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 Template:Nowrap game. The Template:Nowrap or Template:Nowrap games are trivially won by the first player as there is only one space (Template:Nowrap and Template:Nowrap). A game with Template:Nowrap and Template:Nowrap 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 $\frac{{(n+2)}^d-n^d}{2}$ 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

• Template:Annotated link

External links

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