Infinite Hex is a draw

TL;DR

This paper extends Hex to an infinite lattice, proving that unlike the finite version, infinite Hex results in a draw with both players having drawing strategies, and that all game values are finite and local.

Contribution

It introduces infinite Hex, proves it is a draw with finite game values, and shows all positions are intrinsically local, generalizing to simple stone-placing games.

Findings

Infinite Hex is a draw with both players having drawing strategies.

All game-valued positions in infinite Hex have finite values.

Winning play depends only on a finite region of the board.

Abstract

We introduce the game of infinite Hex, extending the familiar finite game to natural play on the infinite hexagonal lattice. Whereas the finite game is a win for the first player, we prove in contrast that infinite Hex is a draw -- both players have drawing strategies. Meanwhile, the transfinite game-value phenomenon, now abundantly exhibited in infinite chess and infinite draughts, regrettably does not arise in infinite Hex; only finite game values occur. Indeed, every game-valued position in infinite Hex is intrinsically local, meaning that winning play depends only on a fixed finite region of the board. This latter fact is proved under very general hypotheses, establishing the conclusion for all simple stone-placing games.