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Tic-tac-toe (also known as noughts and crosses or Xs and Os) is a paper-and-pencil game for two players, X and O, who take turns marking the spaces in a 3x3 grid. The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row wins the game. Players soon discover that best play from both parties leads to a draw. Hence, tic-tac-toe is most often played by young children. 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, 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. The game can be generalized to an m,n,k-game in which two players alternate placing stones of their own color on an mxn board, with the goal of getting k of their own color in a row. Tic-tac-toe is the (3,3,3)-game. Harary's generalized tic-tac-toe is an even broader generalization of tic tac toe. It can also be generalized as a nd game. Tic-tac-toe is the game where n equals 3 and d equals 2. If played properly, the game will end in a draw making tic-tac-toe a futile game.
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