Solving for an Error

by 6-D Pegasus

Rating: everyone
Tags: Pinkie Pie, Slice of Life, Twilight Sparkle, Trixie, My Little Pony: Friendship is Magic, Comedy, Starlight Glimmer
Length: 6485

After seeing Twilight struggling with an impossible problem, Starlight decides to impress her by attempting to solve it. Emphasis on "attempting."

Proofs are a major component of mathematics. But there are are some laws in math that cannot be proved and should NEVER be questioned. Unfortunately, a unicorn named Starlight Glimmer attempts to question a fundamental piece of logic in an effort to help her mentor solve an impossible problem.

Published 2018-11-22 07:20:01 UTC
Last updated 2022-02-05 20:41:29 UTC

Chapter 1: Make an Observation

It was just an ordinary day in Ponyville. Birds were chirping, the sun was shining, and ponies trotted happily around town. Everything was perfectly fine...Until a particular powerful unicorn decided to ignore her mentor's warnings and try the impossible.

And succeeded...

...ish

"Ahhhhhh that was nice."

Starlight trotted happily along the dirt road stretching from the edge of Ponyville back to Twilight's castle. Her entire body felt completely relaxed from the massage she had just had.

Earlier that day, Starlight had promised Twilight that she would complete another friendship lesson with Applejack. Although Applejack didn't need any help with anything at the time, she decided to let Starlight help her buck some of the extra apple trees that she left alone when she had enough. After about a minute, Applejack managed to convince Starlight to do it without using magic. She attempted to teach her how to apple-buck, and eventually Starlight had began to get the hang of it. However, her job ended much earlier than she expected and she left Sweet Apple Acres after learning two very good lessons: to truly appreciate the hard work that somepony does for others...

...and to not teleport right behind Applejack while she was bucking apple trees without giving some sort of warning ahead of time.

Luckily for her, Applejack was already pretty worn out from the morning harvest to have put the normal of energy she usually did in her kick. Starlight shuddered a little at the thought of what possibly would have happened to her if she had done that before noon, when Applejack was fully awake and at her peak. Either way, Applejack felt at fault for causing the accident, and treated her to a quick massage from the Spa ponies. Although it was barely less than ten minutes, to her it felt like an hour.

As Starlight approached the castle doors, she realized that they were wide open. Twilight usually wasn't one to leave anything unfinished, and that included going through a door. Starlight cautiously poked her head through the doorway and, upon seeing nopony waiting to ambush her, proceeded inwards and silently closed the doors behind her. She perked up as the sound of distant voices reached her ears, causing her to freeze where she was. She could make out two voices, but couldn't discern any words or determine who the speakers were. Their voices seemed very familiar, though...

Starlight poured some magic into her horn and arranged two spells to single out the voices. First she cast a spell on herself to amplify all sounds that reached her ears, then layered a background dampener spell on it to block out ambient noise that would interfere with the voices. Immediately, Starlight recognized the voices to be those of her mentor Princess Twilight and Pinkie Pie. It sounded like they were arguing.

"Pinkie, I already told a thousand times, that's IMPOSSIBLE!"

"Oh Twilight, you still can't see the answer, even when it's right in front of you."

"But that doesn't make ANY sense! This piece is the last part I need for my spell and I just CAN'T figure it out!"

Starlight perked upon hearing the word "spell". Eager to find out more, She cast yet another spell, Visionem Sanorum. The surrounding air began to pulse in bright ring shapes emanating from a hallway to Starlight's left. She proceeded down the hallway, slowly following the colorful sound waves until she reached a partially opened door. Cutting off all her current spells, Starlight carefully leaned towards the crack in the doorway to listen in.

"Come on, Twilight! Didn't they teach this BEFORE magic kindergarten?"

"NO! Well, yes I guess... but this is different! Don't you have any idea what you're implying?! Wait, where are you going?"

"My mouth just got dry. That means it's time introduce the main character to the conflict!"

"Pinkie, what are you -"

SLAM!

The door suddenly slammed open with enough force to send Starlight flying to the wall. Fortunately her horn reacted faster than the rest of her body, allowing her to teleport into the room immediately before collision. Unfortunately for Starlight, the law of conservation of momentum decided to continue existing as the lilac unicorn proceeded to trip and roll several feet before smashing into a table at the far end of the room. Starlight shook her head a few times to get rid of the stars that covered her vision and looked up to see a bouncing pink pony inches away from her face.

"Hiya ,Starlight! WatchadoingiwasjusthelpingTwilightwithaspellsheneededhelpyourmanelooksamazingomaigashmytailitstwitchatwitchingyouknowwhatthatmeansnoyouprobablydontbutiknowitssomethingtodowithsomethinglikeneartheendofthechapteranyway BYE!"

And with that Pinkie Pie sped off towards the castle's doors and left for Ponyville, leaving behind her a flaming trail the entire way.

"Oooooookay..." Starlight muttered, rolling her eyes. She rubbed the back of her head with a hoof. "Ugh, so this was what it was like being Spike, huh?" Remembering why had come to the room in the first place, she trotted in through the open door and glanced around the . "Twilight? What's going on?"

Twilight was seated at a small wooden desk across the door. There was loose paper and books scattered over the desk and surrounding floor. "Oh, hey Starlight. Pinkie was just help-...trying to help me with a new spell I've been trying to create."

Starlight blinked. "But...what does Pinkie Pie have to do with new spells? Sure, she IS an Earth pony. But although they may have just as much magic as us unicorns, but they can't cast any spells with it. I mean, it's not like Pinkie Pie studies them in her free time...right?"

"Actually, now that you mention it, she..." Twilight paused and looked at Starlight, who had a very expectant look plastered on her face. "Never mind, it's not important."

Starlight stared at her teacher for a few seconds before shifting her gaze to the piece of parchment Twilight had in front of her. "So, what is this 'new spell' you're talking about?" As if on cue, Twilight's entire face exploded with excitement. She vanished in a bright flash of purple light only to appear out of another flash inches in front of Starlight's face, shoving the paper right in her face.

"Okay, so I have been developing this spell for YEARS! Like, when I say years, I mean YEARS! I began working on it just a few months after Princess Celestia took me in as her private student." Twilight was practically beaming with pride.

Starlight took a few steps back to regain her precious personal space. "But, I've never heard of a spell that should take THAT long to figure out, especially one that would have gotten your interest as a filly! Aren't there, like, books on it?"

Twilight looked like she was expecting that response. "That's why my project is taking forever. Every one of the spells used now have already been invented, developed, and modified. Many basic spells taught in unicorn schools today are really just modifications of preexisting spells. The more advanced spells that are found primarily in books are complex combinations of other spells that had already been developed. Even that 'cutie unmarking' spell you mastered," Starlight shifted uncomfortably, unnoticed by Twilight,"is a mix of several spells occurring near simultaneously while interacting with each other in order to have the final effect. What I am trying to do here is create a new base spell!"

Now Starlight was even more confused. "Wait, what's a base spell? I've studied magic for most my life, and I've never come across even a mere mention of the term."

"That might be my fault. I, umm..." Twilight looked away nervously," may have snuck into the Canterlot archives as a filly and took the only copy of the book that was still intact."

Starlight deadpanned. "Seriously?"

Twilight's cheeks burned bright red. "It wasn't in the Restricted section so I figured it technically shouldn't have been as important as it seems now."

Starlight rolled her eyes. "But how does this relate to it why it's taking so long to finish?"

"You see, when an ordinary spell is created - the kinds that are typically found in scrolls or taught in schools - it involves piecing together several base spells in a unique pattern to create the new spell. When that spell is cast by somepony else, the whole spell is cast at once, so most unicorns never see the role of each base spell individually. It's because of this that most unicorns don't even know that they exist. The bases themselves are very easy to cast, but have practically no visible effect save for a small glow. Many of them don't even work without some form of filled parameter."

"I still don't get it.

Twilight groaned in exasperation, trying to come up with a clever analogy to simplify everything. "I got it. Think of a spell as a word. A word is an arrangement of simultaneously existing letters that combine to form a general definition that doesn't apply to each individual letter. In order for the word to exist, all the necessary letters must be present at the same time and arranged in an exact combination. We can very easily create new words by adding prefixes or suffixes to the original word, adding or deleting letters, or completely rearranging them altogether. These letters are what base spells are to normal spells."

Starlight pondered on that thought for a few seconds before it hit her in the face like a brick. "Ohhhh. Wait, so by creating a new base spell, you're essentially-"

"- adding a new letter to the alphabet of base spells. Yes that is correct."

Starlight gaped in wonder as the full realization of what Twilight was trying to achieve sunk in. "Wow, that's big."

Twilight looked away, uncertainty filling her voice. "But there's one problem."

"What is it?"

"The unicorns who created the first base spells never recorded the process by which they created them, only the way to cast them and how to combine them with others. Almost all the existing base spells were created and documented before even Celestia herself was born. Fortunately I was able to do enough research to configure a fairly accurate model for the procedure. I've even managed to calculate how much thaumic energy I would need; it's the same for every base spell apparently. There's just one final step I need..." Twilight trailed off as her ears drooped down.

"What's the problem with that last step? Maybe I could help you with it!"

Twilight sighed as she looked back up to Starlight. "Starlight, I understand how skilled you are at magic, but I'm afraid this one is out of your range. The last step requires dividing the necessary total amount of thaumic energy for the spell by a quantitative value equal to the number of spells that derive the final product, thus in the process also determining the identi-. "

Starlight raised an eyebrow. "Ponish, please?"

"I divide how much magic I need by how many bases I'm casting. The number of bases is the spell's level."

"Oh."

Starlight nodded in agreement. "I know that. Like, a basic transfiguration spell has a level of 4, so it is made up of four base spells, right?"

"Correct; it consists of a molecular scan, molecular image construction, image alignment, and temporary atomic replacement spell. A simpler spell like telekinesis is level two, since it only requires modified energy transformation and a mental link."

"And teleportation is three, right?"
,
Twilight glanced at the floor in unease. "Well, an ancient version of the teleportation spell had three, and that has been made illegal a very long time ago for ... " Twilight shuddered. "... ethical reasons. The one that has been taught for the last thousand years actually has a number of 15, but all elementary textbooks just dodge the number question."

Starlight pondered over the concept for a bit in her head. "So what's the issue? If you already know the amount of magic you need for the spell, all you need now is just divide that by the ... number of..." Starlight trailed off, her mind trying to fight common sense just to avoid the conclusion.

Twilight nodded back. "Yes, but that's the problem. Because a base spell is totally original, that would set its number at 0."

Starlight's eyes shrank upon realizing what the last step was pretty much asking for. "You don't mean-"

"Yes, Starlight," Twilight looked Starlight dead in the eyes. "I have to divide... by zero."

Chapter 2: Identify the Problem

Starlight stared at Twilight for a good ten seconds, trying to process what she had just said. "So... you've spent ... half of your life... trying to divide by zero?"

Twilight blushed and looked away. "Well it isn't really half my life. I've taken cracks at it whenever I have time or if I just feel like challenging myself, but most days I forget about it entirely."

Starlight rolled her eyes. "I'm going to be completely honest here. If I was in your position, I would have just given up the moment I reached that conclusion. "

Starlight was then forced to backpedal as Twilight once again teleported directly in her face. "But don't you see?! That's the idea! If I could reconstruct the process of creating a base spell and publish the results. The awards! The fame! I might even become a prin-"

Twilight quickly caught herself and blushed. "Hehe, sorry about that Starlight, just got caught up again in the moment." She sat on her haunches before unfurling her right wing and staring at it. "A very old moment..."

Starlight cleared her throat to break the silence before it became any more awkward. "Ahem anyway so about your project..."

Twilight shook her head a little to clear her thoughts. "Right, sorry. I've tried almost everything I could think of! At first, I thought I could just skip the last step all together and cast the construction matrix by itself."

In a bright flash, she teleported to the other side of the room and gestured with a hoof to the surrounding walls and floor. "You can tell how well that went."

Starlight glanced around the room again, paying much closer attention to detail, and did a double take upon noticing that what she had initially thought were intricate wall and floor patterns looked very similar to char marks from an explosion. She cast a basic scanning spell on the room and blanched at the results. Two of the crystal walls and parts of the ceiling had been magically regrown in the last hour.

Starlight turned back to Twilight with a confused expression. "Wait, I'm sensing that you magically repaired the room, so why are there still so many scorch marks?"

Twilight laughed nervously, her wings twitching ever so slightly. "Yeaahhh about those. Let's just say that those are from ONLY my last attempt. I've been at this for four hours already."

Starlight's jaw dropped. That's a lot of damage. "Why are you suddenly so obsessed with this?! In all the time I've known you, I've never seen you spend so much time on one task before on..." She thought a little more about what she had just said. "Forget I said that. What I meant to say is that whenever you got obssessed over something, it was because of some new and sudden development, like when we found Starswirl's old journal, or that one time you told me about when you thought Celestia was going to give you some big test but ended up just sending you to the Crystal Empire to deal with Sombra. Why did this come up just now?"

Twilight turned away and muttered under her breath. "I don't know about obsessing, but..."

"Whatever you say."

Twilight rolled her eyes. "To answer your first question, Spike and I were almost done reshelving the castle library yestersay afternoon when he came across one of the journals I used to keep as a filly. Most of it were just notes from Celestia's lectures during my time as her student, bu-"

"Wait, Celestia actually taught you herself?!" Starlight interjected, trying to form the image in her mind of the solar diarch standing in front of a chalkboard with a pointer stick in hoof. She failed. "I mean, I knew you were her 'personal protégé', but I always thought it was a metaphor for you being the valedictorian."

Twilight giggled briefly to herself before continuing. "Yes, Starlight, Celestia was my teacher, what did you think? That's not the whole truth though; even as her personal protégé, I still attended classes with the other ponies at CSGU, but those were just for subjects like math, history, and Ponish. I also took my General Education classes later on with everypony else, but every Tuesday I would have weekly sessions with the princess for advanced magic exploration. But yeah it also meant I was technically the valedictorian."

A thought came to Starlight. "Did you ever try asking the princess for help?"

"I did." Twilight groaned in exasperation. She trotted back to her desk and sat down. "Every. Single. Week. She just kept chuckling and telling me that, while it was nice that I was trying to go beyond expectations, that she was already proud of me and to not be blinded by my ambitions. Now that I think of it, she always seemed a little sad every time she told me that."

Starlight followed Twilight to her desk and sat down on the ground next to her. She put a hoof to her chin in thought. "What if you didn't need to blatantly divide by zero?"

Twilight glanced at Starlight in confusion. "What do you mean? "

"I'm saying what if you just figure out what the value is going to be instead of what it is straightforward?" Starlight trotted over to one of the nearby bookshelves, scanned the titles of the top row, and levitated out a thick, dusty tome. She started for Twilight with it in tow. "You can just plot out the function for the total magic divided by a number. If you track the value of the function as that number approaches zero, you might be able to calculate the-"

"Limit?" Without even throwing a glance behind her, Twilight lit her horn and snatched the book out of Starlight's magic aura, opening it and turning precisely to the third chapter before promptly shoving it back into Starlight's face. "Yes, I've studied singlevariable Hoofulus years ago, Starlight. Oh how I wish it was just as simple as that!"

Twilight leaned in her seat until she could rest her head on the desk. Her magic cut off, letting the book fall onto Starlight's face. "The function's behavior matches that of a normal inverse function, since it's basically a constant divided by a number. But if you try taking the limit as that number approaches zero, it just keeps going without bounds! It doesnt exist!"

An irritated Starlight levitated the heavy book off her face before teleporting it into one of the Dragonland's many active volcanos. "What about a reverse sequence? "

Twilight threw her hooves up in annoyance. "Yes I tried a sequence! The stupid thing just ended up diverging. I though of trying a series as well out of desperation, but then I remembered that because of the nonexistent limit, it would have ended up diverging as well because of the divergence test."

"The what?"

Confused, Twilight looked at Starlight, who in turn looked away in embarrassment. "You don't know about testing for divergence in series? Which university did you go to?"

Starlight blushed. "I never went to any. After high school i told my dad that I was going on a journey of self-discovery, then spent the next few years researching and experimenting with my unmarking spell, developing my... erm... ideology, and creating the foundations for my village."

Now it was Twilight's turn to drop her jaw. "Are you serious?! But your magic is so powerful! Did you ever have some sort of professional magic teacher during high school?!"

"Nope, none at all." Starlight shrugged and lit her horn. Twilight felt the surrounding air getting dryer and dryer until, to her astonishment, a strange sort of mist became visible in front of Starlight. After a few seconds, the mist had coalesced into a small ball of water about the size of her eye. "I should probably rephrase something I said earlier. Whenever I've said how long I've studied magic, I meant that's how long I've PRACTICED it. Magic always felt natural to me; as long as I knew what I wanted to do, I could figure out how to do it. When I was younger my biggest problem was really just knowing the solution. Sunburst would always come up with solutions for me that I could implement in the blink of an eye."

Starlight's horn grew a few lumens brighter, and the ball of water slowly morphed and froze into a solid block of ice. It rose higher and higher until it reached the ceiling, where it burst into a minute flurry of snow. "To do this, you would probably research individual spells and determine what order to cast them, then experiment until you succeed. I simply... do it. I don't worry about the specifics of how it works and just trust my magic and my emotions to work together with what I'm trying to accomplish. As long as it's nothing too complicated, like my unmarking spell."

Starlight then looked Twilight dead in the eye. "Maybe you it's impossible to approach this problem through math. Maybe you can use magic to solve it. Based on what I can read from your loose paper, you've only been attempting mathematical approaches, then magically implementing the last step."

"Hmmmmm that could work." Twilight pondered the idea in her head for a few seconds. "Well if you think it's worth a shot, we could... maybe..."

A look of horror flashed on Twilight's face. Starlight cocked her head in confusion. "We could... what? Is something wrong?"

With a flash of light, a small clock materialized in front of Twilight, confirming her fears. "No no no no no no no no no no I was so focused on this problem that I completely lost track of time! I was supposed to start teaching another history lecture ten minutes ago!"

Twilight quickly disappeared in burst of magic, leaving Starlight to awkwardly stand by herself before she reappeared in front of her. "I'm really sorry, I'll talk more about this with you tomorrow if you want. Anyways I have to go. I'll see you at the castle tonight!!"

With one final mini explosion, Twilight teleported to the school. Starlight glanced around the room at the all the scorch marks before turning to leave.

Impossible, you say?

Starlight paused mid-trot right outside the room and turned to face the desk once more. She lit her horn and all the scattered paper and books glowed in an identical light before neatly stacking themselves on the desk.

Not if I have anything to do about it.

Starlight took one last glance around the hallway for any prying ponies. Satsfied, she turned back to the room and trotted inside. The door was wrapped with a light teal light before slowly closing behind the unicorn.

Author's Note

Wrote this chapter on phone while we drove to vacation, so not sure if flow is different

Chapter 3: Form a Hypothesis

"What the hay?!"

Trixie's entire wagon shook violently from the force of a nearby explosion, throwing her out of her hammock. She remained on the ground for a few seconds, silently cursing herself for thinking parking her wagon under Twilight's castle was a good idea. Then again, the privacy and reliable shade were always quite worth it.

She quickly clambered back on her hooves and brushed herself off, giving a brief look of annoyance on the newfound mess on her wagon floor. After magically reorganizing any fallen trinkets, Trixie bucked open the doors to her wagon and trotted outside to look for the pony she would exact revenge upon for the loss of a good nap. What she was not expecting to see, however, was a large cloud of smoke billowing out of Twilight's castle. Specifically, out of Starlight's window.

Hmm... that's not good. Umm what do I do what do I do what do I do... Knowing how powerful of a unicorn her friend was, Trixie was certain that Starlight was more than capable of taking care of whatever shenanigans Ponyville had to offer. However, there was still that feeling lingering in the back of her mind that she should at least check to make sure her friend was alright.

Ho humm, might as well go help. Celestia knows where she'd be without me now. She rolled her eyes and grinned as she levitated her signature cloak and hat off their stand onto her body. She stepped off the wagon and gave a quick wave of her horn, splashing a camouflage spell over her wagon to ensure no peering ponies let curiosity get the better of them. Once she was satisfied, she trotted towards to the front door of the castle.

Okay Starlight, just a few more layers...This time it'll work... The lilac unicorn gritted her teeth as she poured more magic energy into her horn. The glow around it was almost blinding, casting her surroundings in a bright, teal hue. Pure threads of magic spilled from her horn, swirling around her like a mystical whirlwind.

Come on you, stay together... Starlight gritted her teeth to ward off the rising headache and furrowed her brows in concentration. Beads of sweat trickled down her forehead, and she panted for breath. Slowly but surely, the magical strings began to stitch together in front of her, forming a giant sigil of arcane energy that spanned the length of her room. Her eyes widened.

Yes yes yes yes yes! This could be it. The key to the door. The last piece of the puzzle. The answer sheet hidden upside down on the back of the test. All she needed to do now was activa-

CRACK!

Starlight's ears twitched at what was considered the worst sound a spell caster could hear. It was the magical equivalent of the sound of bits tumbling down a drainage grate. In her case, it was the sound of a magical tear ripping through the fabric of her spell.

Something's wrong... Starlight peeked an eye open and, to her horror, noticed a large, glowing crack slowly growing along the middle of the spell circle. She double checked her algorithm and quickly realized something extremely important was missing. Something big. Something obvious. Something just beyond her reach. And without it, her spell was rapidly falling apart. More cracks splintered off the first, quickly running across the flickering sigil of light.

Starlight's eyes widened in panic. No no no no no! Please, not when I'm so close! She pleaded with the spell to stay together, offering more of her magic in an attempt to stabilize the broken sections. The glow around her horn doubled, then tripled in brightness. At first, it appeared her efforts would be in vain; the cracks continued to grow until they almost covered the entirety of the spell. However, the spread of the cracks gradually began to slow, and a flicker of hope crossed the unicorn's straining mind.

She inhaled sharply, filling her lungs with more precious oxygen, and gritted her teeth as she poured all her energy into maintaining the unstable regions of her spell. Her head was beginning to ache from the strain, but she paid it no attention. For a minute, it seemed she and the cracks were at an impasse, like two ponies in an even game of tug of war. Soon, she noticed the cracks beginning to recede, slowly but surely. Her eyes lit up with a newfound fire as her magic raced along the glowing tracks, mending them and chasing them back to their origin. The original crack was now no longer than her foreleg. Just a little bit further and-

BANG! "Starlight! The Great And Powerful Tri-"

Starlight's focus completely collapsed from the momentary distraction. Her magic cut off for just a split second out of surprise, but that small slip up gave the cracks the ground they needed to overcome her defenses. In mere seconds, the cracks spread out through the entirety of her spell, giving the appearance of a bright, pulsating spiderweb. Starlight realized her spell was beyond any hope of recovery and, after only a moment's decision, abandoned it altogether. She dived towards the only other living pony in the room and instinctively cast a crystal shield spell around them both as the room was bathed in a blinding white light.

Starli- ... are yo- ... -ight...

A voice rang out faintly in Starlight's head, shifting in and out of focus like watching a distant searchlight through a thick fog. She concentrated her will on it, trying to bring the sound into focus.

Wake- ... -me, Trixie... WAKE UP!

Two recognizable words cut straight through the fog clouding her mind. She focused on them and obeyed their command, dispelling the fog just enough to be able to peak open her eyes.

A pair of wide, grayish violet eyes stared back at her from less than a hoof length away.

"AH!"

Starlight's shout elicited a surprised "Eeep!" from the pony occupying her personal space, who jumped back just enough for her to briefly take in her surroundings. Her floor and walls were covered in scorch lines, and some of her furniture had been knocked over. One of her bookshelves appeared in immediate danger of toppling over. The protection spells on all her precious kites remained intact however, as she noted with satisfaction that none of them had a single mark of damage.

She turned her attention back to the pony in front of her and did a quick once-over. Lilac blue fur, magician's cloak. She only knew one pony who fit that description.

"Trixie?! What- oh my gosh, are you okay? Why are y-... wait, you... YOU! You're the one who interrupted my spell!"

Trixie's eyes widened in surprise. "That was your spell?! Trixie thought you were in some imminent, magical danger!"

Starlight let out a dry laugh. "When it comes to me Trixie, I don't think those two are entirely mutually exclusive."

Trixie cocked her head.

Starlight rolled her eyes. "Meaning both can be true."

"Psssst, hey Starlight." Trixie leaned in towards Starlight's ear. "You know you can speak Ponish when it's just the two of us. You can save the textbook language for Twilight."

"Oh stop it." Starlight playfully pushed Trixie away, but failed to suppress a faint giggle. "You haven't even experienced a fraction of Twilight's true vocabulary."

"And Trixie hopes to keep it that way." The light blue unicorn trotted over to the other end of the room where her hat had apparently been blown to and picked it up with her magic. She gave it a few shakes to clear off any dust and plopped it unceremoniously on her head. As she did, however, she noted that the scorch marks lining the walls held an interesting pattern. "Starlight, how did you manage to destroy your walls in the shape of runes?"

Starlight glanced around her room and noted with fascination, and a hint of dismay, that her walls indeed held scorch patterns matching the runes cast in her spell circle.

"I cast those runes so the magic contained in the circle would pool into them and act as the spell's supporting structure." She scanned the room, counting the number of rune patterns burned into her walls, and confirmed there were less of them than the amount she made for the spell. "I think a few of them ruptured, so their collected magic flowed into the other runes, causing them to overload."

She walked over to one of the marks on her walls and tapped a hoof on it. There was still a trace amount of smoke trailing from within. "Once the magic contained in the runes overwhelmed the circle's container, the circle broke and sent all the runes exploding outwards. Normally they would just dissipate into the air if I cast this outside, but since I did this in an enclosed space", Starlight let out a groan of annoyance, "they just ended up burning into my walls. I'll have to clean this mess later..."

Starlight thought back to when she first heard it crack and mentally went over how unstable the spell felt from then on. It must have cracked from the first rune failing. The spell was already done for, I was just overloading the other runes by trying to keep the circle intact. If I had just let the spell go as soon as it cracked, it would have collapsed a lot more peacefully...

"Trixie to Starlight!"

"Wha-?" Starlight shook her head. "Sorry, I was just thinking about something. Did you say something?"

Trixie raised an eyebrow. "Trixie was wondering what you were doing that would need a spell this powerful."

Starlight pondered for a second. "And if I said you don't need to worry about it?"

"Then Trixie will politely ask about it..."

Starlight let out a sigh of relief.

"... the entire day."

Starlight quickly sucked it back in and let it back out as a sigh of exasperation. "Alright, fine. I was trying to help Twilight with a new project she had. She's trying to invent an entirely new base spell which, if she does it successfully, could allow for the creation of thousands of new spells, as well as possibly make some existing ones far easier to cast."

Trixie again cocked her head in confusion. "Trixie has never heard of a base spell before, is that something to do with making new spells?"

"I... think? I've never heard of them either before, might have Twilight to blame for that. But anyways, yes, Twilight's trying to invent a new building block for spell creation, which apparently hasn't been done since before Luna's banishment." Starlight levitated a book from a messy stack sitting at her desk and opened it in front of them.

"It turned out Starswirl the Bearded was the last one to create a base spell after discovering the first version of the teleportation spell worked by creating an exact replica of the target, sending that to the destination, then sending the original to some closed bubble outside our dimension, eventually leading to the original pony's death. That version was much easier to cast and was how most unicorns got around until, somehow, one of Starswirl's students escaped the bubble alongside the copy after their first time casting it. Once Starswirl worked out what was going on, he combined his magic with that of an ancient artifact to erase the memory of how to cast the spell from all other ponies' minds, and all physical traces of the spell's existence was wiped out. Then, with the help of Celestia and Luna's parents, he did something to our world's field of magic to make sure all attempts to cast it would fail. He worked tirelessly for many moons to craft a new base spell, with which he forged a new teleportation spell that folded the fabric of space-time to directly transport matter from one location to another without destroying the original. As it sounds, this version was far more difficult to task and required far more magical energy. In most cases, only a unicorn whose special talent is in magic has the natural magical reserves to cast it. You can still cast it without this affinity, but you'd almost completely exhaust yourself from the effort. Even with a magic cutie mark, the spell can still drain your reserves if not cast absolutely perfectly. That's why you rarely see unicorns casually traveling by teleportation."

After a few seconds of silence, Trixie registered the story was over. "Wait, if he erased all traces of its existence, how do you have... this?"

Starlight closed the book and revealed the cover, eliciting a gasp from Trixie. "After casting the Erasure, Starswirl documented everything that happened in his personal journal. After realizing I can somewhat understand his horn-writing, Twilight asked me to help her translate the rest of Starswirl's journal. After consulting with Princess Celestia and Luna, they all entrusted me with one of the few copies of its translation, since it contains a lot of details left out of our history books."

She floated the book over to her desk and turned back to her friend. "I'm thinking that if I can figure out what calculations Starswirl made during that period, I'd be on the right track to crafting a new base spell. The problem that Twilight, and now I, have ran into is that among these calculations seems to be an unavoidable... division by zero."

Trixie raised an eyebrow. "Trixie may not have Twilight's brain, but isn't that impossible?"

"Mathematically, no actually, there's a concept called limits that lets you theoretically 'guess' a division-by-zero problem by dividing two numbers close to zero, known as an upper and lower limit. You would then divide numbers increasingly closer to zero to see what value this division approaches as the limits approach zero. Unfortunately it's not something I ever studied; I was always just into the spellcasting. Twilight knows more than me about the topic, but even she seems to be stumped."

Starlight's stomach suddenly grumbled, bringing a blush to her face. Trixie noted from the shadow cast by the sun that it was roughly noontime and deadpanned. "Starlight. When did you last eat?"

Starlight's blush deepened and she sheepishly grinned. "Umm... yesterday?" Trixie opened her mouth to speak, but Starlight cut her off. "You know what, I think I do need a break from this anyways. Maybe I'll be able to think about this better after some fresh air and a full stomach. You want some hayburgers? We can talk about this more over lunch." She trotted over to her door and held it open, waiting for Trixie to respond.

Trixie held her deadpan stare for a few more seconds before rolling her eyes and grinning. "Alright fine, Trixie had a light breakfast anyways." As she trotted past Starlight, she fixed her with a cold glare. "But if you spill ketchup on my cloak again, you'll be finding several, and I mean several smoke bombs hidden in your room tonight."

Starlight closed the door behind her as they trotted away from her room, their sounds of conversation becoming more distant and muffled the farther went.

With nopony around to hear it, Starlight's bookshelf finally fell forward with a resounding thud.