Tic-tac-toe (American English), noughts and crosses (Commonwealth English and British English), or Xs and Os/“X’y O’sies” (Ireland), is a paper-and-pencil game for two players, X and O, who take turns marking the spaces in a 3×3 grid. The player who succeeds in placing three of their marks in a diagonal, horizontal, or vertical row is the winner. It is a solved game with a forced draw assuming best play from both players.

Gameplay

In order to win the game, a player must place three of their marks in a horizontal, vertical, or diagonal row.

The following example game is won by the first player, X:

Players soon discover that the best play from both parties leads to a draw. Hence, tic-tac-toe is most often played by young children, who often have not yet discovered the optimal strategy.

Incidence structure for tic-tac-toe.

Because of the simplicity of tic-tac-toe, it is often used as a pedagogical tool for teaching the concepts of good sportsmanship and the branch of artificial intelligence that deals with the searching of game trees. It is straightforward to write a computer program to play tic-tac-toe perfectly or to enumerate the 765 essentially different positions (the state space complexity) or the 26,830 possible games up to rotations and reflections (the game tree complexity) on this space. If played optimally by both players, the game always ends in a draw, making tic-tac-toe a futile game.

The game can be generalized to an m,n,k-game in which two players alternate placing stones of their own color on an m×n board, with the goal of getting k of their own color in a row. It can also be generalized as a nd game. Tic-tac-toe is the game where n equals 3 and d equals 2. It can be generalised even further by playing on an arbitrary incidence structure, where rows are lines and cells are points. Tic-tac-toe is the game given by the incidence structure shown to the right, consisting of nine points, three horizontal lines, three vertical lines, and two diagonal lines, each line consisting of at least three points.

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