Almost optimal pairing strategy for Tic-Tac-Toe with numerous directions

TL;DR

This paper proves that Breaker can force a draw in a generalized Tic-Tac-Toe game with many directions using an optimal pairing strategy, improving previous bounds and establishing near-optimal conditions.

Contribution

It introduces an almost optimal pairing strategy for Breaker in multi-directional Tic-Tac-Toe, refining previous bounds and demonstrating near-optimality of the main term.

Findings

Breaker can force a draw with m=2n+o(n) in the game.

The result improves the previous bound of m≥3n.

The main term of the bound is shown to be optimal.

Abstract

We show that there is an $m=2n+o(n)$, such that, in the Maker-Breaker game played on $\Z^d$ 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 who showed that such a pairing strategy exits if $m\ge 3n$. 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.