Paper / 2025
An Inductive Proof that Lights Out Configurations are Invertible, and a Parity-Invariance Result
Elementary graph arguments for complementing every Lights Out configuration and proving that the parity of successful move sequences is fixed.
- Authors
- Keivan Mirzaei
- Status
- Preprint
The Lights Out problem
Each vertex of a finite simple graph carries a light. Pressing a vertex toggles that vertex and each of its neighbors.
The paper gives an elementary inductive proof that any starting configuration can be turned into its complement: every light changes state.
Parity invariance
For a fixed starting configuration and any attainable target configuration, all sequences of presses reaching that target have the same parity. Equivalently, two solutions to the same target differ by an even number of presses.
Both results are established through elementary arguments about the graph.
Preprint history
First submitted to arXiv on September 22, 2025; revised on March 20, 2026.