Transfinite game values in infinite draughts

TL;DR

This paper demonstrates that in infinite draughts, every countable ordinal can be realized as a game value, showing the rich and complex strategic structure possible in an infinite setting.

Contribution

It proves that all countable ordinals can be represented as game values in infinite draughts, revealing the depth of strategic possibilities in infinite combinatorial games.

Findings

Every countable ordinal is a game value in infinite draughts.

Positions exist where the game length can be controlled by Black to match any countable ordinal.

Infinite draughts exhibits a wide range of transfinite strategic complexities.

Abstract

Infinite draughts, or checkers, is played just like the finite game, but on an infinite checkerboard extending without bound in all four directions. We prove that every countable ordinal arises as the game value of a position in infinite draughts. Thus, there are positions from which Red has a winning strategy enabling her to win always in finitely many moves, but the length of play can be completely controlled by Black in a manner as though counting down from a given countable ordinal.