On the Nature and Complexity of an Impartial Two-Player Variant of the Game Lights-Out

TL;DR

This paper investigates a two-player impartial variant of Lights-Out played on graphs, analyzing its complexity and computing Nimbers for specific graph classes using recursive algorithms.

Contribution

It introduces a new two-player version of Lights-Out and develops methods to compute Nimbers on grid and Petersen graphs, expanding understanding of its combinatorial structure.

Findings

Nimbers for 2 x n grid graphs computed

Nimbers for certain Petersen graphs determined

Recursive algorithms effectively analyze game complexity

Abstract

In this paper we study a variant of the solitaire game Lights-Out, where the player's goal is to turn off a grid of lights. This variant is a two-player impartial game where the goal is to make the final valid move. This version is playable on any simple graph where each node is given an assignment of either a 0 (representing a light that is off) or 1 (representing a light that is on). We focus on finding the Nimbers of this game on grid graphs and generalized Petersen graphs. We utilize a recursive algorithm to compute the Nimbers for 2 x n grid graphs and for some generalized Petersen graphs.