstream is either third or fourth, then statement 6 can’t be true, because that would force Gallus and Yona to be on the same side, which could only happen if Silverstream was either first or last. And that forces Gallus to be opposite Yona.” “OK, let’s take another look at all those false statements,” Gallus suggests. “The last two statements are false; we know that now. That means that I’m on the other side of both Yona and Ocellus … so they have to be on the same side of Silverstream. And that means that statement 3 is true: Ocellus is higher than Yona.” “And if statement 4 was true as well,” Ocellus adds, “then there would be four of us on the same side of Silverstream: myself, Yona, Smolder, and whoever was between Yona and Smolder … but then Silverstream couldn’t be third or fourth in that case. So statement 4 is false, and Yona is opposite Smolder.” “But statement 5 can’t be true, either,” Sandbar states. “That would also force Silverstream to be somewhere other than third or fourth. And that means Smolder is opposite me.” “Hey, can I work on this?” Silverstream asks. “They were my statements, right?” The others laugh. “Our apologies,” Gallus states with a wry smile. “By all means.” “OK, OK … so all the last four statements are false, so Ocellus is opposite Gallus who’s opposite Yona who’s opposite Smolder who’s opposite Sandbar. So … it’s like, every other student is on the same side, right? Because two wrongs make a right, or something like that?” “Sounds about right to me,” Smolder admits. “Cool! … so that means that Ocellus, Yona, and Sandbar are all on one side of me while Gallus and Smolder are on the other side. That means that Gallus and Sandbar are on opposite sides, so statement 1 is false; Sandbar must be higher than Gallus … and that means that Ocellus, Yona, and Sandbar are one, two, three in some order, I was fourth … aww, I was only fourth? … and Gallus and Smolder are five, six in some order.” “We’ve all been doing well on all these quizzes,” Ocellus reassures. “I’m sure we all passed with flying colors.” “I hope so,” Smolder adds with a tinge of derision. “Calm down, everyone,” Sandbar says. “Let’s finish this, huh?” “Yona finish,” the yak replies. “So, Sandbar, Ocellus, and Yona all above Silverstream … so statements 2 and 3 are true, so Sandbar first, Ocellus second, and Yona third.” “That just leaves Smolder and myself,” Gallus states, “… but since Yona was third, Smolder couldn’t be fifth since statement 4 is false. That means I was fifth, and Smolder was sixth.” He turns to Smolder. “No offense meant.” “Meh … none taken,” Smolder reassures. “I don’t think professor Fluttershy has ever given a student a bad grade.” Ocellus muses for a moment. “About what I’d expect from the bearer of the element of Kindness.” “So … that’s it, right?” Silverstream asks. “I mean, we did it, didn’t we.” “Looks like it,” Gallus answers. “In order: Sandbar-Ocellus-Yona-Silverstream-me-Smolder. I hate to say it, but I’m starting to enjoy doing these.” “Wow,” Yona says, “so which professor bearer of element of Miracles?” The group shares one final laugh before eventually retiring for the night.
Puzzle 9: Ocellus is Stumped“Now everycreature take a number,” Pinkie instructs the class, “and starting tomorrow, you’ll all take turns baking cupcakes for the class.” “I’ve never baked anything before,” Sandbar reminds his professor. “I’m not sure my batch will turn out OK.” “It’s not about the quality, silly!” Pinkie reassures. “It’s about the act of doing something for your friends. And if they don’t come out exceptionally well, then you have a fun story to share.” “I’m with Sandbar on this,” Gallus states. “What if my ‘fun story’ gets told in a hospital ward?” Smolder dismisses Gallus’s concern: “Meh … I can taste-test if everyone else wants. We dragons have strong stomachs; it takes a lot to make us sick.” “Is that a challenge?” Gallus slyly asks. The group laughs at the veiled threat. “See … even this discussion is fun,” Pinkie declares. “Now come on up, one number per student.” Each student, in turn, randomly selects a number from Pinkie’s desk. Each peruses his or her own number, without seeing any other number selected. “Okie-dokie, that’s all for today! See you bright and early tomorrow!” Having concluded classroom proceedings for the day, Pinkie pronks her way out the classroom door. “At least she’s not going on about the walls again,” Silverstream states. “So when everyone else go?” Yona asks. “Yona second,” she adds as she shows off the ‘2’ to the others. Smolder leans over to look at Sandbar’s number. “My number is three less than Sandbar’s,” she states. Silverstream opens her mouth to speak, but Gallus stops her and turns to Ocellus. “OK, you’re the puzzle-lover here. I’m guessing you can figure out everyone’s number just based on those statements, right?” Ocellus looks at her number and ponders the previous statements. “I … I …” “Is everything OK?” Sandbar asks. “I … I don’t know.” “You don’t know if you’re OK?” Silverstream asks. “No … I don’t know everyone else’s number. In fact, I don’t know anyone else’s number, besides my own and Yona’s. That’s not like me.” “ ‘Huh’,” Gallus says, taking a second look at his own number. “If that’s the case, then I know everyone’s number.” “YOU DO???” the other students ask in unison. “I do … and if I add that my number is higher than Ocellus’s, everyone can now figure out who has what number.” Author's Note So … what number does each student have? (All numbers are from 1 to 6.) (Admittedly a bit on the easy side ... I'll try to make the next one more challenging.)
Puzzle 12 solution Unicorn A said “None of the others have the same hometown as me.” Unicorn B said “F and I have the same hometown.” Earth pony E said “C is from Utopia.” Earth pony F said “D and E have different hometowns.” The students ponder the four statements, aware that the two pegasi, C and D, said nothing, and also aware that of each pair of like-type ponies, one was truthful and one lied. “There doesn’t seem to be much to go on,” Silverstream states to the group. “When I solve puzzles like this,” Ocellus states, “I usually start with statements one makes about one’s self.” “Like what B said?” Yona asks. “Exactly. B said that she had the same hometown as F. Let’s start there.” The group ponder in silence for a minute, before Sandbar speaks up. “If B was truthful, she’d be from Utopia … and so would F.” Smolder continues: “And that would make her a liar. It would also force E to be a truth-teller, and that would mean E was from Elysium.” “But if F was a liar,” Silverstream adds, “then D and E would have the same hometown … they would both be from Elysium.” “And if E truthful,” Yona says, “C from Utopia.” Gallus concludes the train of thought: “So C would be from Utopia and D would be from Elysium … neither one would be a truth-teller, and that’s contrary to what we were told from the outset, that one of the two pegasi is truthful.” “OK then,” Ocellus states, “that means that B is a liar and A is truthful. So A is from Utopia … the only pony from Utopia, as it turns out.” “But if no one else is from Utopia,” Sandbar says, “then E lied. That means F is truthful, so F is from Elysium.” “But B lied,” Smolder adds, “so B isn’t from Utopia. And if F is from Elysium, B isn’t … B must be from Paradise.” “And since E lied,” Silverstream states, “that means E isn’t from Elysium, right? But no one else is from Utopia, so E must be from Paradise, too.” “F told the truth,” Gallus adds, “so D and E do have different hometowns. If E is from Paradise, D isn’t … and D isn’t from Utopia either, since A was truthful. That means D is from Elysium … and would have lied if she said anything.” “Then C truthful,” Yona says, “so C from Paradise.” “So that’s the answer, right?” Ocellus asks. “A is from Utopia, B is from Paradise, C is from Paradise, D is from Elysium, E is from Paradise, and F is from Elysium.” Coccyx clops his hooves together in applause. “BRAVO, everyone! That was some incredible deductive reasoning you demonstrated. I am most impressed!” “These towns really do sound weird,” Smolder states. “I can’t speak for other dragons, but I get the feeling most wouldn’t care about these ponies’ oddball behavior.” “Still, it’s incredible that ponies would go to such lengths for any reason,” Sandbar adds. Ocellus adds: “You haven’t even heard about their alicorns.” “ALICORNS???” The group turn to Ocellus, then to Coccyx. “These out-of-the-way villages in some remote corner of Equestria have princesses?” Gallus asks incredulously. “Not princesses, alicorns,” Coccyx corrects. But not in the way that you think.” “Could you tell us more about these ‘alicorns’?” Sandbar asks. Coccyx chuckles and smiles. “This, my friends, leads to a whole new level of logic puzzles.”
IMPORTANT ANNOUNCEMENTThank you to everyone who has read this collection of MLP-themed logic puzzles. Due to copyright infringements, I will be deleting this thread effective at 8pm EST. I only learned about this shortly before heading off to work, so I will be unable to take action until I get home later today. More details to come … . . . . . . . . . . . OK, about that announcement: (Seriously, though, if anyone can let Raymond Smullyan know that his work inspired me to write pony-themed puzzles …) I'm going to get to at least 20 before all is said and done. After that … we'll see ...
Puzzle 20: Double Vision, Part 2After several minutes of prepping in a separate room, Ocellus asks Smolder one last time: “You know what you have to say?” “I got it,” the young dragon insists. Satisfied, Ocellus returns to the dining nook and explains the next challenge to the others, with Smolder following close behind. “OK,” she begins, “for these five rounds, you’ll each see two Smolders, one of which will be me. Each of us will have a card from 1 to 4, and of course, we won’t have the same card. You’ll have to figure out which of us is really Smolder, as well as which card each of us has.” The foursome look at each other quizzically, before Silverstream addresses the changeling: “Uh … how is this any different from what we just did?” “One small change, but it makes a big difference: this time, if I have the 1 or 3 then my statements will be false, and if I have the 2 or 4 then my statements will be true.” Ocellus’s comment is met be more strange looks. “That exactly how first five rounds went,” Yona points out. “Oh … yeah, that part’s the same.” Ocellus begins to blush as she continues: “What’s different is—” “What’s different,” Smolder interrupts, “is that if I have the 1 or 3 then my statements will be true, and if I have the 2 or 4 then my statements will be false. It’s the opposite of Ocellus.” “That … really doesn’t sound all that much more challenging,” Gallus states. “Hey, I’m game either way,” Sandbar replies. “Go ahead and start.” The foursome collectively close their eyes, and the rounds begin: [In each round, one Smolder appears on the group’s left (L) and one appears on their right (R).] [Round 1] L: “I am Smolder. I have the 1.” R: “I am Smolder. I have the 2.” [Round 2] L: “I am Smolder. Ocellus has the 3.” R: “I am Ocellus. Smolder does not have the 4.” [Round 3] L: “I have the 1. The other card is not the 2.” R: “I have the 3.” [Round 4] L: “The sum of our numbers is 5. My number is the smaller of the two.” R: “One of our numbers is double the other. My friend’s card is not the 1.” [Round 5] L: “Either I am Smolder or I have the 3.” R: “I am Smolder and I do not have the 4. Neither of us has the 2.”
