Potential-Based Strategies for Tic-Tac-Toe on the Integer Lattice with Numerous Directions

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 S ⊂ Zd which determines the set of winning lines. We show, by using the Erdős–Selfridge theorem and a modification of a theorem by Beck about games played on almost-disjoint hypergraphs, that for the special case when the coordinates of each direction-vector are bounded, i.e., when S ⊂ {~v : ‖~v‖∞ 6 k}, Breaker can win this game if the length of each winning line is on the order of d 2 lg(dk) and d2 lg(k), respectively. In addition, we show that Maker can build winning lines of length up to (1+o(1))d lg k if S is the set of all direction-vectors with coordinates bounded by k. We also apply these methods to the n-consecutive lattice points game on the Nd board with (essentially) S = Zd, and we show that the phase transition from a win for Maker to a win for Breaker occurs at n = (d+ o(1)) lgN .