Impartial Chess on Integer Partitions

TL;DR

This work extends impartial chess from rectangular boards to Young diagrams, providing a comprehensive SG-value analysis for all standard chess pieces on partition families and situating the results within the Conway-Gurvich-Ho classification framework. By defining moves on partitions, introducing DAG-based partition- and game-equivalence, and deriving explicit SG-values and P-positions for King, Rook, and Queen (with reductions for Pawn and Knight to Downright), the authors reveal new CGH regions and connect partition games to classical Nim and Wythoff structures. The study also investigates misère versions via a truncation operation that preserves moves and links misère play to normal-play on modified partitions, along with an overarching CGH-type scheme for normal/misere interaction. Overall, the paper broadens the scope of impartial combinatorial game theory to partition-based boards, offering new analytical tools and concrete results across multiple piece-types and partition families. The results advance understanding of how classical chess-inspired moves behave in non-rectangular combinatorial geometries, with potential implications for broader partition- and lattice-based games.

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.

Figures (6)

• Figure 1: The Young diagrams of $\lambda=\llbracket 5, 4^2, 2, 1^2 \rrbracket$ (left), its conjugate $\lambda'=\llbracket 6, 4, 3^2, 1 \rrbracket$ (middle), and the subpartition $\lambda[0,2]=\lambda'[1,1]=\llbracket 3,2,2 \rrbracket$ (right).
• Figure 2: Partitions corresponding to $^{6}$, $_{6,4}$, $_{6,4}$ and $^{2}_{3,2}$.
• Figure 3: Visual representation of Sprague-Grundy values (left), and $\mathcal{P}$/$\mathcal{N}$ value (right) for $(\symking,\lambda)$ with $\lambda\le \llbracket 5,5,3,2,1 \rrbracket$.
• Figure 4: Partition equivalence depends on equality, not isomorphism, of DAGs.
• Figure 5: Knight and Pawn on $\lambda=\llbracket 14^2,13,12^4,9,5,3 \rrbracket$, shaded as and , are game-equivalent to Downright on $\varphi_\symknight(\lambda)=\llbracket 6,5,4,2,1 \rrbracket$ and $\varphi_{\sympawn}(\lambda)=\llbracket 14,13,11,9,8,7,6,2 \rrbracket$, resp.

Theorems & Definitions (36)

• Definition 2.1: Partition Families
• Definition 3.1
• Proposition 3.6
• proof
• Proposition 3.7
• proof
• Theorem 3.8: Berlekamp
• proof : Proof of \ref{['thm:SGKingrectangle']}
• Proposition 3.9
• proof