Simple

by Clacksphob

First published 30th of September, 2013

For too long now had Rarity just accepted that there was one thing she couldn't do to save her life.

This is a story of a young mare who failed one task over and over again. It wasn't an important task, it wasn't even a particularly hard task, but no matter how hard she tried, it was not to be. She resigned in the knowledge that she'd never come clean, never dare looking back. But nopony can just run away from their troubles...

Unfinished Business

It wasn't often that Rarity considered herself truly lacking. Miserable, unsuccessful, inconfident, certainly; but she had little concerns about personal achievements outside her forte. However, this time it was different. The almost comically abysmal losing streak of thirty-five games of Tic-Tac-Toe against Pinkie Pie - certainly impressive considering her maturity, in a very morbid sense - ground on her for a long time ever after, up until this very day. It was a children's game, strategic at that. It wasn't up to chance who won. In fact, everypony around her always claimed the game couldn't be won if both players go at it seriously. However, she never saw how this wretched pastime could be so utterly predictable to others. Worse yet, Rarity most definitely wasn't stupid, she constantly reminded herself. She couldn't be: she could use her magic to thread together the finest garments, she was the one who found out the secret identity of Gabby Gums, she could read other ponies and even managed to fool not only the high society of Canterlot but also, in hindsight regrettably, her friends with the most outrageous claims.

Rarity took a sheet of paper and a pencil. A rubber too, just in case. Mentally bracing herself for this seemingly gargantuan task, she drew up a three by three grid. She would open Pandora's Box, and there better be plenty of hope left inside. Suddenly, the first wave of panic struck.

"Oh Celestia, where do I start...?"

She could just go ahead and take the center for X, but, while symmetric and aesthetically balanced, would it be a smart move? Any other field raised the same issues. It couldn't possibly matter where she started, any place would be fine because the other side could never be pretictable; especially not if she started in the middle for maximum symmetry. Pinkie Pie managed to best her countless times, and she wasn't exactly the pony to plan ahead much. So perhaps she should pick a field at random?

Ultimately, she picked the bottom left, which she marked with an 'x'. If she could win against herself - "Random Rarity" (X) versus "Classic Rarity" (O), so to speak - then at least she would have proof that having no strategy is better than her trivial attempts to form a straight line, a feat she could easily accomplish in her daily life, but became the single greatest obstacle in her life as soon as this was subjected to taking turns with somepony else. In vein with her old ways of playing, she put the O in the center, the place around which everything else arranged itself. This would be the key to see how Pinkie did it. Almost laboriously, she forced herself to put the next X in the top left, rather than the top right for symmetry. Pinkie wouldn't play symmetrically, she would simply pick out any random field - and the top left it became. But now the grid was imbalanced to the left - she put the O to the center right. Now all that X would have to do is claim the center left and win - but this isn't how pinkie would have thought, not at all. Middle top it was instead.

Anxiously, Rarity stared at the paper. Did she really find a way to beat Pinkie Pie? By merely playing as always? A preposterous idea. Had this been the way to go she'd have won at least one game of the dozens and dozens of times the Earth Pony had inflicted her will upon her. This was suspicious, wrong even. Pointless. That was the word she was looking for - pointless. No matter how much she'd practice, there was no way she'd ever be able to uncover this enigma. Tears began to well up inside her, as she ripped the paper into pieces, and strewed them across her room, sobbing, before sweeping up the remains of her failed research. She needed her couch, it was too strenuous a flight of emotion to deal with to stand while wallowing in the tragedy of her situation.

Only a short while later, Twilight Sparkle dropped by Carousel Boutique for a neighbourly visit.
"Hey Rarity, how are you?" she called. Hearing only faint sobbing and bawling, Twilight started looking for her friend. Granted, it was possible Rarity was just in another fit of melodrama, but she'd rather not take any chances.
"Rarity, what happened? Are you okay?"
"Oh Twilight... i-it's nothing, I-I just-", Rarity squeaked in despair. "I tried to prac-practice playing Tic-Tac-Toe, a-and I just can't! How am I ever going to cope? Am I really too stupid for such trite a game?"
The purple alicorn consoled her friend, and after a while Rarity caught herself again. It was good to have a friend closeby who is able to cheer you up.
"Rarity, I could teach you how the game works, it'll even get boring quickly after you've got the hang of it."
"Would you really do that for me, Twilight?"
"Sure, let me just draw up a grid and we'll start right away..."

Author's Notes:

Perhaps a bit too much, but I liked the idea. Now you could chalk up Pinkie's amazing win streak to her fourth-wall powers, but those don't exactly give an edge in Tic-Tac-Toe, do they...?

Maths

"Alright Rarity, here we have the grid. There is a mathematical limit to combinations that will result in a win in games like this one, since the grid here is a small one we could almost go through all of them, but that won't lead us anywhere. Instead, we'll ignore rotations and inversions, so in the end there are 3 ways to make a line: 2 horizontal or vertical, and one diagonal-"

"But Twilight", interrupted the unicorn, "surely there must be more ways to win than three?"
"That's what I meant by rotations and inversions. For example...", Twilight began to draw up a couple of states; it came almost disgustingly easy to her, "there's no topological difference between these states, whether the row is horizontal or vertical, if it is along one edge, you can say it's the same constellation. So if you know how to build one like these, you can apply that to any similar state."

"But Twilight", began Rarity, "I don't even know how to start a game!"
"I'm getting there, Rarity. It'll all be clear in a moment, I promise. Now, the complete number of states the game can adopt is 3^9, that is each of the nine spaces can have one of three states - claimed by X, claimed by O, or not claimed. You can adapt this formula to any similar game with any number of players, if we played with Spike and Fluttershy, on a twelve by eight grid, there'd be 5^120 states! Isn't that amazing?"

"But Twilight", said a puzzled Rarity, "How in Equestria is that going to help me? I really just need to know how to begin the game first!"
"Trust me, this may not seem relevant, but you will need this. So as I was saying, we have for the base the number of players plus one, since a space can be empty in addition to claimed by either player, and for the exponent we have the total number of spaces. So exponentiation is pretty much multiplying a number by itself several times. But this isn't where this all stops; since for the game it doesn't matter if we start in one corner or another - because we can clearly just rotate the paper, see? - this means that any moves our opponent makes are in relation to the others. Because we can rotate - and by extension, reflect - a state, this cuts down on the numbers of total combinations, as we already mentioned over the course of this seminar, so that in total..." Twilight didn't notice Rarity's mind had already begun to wander off. In the end, it didn't much matter, did it? It's not like she had to be ashamed of herself for not having grasped this simple game at an age where she already was a successful businessmare.

Having borrowed a raincloud for personal use from Rainbow Dash, who tactfully didn't ask for its purpose, Rarity trudged across Ponyville, drenched by the constant downpour upon her now thoroughly wet mane, tail and coat, a deep scowl on her face. She had to find a way, and merely succumbing to the despair did not yield any results aside from venting, but venting could only take you so far. Since Twilight, though well-intentioned and certainly educated enough, managed to turn a little explanation of a game into a lecture on maths, Rarity may need somepony more blunt to keep the explanation strict to the point. She already had a certain farm pony in mind, and thus headed towards Sweet Apple Acres, hoping that Applejack was able to clear the fog for her.