The Sprague-Grundy function for some selective compound games

TL;DR

This paper investigates the behavior of Sprague-Grundy functions in a specific class of selective compound Nim games, revealing chaotic patterns at small values and predictable behavior at large heap sizes.

Contribution

It provides new insights into the behavior of Sprague-Grundy functions in selective compound games, especially highlighting the transition from chaos to predictability.

Findings

Chaotic behavior of Sprague-Grundy functions at small heap values.

Predictable Sprague-Grundy behavior when one heap is large.

Analysis applies to a class of almost disjoint selective compound Nim games.

Abstract

We analyze the Sprague-Grundy functions for a class of almost disjoint selective compound games played on Nim heaps. Surprisingly, we find that these functions behave chaotically for smaller Sprague-Grundy values of each component game yet predictably when any one heap is sufficiently large.