Impartial Chess on Integer Partitions

TL;DR

This paper generalizes impartial chess games played on Young diagrams, analyzing winning positions and Sprague-Grundy values, and classifies these games on various sets of partitions, extending classical combinatorial game results.

Contribution

It introduces a framework for impartial chess on Young diagrams and classifies these games, extending known results to new classes of partitions.

Findings

Classified impartial chess games on Young diagrams.

Derived Sprague-Grundy values for all chess pieces.

Connected these games to classical Nim variants.

Abstract

Berlekamp proposed a class of impartial combinatorial games based on the moves of chess pieces on rectangular boards. We generalize impartial chess games by playing them on Young diagrams and obtain results about winning and losing positions and Sprague-Grundy values for all chess pieces. We classify these games, and their restrictions to sets of partitions known as rectangles, staircases, and general staircases, according to the approach of Conway, later extended by Gurvich and Ho. The games $\rm {R\small OOK}$ and $\rm{Q\small UEEN}$ restricted to rectangles are known to have the same game tree as $2$-pile $\rm N{\small IM}$ and $\rm W{\small YTHOFF}$, respectively, so our work generalizes these well-known games.