Lights Out on graphs

TL;DR

This paper models the Lights Out game on graphs using linear algebra over finite fields, introduces a new invariant for state transformation, and explores variants with different states, providing both theoretical insights and practical circuit implementations.

Contribution

It introduces a separating invariant for Lights Out states, extends analysis to multi-state variants over rings, and discusses the NP-hardness of minimal solutions.

Findings

Invariant determines state transformability.

Multi-state variants analyzed over rings.

Finding minimal solutions is NP-hard.

Abstract

We model the Lights Out game on general simple graphs in the framework of linear algebra over the field $\mathbb F_2$. Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the invariant of both states agrees. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis resides on systems of linear equations over the ring $\mathbb Z_n$. Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game.