Nim on Integer Partitions and Hyperrectangles

TL;DR

This paper introduces two variants of Nim, PNim and RNim, involving integer partitions and hyperrectangles, providing formulas and bounds for their Sprague-Grundy values and classifying their positions within established game hierarchies.

Contribution

The paper defines PNim and RNim, derives Sprague-Grundy value formulas, and classifies these games within the Conway-Gurvich-Ho hierarchy, advancing combinatorial game theory.

Findings

Derived tight upper bounds for Sprague-Grundy values in PNim.

Provided explicit formulas for Sprague-Grundy values in RNim.

Classified both games within the Conway-Gurvich-Ho hierarchy.

Abstract

We describe PNim and RNim, two variants of Nim in which piles of tokens are replaced with integer partitions or hyperrectangles. In PNim, the players choose one of the integer partitions and remove a positive number of rows or a positive number of columns from the Young diagram of that partition. In RNim, players choose one of the hyperrectangles and reduce one of its side lengths. For PNim, we find a tight upper bound for the Sprague-Grundy values of partitions and characterize partitions with Sprague-Grundy value one. For RNim, we provide a formula for the Sprague-Grundy value of any position. We classify both games in the Conway-Gurvich-Ho hierarchy.