The following was a conjecture for several years. Put a light bulb and a switch at every vertex of an $m\times n$ grid ($mn$ vertices in all). Each bulb can be on or off. Each switch changes the state of the bulb at its vertex and all its neighbors. (A neighbor is a vertically or horizontally adjacent vertex.) Then whatever the intial set of bulbs that are lit, it is possible to turn all the lights off by some set of switches. Sutner showed in 1989 that the corresponding result is true for any graph. Caro gave a simpler proof in 1996. The proof is an elegant application of linear algebra mod 2. Looking at grids adds an extra layer of complexity that obscures the underlying theory. One reference is http://mathworld.wolfram.com/LightsOutPuzzle.html.

The following was a conjecture for several years. Put a light bulb and a switch at every vertex of an $m\times n$ grid ($mn$ vertices in all). Each bulb can be on or off. Each switch changes the state of the bulb at its vertex and all its neighbors. (A neighbor is a vertically or horizontally adjacent vertex.) Then whatever the initial set of bulbs that are lit, it is possible to turn all the lights off by some set of switches. Sutner showed in 1989 that the corresponding result is true for any graph. Caro gave a simpler proof in 1996. The proof is an elegant application of linear algebra mod 2. Looking at grids adds an extra layer of complexity that obscures the underlying theory. One reference is http://mathworld.wolfram.com/LightsOutPuzzle.html.