Nd game

An nd game, also called an nk game, is a generalization of the classic combinatorial game tic-tac-toe to boards of higher dimensions, played by two players who alternate placing their own symbol on the cells of an n dimensional hypercube shaped board. A player wins by forming an unbroken line of a fixed target length within the board that contains only their own symbol, and if every cell fills up without either player achieving such a line, the game ends in a draw. Ordinary tic-tac-toe is the simplest familiar case, a three by three board played in two dimensions, while a well known three dimensional extension called Qubic is played on a four by four by four cube and requires a line of four in a row to win. Mathematicians studying the family of games have identified trivial cases, including games where the winning line length is zero or where both the board size and line length equal one, in which the first player automatically wins because only a single space exists to play in, and games confined to a single dimension with a winning line longer than one space, which can never be won by either player because a single blocking move by the opponent is always enough to prevent a line from forming.