Twilight Sparkle rounded the cloud and swung around to the left, heading for the next cloud as quickly as possible. Passing that, she slalomed around it, flying for the finish line. She pushed herself as hard as possible, and flew past the finish markers in what she hoped was record time.
Rainbow Dash was lazily flapping her wings at the end of the race track, and hit the stop button on her watch as soon as Twilight flew past. "Not too shabby, Twilight!"
Twilight slowed down to a stop, and flew back to where Rainbow Dash was hovering. "How'd I do?"
Rainbow playfully hoofed Twilight in the shoulder. "You're making good progress. You shaved three seconds off your personal best."
Twilight grinned. "I'm really getting the hang of these wings now."
"It must be your awesome trainer!" Rainbow puffed out her chest.
Twilight chuckled to herself. For some reason the book of Greek philosophy that she'd read when visiting the human world popped into her head. Particularly the passage by a certain Zeno of Elea. She decided to play a little trick on her friend. "You know, I think I'm nearly as fast as you now."
Rainbow Dash burst into laughter, holding her belly to stop it splitting. After a few minutes she calmed down, and wiped the tears from her eyes. "Good one, egghead!"
A sly smile crept onto Twilight's face. "I'm serious," she said. "Forget these obstacle courses, but on a straight track, if you gave me just a small head start, I could beat you."
Rainbow laughed again, but soon stopped when she saw Twilight's smug expression. "You're really serious?"
Twilight nodded.
Rainbow was stunned — she would never boast about it, but she was one of the fastest fliers in Equestria. "Twilight, you do remember that I've trained with the Wonderbolts and competed at the Equestria Games? I was even crowned Equestria's Best Young Flier a few years ago!" Who was she kidding? Of course she would boast about it!
"But you said yourself, I'm making good progress."
Rainbow rolled her eyes. "Yeah — good for a pony who's only had wings for a few months."
Twilight flew in a little circle around Rainbow Dash. "Well, I still believe I could beat you."
Rainbow was getting tired of this conversation and decided to put an end to it. "What sort of race are we talking about then?"
"A hundred yards," answered Twilight confidently.
Rainbow smiled. She could cover that distance in seconds. The race was in the bag. "And how big a head start do you need?"
Twilight thought for a moment, pretending to do some calculations in her head. "Ten yards."
Rainbow Dash laughed louder than ever. "I thought you'd ask for fifty. There's no way you could beat me with just ten yards' head start," she told her friend, "but if you want to humiliate yourself, feel free to try."
"On the contrary," said Twilight, "I will win, and I can prove it to you."
"Go on then," Rainbow Dash replied, with less confidence than she felt before. She knew she was the superior athlete, but she also knew that Twilight was smart, and wouldn't make such a claim unless she had reason to do so.
Twilight hovered in front of Rainbow Dash and started her explanation. "Suppose that you give me the ten yard head start. Would you say that you could cover those ten yards between us very quickly?"
Rainbow nodded and gestured dismissively. "Very quickly."
"And in that time, how far would I have gone, do you think?"
Rainbow looked at the stop watch, and looked at Twilight, trying to judge her speed over an obstacle-free course. "Perhaps five yards — no more."
"Very well," continued Twilight, "so now there is five yards between us. And you would catch up that distance very quickly?"
"Very quickly indeed!" Rainbow replied with growing confidence.
"And yet, in that time I would have gone a little way further. So that now you must catch that distance up, yes?"
Rainbow's confidence began to falter. "Ye-es," she answered slowly.
"And while you are doing so, I will have gone a little way further again, so you'll need to catch up the new distance," Twilight continued.
Rainbow said nothing.
"And so you see, the whole time, you must be catching up the distance between us. And I — at the same time — will be adding a new distance, however small, for you to catch up again."
Rainbow's jaw dropped. Everything Twilight said seemed to make sense. Once she regained control of her facial muscles, she had no choice but to admit it. "Yes, that must be right."
"And so you can never catch up," Twilight concluded.
Rainbow Dash was speechless. She knew she was faster than Twilight Sparkle, and should be able to easily beat her in a race, even after giving her a small head start. But she couldn't find fault in Twilight's logic.
Twilight flew in another circle around her friend. "So, should we race then?"
Rainbow Dash looked around at the other pegasus ponies training on the flying course. If she lost a race to Twilight Sparkle in front of all these ponies, she'd never hear the end of it. No longer would she have the reputation of the best flier in Ponyville. She sighed defeatedly. "Maybe some other time."
Rainbow would have to step up her daily training regime.
Author's Note
Zeno of Elea was a Greek philosopher — a predecessor and inspiration for later philosophers such as Plato and Socrates.
Zeno's paradox of Achilles and the tortoise is a classic mathematical problem dealing with infinitesimal quantities. Although it's no longer regarded as paradoxical, it was at the time regarded as a serious philosophical problem. There's a nice little video about it on YouTube here.
