Impartial Chess on Integer Partitions

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 $\mathrm{R}\mathrm{OOK}$ and $\mathrm{Q}\mathrm{UEEN}$ restricted to rectangles are known to have the same game tree as $2$-pile $\mathrm{N}\mathrm{IM}$ and $\mathrm{W}\mathrm{YTHOFF}$, respectively, so our work generalizes these well-known games.