Combinatorial Games: Tic-Tac-Toe Theory
Combinatorial Games: Tic-Tac-Toe Theory is a monograph on the mathematics of tic-tac-toe and other positional games, written by József Beck. It was published in 2008 by the Cambridge University Press as volume 114 of their Encyclopedia of Mathematics and its Applications book series (). Topics A positional game is a game in which players alternate in taking possession of a given set of elements, with the goal of forming a winning configuration of elements; for instance, in tic-tac-toe and gomoku, the elements are the squares of a grid, and the winning configurations are lines of squares. These examples are symmetric: both players have the same winning configurations. However, positional games also include other possibilities such as the maker-breaker games in which one player (the "maker") tries to form a winning configuration and the other (the "breaker") tries to put off that outcome indefinitely or until the end of the game. In symmetric positional games one can use a strategy-stealing argument to prove that the first player has an advantage, but realizing this advantage by a constructive strategy can be very difficult. According to the Hales–Jewett theorem, in tic-tac-toe-like games involving forming lines on a grid or higher-dimensional lattice, grids that