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Under the Hood
Let us explore these two kinds of formal structures one at a time. First, there is the foundational formal structure that lies "under the hood" of the rules of Tic-Tac-Toe. Does such a structure exist? Is it different than the stated rules of play? There is, in fact, a core mathematical logic that is part of every game but that is not necessarily expressed directly in the stated rules of the game that a player must learn. To understand this point, take a look at a game thought experiment by Marc LeBlanc. The game is called 3-to-15.[1]
Rules for 3-to-15:
Two players alternate turns.
On your turn, pick a number from 1 to 9.
You may not pick a number that has already been picked by either player. If you have a set of exactly 3 numbers that sum to 15, you win.
What does this game have to do with Tic-Tac-Toe? At first glance, 3-to-15 doesn't seem anything like Tic-Tac-Toe. Instead of making Xs and Os, players are picking numbers. There is not even mention of a grid. However, the "punch line" of the game is that 3-to-15 is in fact a kind of Tic-Tac-Toe. If you think you have it figured out, look at the diagram across the page. 3-to-15 is a "magic square" puzzle, in which the numbers in any horizontal, vertical, or diagonal row add up to 15. By picking numbers from the magic square in the fashion proscribed by the rules, players are actually playing a game of Tic-Tac-Toe. Or are they? What do the two games have in common? The underlying rules found in both Tic-Tac-Toe and 3-to-15 look something like this:
Two players alternate making a unique selection from a grid array of 3 by 3 units.
The first player to select three units in a horizontal, vertical, or diagonal row is the winner.
If no player can make a selection and there is no winner, then the game ends in a draw.
These "rules" resemble both the rules of Tic-Tac-Toe and 3-to-15, with some significant differences. For example, the rules don't mention how the player makes a selection from the array of choices, or how to record a player's action. The rules above are a kind of abstraction of both games.
Questions remain: is 3-to-15 a variant of Tic-Tac-Toe or a different game entirely? If it is a different game, what does it share with Tic-Tac-Toe? What does all of this say about the "rules" of Tic-Tac-Toe? We answer these questions later in this chapter. For the time being, just note that there are in fact formal aspects of games such as Tic-Tac-Toe that lie underneath the stated "rules of play."
The punch line
[1]Marc LeBlanc, Game Developers Conference 2000.
