Compound Node-Kayles on Paths

TL;DR

This paper analyzes twelve versions of compound Node-Kayles on paths, solving most under normal and misère play, with one remaining unsolved version, advancing understanding of this combinatorial game.

Contribution

It provides a comprehensive analysis of twelve compound versions of Node-Kayles on paths, solving ten and identifying the only remaining unsolved case.

Findings

Most compound versions are solved under normal play.

Most compound versions are solved under misère play.

One version remains unsolved.

Abstract

In his celebrated book "On Number and Games" (Academic Press, New-York, 1976), J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node-Kayles game on paths. For usual disjunctive compound, Node-Kayles has been solved for a long time under normal play, while it is still unsolved under mis\`ere play. We thus focus on the ten remaining versions, leaving only one of them unsolved.