Matrix, Graph, Lights Out!, Sylvester Equation
Lights Out!\ is a game played on a $5 \times 5$ grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out!\ on Cartesian products of graphs are resolved.
Curtis, Bryan A.; Earl, Jonathan; Livingston, David; and Shader, Bryan L..
"Resolution Of Conjectures Related To Lights Out! And Cartesian Products",
Electronic Journal of Linear Algebra,
Volume 34, pp. 718-724.