A Pairing Strategy for Tic-Tac-Toe on the Integer Lattice with Numerous Directions

Abstract

We consider a tic-tac-toe game played on the $d$-dimensional integer lattice. The game that we investigate is a Maker–Breaker version of tic-tac-toe. In a Maker–Breaker game, the first player, Maker, only tries to occupy a winning line and the second player, Breaker, only tries to stop Maker from occupying a winning line. We consider the bounded number of directions game, in which we designate a finite set of direction-vectors ${\cal S} \subset{\Bbb Z}^d$ which determine the set of winning lines. We show by a simple pairing strategy that Breaker can win this game if the length of each winning line is at least $3|{\cal S}|.$ It should be noted that Breaker's winning strategy can be used as a drawing strategy for Player 2 in the strong version of this game.