Nd Game

A ''n''^''d'' game (or ''n''^''k'' game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions. It is a game played on a ''n''^''d'' hypercube with 2 players. If one player creates a line of length ''n'' of their symbol (X or O) they win the game. However, if all ''n''^''d'' spaces are filled then the game is a draw. Tic-tac-toe is the game where ''n'' equals 3 and ''d'' equals 2 (3, 2). Qubic is the game. The or games are trivially won by the first player as there is only one space ( and ). A game with and cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line.

Game theory

An ''n^d'' game is a symmetric combinatorial game.

There are a total of $\frac$ winning lines in a ''n''^''d'' game.

For any width ''n'', at some dimension ''d'' (thanks to the Hales-Jewett theorem), there will always be a winning strategy for player X. There will never be a winning strategy for player O because of the Strategy-stealing argument since an n^d game is symmetric.

See also

*

References

External links

Higher-Dimensional Tic-Tac-Toe
from the PBS Infinite Series on YouTube

Tic-tac-toe
Unsolved problems in mathematics

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