Cofinite Induced Subgraphs of Impartial Combinatorial Games: An Analysis of CIS-Nim

TL;DR

This paper analyzes cofinite induced subgraphs of Nim, revealing that CIS-Nim inherits a period-two scale invariance from Nim despite structural differences in winning strategies.

Contribution

It provides an analytical study of CIS-Nim, showing how these subgraphs retain certain invariance properties of the original Nim game.

Findings

CIS-Nim inherits period-two scale invariance from Nim.

Winning strategies in CIS-Nim can differ significantly from Nim.

Structural differences in CIS-Nim do not eliminate certain invariance properties.

Abstract

Given an impartial combinatorial game G, we create a class of related games (CIS-G) by specifying a finite set of positions in G and forbidding players from moving to those positions (leaving all other game rules unchanged). Such modifications amount to taking cofinite induced subgraphs (CIS) of the original game graph. Some recent numerical/heuristic work has suggested that the underlying structure and behavior of such "CIS-games" can shed new light on, and bears interesting relationships with, the original games from which they are derived. In this paper we present an analytical treatment of the cofinite induced subgraphs associated with the game of (three-heap) Nim. This constitutes one of the simplest nontrivial cases of a CIS game. Our main finding is that although the structure of the winning strategies in games of CIS-Nim can differ greatly from that of Nim, CIS-Nim games inherit a type of period-two scale invariance from the original game of Nim.