Lights Out! A game of combinatorics and linear algebra

TL;DR

This paper explores a triangular variant of the Lights Out game, using combinatorics and linear algebra to characterize solvable configurations, analyze pattern propagation, and establish a criterion for solutions based on tile coverings.

Contribution

It introduces the concept of quiet patterns, models the game with linear systems over Z2, and provides a new criterion for the existence of solutions based on tile coverings.

Findings

Characterization of configurations with solutions using quiet patterns

Linear algebraic model over Z2 for the game

Solution existence linked to the parity of tile coverings

Abstract

In this work, we study a triangular variant of the Lights Out game, proposed in the 2025 Capixaba Mathematics Olympiad. We present a combinatorial description of the game, formally characterize its operations, and introduce the notion of a quiet pattern, which determines which configurations admit a solution and how many solutions they possess. We then analyze the geometry of quiet patterns and describe the propagation mechanisms that generate patterns for larger board sizes. Finally, we model the problem using linear systems over the field Z2, obtaining a matrix associated with the game and a combinatorial criterion for its invertibility. This criterion shows that the game admits a solution for every configuration if and only if the number of coverings of the triangular board by 1 x 1 and 2 x 1 tiles is odd.