An Inductive Proof that Lights Out Configurations are Invertible, and a Parity-Invariance Result

We give an elementary inductive proof of a classical result for the \emph{Lights Out problem} on graphs: from any configuration of vertices, one can reach the complementary configuration by a sequence of moves, where a move consists of toggling a vertex and its neighbors. We also prove, again by a purely elementary argument, a parity-invariance property: once an initial configuration is fixed, the parity of the number of presses required to reach an attainable configuration is determined by that configuration. In particular, any two solutions leading to the same attainable configuration differ by an even number of presses.