APAGUE AS LUZES! UM JOGO DE COMBINATÓRIA E ÁLGEBRA LINEAR

A GAME OF COMBINATORICS AND LINEAR ALGEBRA

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 the kernel, which determines which configurations admit a solution and how many solutions they possess. We then analyze the geometry of kernel 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 1x1 e 2x1 tiles is odd.

Keywords: linear algebra; combinatorics; matrix