Nim on Integer Partitions and Hyperrectangles

2025/06/05 by Gottlieb, Eric, Krnc, Matjaž, Muršič, Peter
#05A17 #91A46 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.04991

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.

Citations

• Sprague-Grundy values and complexity for LCTR
• Game positions of Multiple Hook Removing Game
• Multiple Hook Removing Game Whose Starting Position is a Rectangular Young Diagram with the Unimodal Numbering
• On tame, pet, domestic, and miserable impartial games

Related

• Fractal Sequences and Restricted Nim
• Slow k-Nim
• More about Exact Slow k-Nim
• On the Sprague-Grundy function of Exact k-Nim
• Row Impartial Terminus