Tic-Tac-Toe on an Affine Plane of order 4

TL;DR

This paper proves that in tic-tac-toe played on an affine plane of order 4, the first player has a guaranteed winning strategy, using combinatorial design theory to provide an explicit, human-verifiable proof.

Contribution

It offers the first human-verifiable proof that the first player can win tic-tac-toe on an affine plane of order 4, employing combinatorial techniques.

Findings

First player can force a win on affine plane of order 4

Provides explicit, human-verifiable proof of the first player's winning strategy

Uses combinatorial design theory to analyze the game structure

Abstract

The game of tic-tac-toe is well known. In particular, in its classic version it is famous for being unwinnable by either player. While classically it is played on a grid, it is natural to consider the effect of playing the game on richer structures, such as finite planes. Playing the game of tic-tac-toe on finite affine and projective planes has been studied previously. While the second player can usually force a draw, for small orders it is possible for the first player to win. In this regard, a computer proof that tic-tac-toe played on the affine plane of order 4 is a first player win has been claimed. In this note we use techniques from the theory of latin squares and transversal designs to give a human verifiable, explicit proof of this fact.