Asymptotically Optimal Pairing Strategy for Tic-Tac-Toe with Numerous Directions

We show that there is an m = 2n + o(n), such that, in the Maker-Breaker game played on Z where Maker needs to put at least m of his marks consecutively in one of n given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg [15] who showed that such a pairing strategy exits if m ≥ 3n. A simple argument shows that m has to be at least 2n+1 if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.