Nd game

A n^d 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 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. Qubic is the game. The or games are trivially won by the first player as there is only one space. 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 theory | combinatorial game].
There are a total of winning lines in a n^d game.
For any width n, at some dimension d, 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.