"Hey, everypony?"
The colts and fillies stopped eating their breakfasts to look at Twilight inquisitively.
"I have a question. Does anypony here know if, when a pony is in full gallop, all four of their hooves leave the ground at once?"
There was silence for a moment.
"Um, ah think they do?" offered Apple Bloom.
"Are you sure?" asked Sweetie Belle.
"But if all four of your hooves leave the ground at once," began Pipsqueak, "wouldn't you fall over?"
"How strange," remarked Rarity. "I have never considered this before. You would think that the answer is obvious, but I can't actually remember if they do or not!"
"Question," said Rainbow Dash. "Who cares?"
"Ah do!" declared Applejack.
"Well, if you care so much, why don't you just take off running and we'll tell you if they do or not!"
"Now that you mention it, maybe ah will!" said Applejack. "Ah didn't really care about this before, but now that Twilight brings it up, it's driving me crazy that ah don't know the answer!"
"There's just one problem," said Twilight. "Our legs become too hard to follow when we gallop. It's practically a blur. We can't find out the answer by simply watching somepony run by.
"But, I do know of a way that we can find out. But, I'll need a lot of help to make it work."
"Ah'll help," declared Applejack.
"Good," said Twilight. "Why don't you be the runner?"
"We're coming too!" declared Apple Bloom.
"Alright, as soon as everypony finishes breakfast, meet me outside," announced Twilight.
Dinky was the first to come outside and behold what Twilight had planned.
"Cameras?" she asked.
"Yep!" answered Twilight. "Twenty-four of them, all in a row! Ordinarily it would have cost me quite a few bits to buy them, but..."
"Everything's free here," finished Dinky. "But, what are they for?"
"We're going to take pictures of Applejack!"
"What? Why?"
"Well, you see," said Twilight, "We'll have to have somepony behind each and every single camera. We will then have Applejack gallop as fast as she can past the cameras. At the exact moment she passes in front of a camera, the pony behind it will take a picture!"
"And you're just hoping to catch a picture of Applejack with none of her hooves touching the ground?"
"Well, no. It should only take Applejack just a few seconds to run past every camera, but because there are so many cameras there should be a fraction of a second between each picture, assuming everypony does their job correctly. If her all of her hooves do leave the ground at the same time, one of the pictures should show it."
"Really?" said Dinky in interest. "This sounds like fun!"
"I was hoping it would."
"Come on, Rainbow Dash," said Twilight coaxingly. "We need just one more pony on a camera. All you have to do is push a button."
"Come on, just find somepony else," complained Rainbow Dash. "I told you, I don't really care what the answer is one way or the other. I already know that pony's hooves don't touch the ground when they fly."
"Are you sure you don't want to help, Rainbow Dash?" asked Scootaloo. "You can be next to me..."
Rainbow rolled her eyes. "Alright, I'll do it, but only as a favor to you all."
"Are you ready, Applejack?" asked Twilight.
"Ready as ah am for Granny's cookin'!"
"Are you nervous?"
"Nah."
"Oh. Good then! Okay, here we go...
"Ready," began Twilight, speaking loudly enough for the others to hear, "set... go!"
Applejack shot away at speed and began running past the row of cameras, which began rapidly flashing one by one.
As soon as she was past the last camera, Applejack skid to a stop.
"Well," she huffed. "That sure was weird lookin'. All those lights flashing in a line."
"Now all we have to do is gather up the photos and wait for them to develop," announced Twilight.
"So... is that it?" asked Diamond Tiara.
"Hopefully," answered Twilight. "I think everypony did a pretty good job at taking a picture at the right moment. Everypony can do whatever they like in the meantime."
"This was so cool!" declared Snips.
"And we haven't even got an answer yet!" said Snails.
As the colts and fillies began to wander off, a subtle cough drew Twilight's attention. She turned to find herself face to face with Diamond Tiara.
Twilight smiled nervously. "Um, can I help you?"
"So, this was your lesson?" asked Diamond.
"Well, yes? I know it may not be what you're used to, but it is interesting, right?"
"I suppose so," said Diamond, "but what use is it?"
"What do you mean?" asked Twilight.
"I mean, how does learning whether or not a pony's hooves leave the ground actually help us in any way?"
"It... it's an example of using our brains to solve a problem?"
"But at the end of the day, the answer doesn't really matter, does it?"
"Well, everypony else seemed to like it. What would you rather have me teach?"
"Do you really want to know?"
"Well, yes?"
Diamond Tiara took a breath.
"Okay," she began. "So, I want to go to school in Canterlot when I'm older. I don't know if you've noticed, but there aren't exactly a plethora of options in Ponyville when it comes to higher education. If I'm going to find schooling in Canterlot, I have to prove that I'm smarter than the average pony!"
"Okay..." answered Twilight, prompting Diamond to continue with a hoof.
"A few weeks back, Miss Cheerlie, our school teacher, started teaching us about square roots."
Twilight let out a sigh. Oh no, she thought.
"It's easy enough. The square root of twenty-five is five. Sixteen is four. Nine is three. But then- then, I asked what the square root of eight was. And Miss Cheerlie told me 'I don't know'!
"Do you know what the most annoying part of square roots is? Some of the answers, there's no end to the numbers! They go on and on for infinity, and they're not even in any particular order, its just all random! And worst of all, everypony keeps telling me that there is no easy way to calculate them! That's soooo annoying!"
"Annoying, but true," said Twilight.
"But it's not true!" cried Diamond. "You can do it! I've seen you give the square roots of crazy numbers without even using a quill and paper! There must be some sort of secret formula that you have! How do you do it?"
"I... don't know if I can explain?" answered Twilight nervously.
"Try anyway," insisted Diamond.
"But there is no 'secret formula'," said Twilight.
Diamond Tiara sighed. "Is it because you think I'm too stupid to get it?" she asked wearily.
"No! I just mean-"
Twilight stopped and let out a huff.
"Okay, Diamond. I kind of lied. There is as special formula you can use. And the best thing about it is that it doesn't require anything more complicated than division."
Suddenly, Diamond Tiara's eyes lit up and she looked at Twilight in near disbelief.
"You're kidding me," she said.
"I'm not. The secret formula..."
"Yes?" asked Diamond impatiently.
"Is..."
"Is what?!"
"Number divided by guess, plus guess, all divided by two. And then repeat."
There was a moment of silence.
"Okay, you know that I don't know what half of that means," said Diamond.
"The number means the number that you are trying to find the square root of," explained Twilight. "The guess is a number that you think is close to the real answer."
"Okay..." said Diamond Tiara. I guess that all makes sense. So, the result is actually the square root of the number?"
"No," said Twilight. "The real answer is infinitely long, so it would take you an infinite amount of time to calculate. But, the answer is guaranteed to be much closer to the real answer than your first guess was. So all you have to do then is use the formula again, but use that answer as your new guess."
"And you keep on repeating the whole process until you get the precision you want!" cried Diamond in realization. "Right?"
"Right."
"Number divided by guess, plus guess, all divided by two. And then repeat," said Diamond to herself, trying to commit the formula to memory. "If this works... you'll be my best friend for forever."
"Um, your welcome?"
"I'm going to try it right now!"
Twilight watched Diamond run away in excitement.
"Well, I suppose now I have to find something to do with myself for a bit," Twilight said to herself.
"Twilight!"
Twilight turned her unicycle around toward Sweetie Belle's voice and rode to meet her.
"What's up?" she asked.
"What did you do to Diamond Tiara?" asked Sweetie Belle suspiciously.
"What? I didn't do anything to her. I just explained a math thing to her!"
"So why is she writing crazy huge complicated equations on a chalk board like a mad scientist?"
"What?"
"Come see!"
Twlight hopped off of her unicycle and carefully set it down in a safe place before following Sweetie Belle into the Fort.
Inside she saw Diamond, who was indeed writing on a chalk board. Several of the others were watching in confusion or amazement.
"What did you do?" repeated Sweetie. "Teach her calculus?"
"Actually, in a manner of speaking, yes, a little bit," admitted Twilight. "She doesn't know it though... But there's no actual calculus symbols in her equations. Those are just really long division equations," said Twilight. "Really, really long ones..."
"Twilight!" cried Diamond upon hearing her voice. She stepped away from the chalkboard and ran up to Twilight.
"You were right, your formula is amazing!"
"Thank you?"
"But!" cried Diamond, startling everypony around her.
"But," she continued, "it gives me really freaky answers when I try to solve a negative number."
"Oh, Celestia's MANE, can we not get into this right now?" moaned Twilight.
Suddenly, an idea occurred to her.
"Hold on, I forgot something."
Twilight ran out of the gym and outside for a minute. When she returned, she was levitating a stack of photographs in front of her.
"I have an announcement everypony!" she cried. "I am holding photographic proof that Applejack's hooves did all leave the ground at the same time!"
"Let me see!" cried Applebloom.
"It's this photo right here," said Twilight.
"Whoa, cool! She really does!"
"And, if we take the stack of photographs and flip through them rapidly, it creates an illusion and it looks like the picutures are actually galloping!" said Twilight.
Drat, thought Diamond Tiara as she watched Twilight get swamped with colts and fillies who wanted to see the photos. I guess I'll have to ask Twilight about this some other time...