Press a light to switch it and its neighbors, and try to turn every light off.
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Two facts make this work. The order of presses doesn't matter, and pressing a light twice is the same as not pressing it at all. So a solution is just a set of lights, each pressed once.
Work down the board one row at a time. Under every light that is on, press the light just below it. That turns it off without touching the rows above. At the end, only the bottom row can still have lights on.
Tiger Electronics sold Lights Out in 1995, a hand-held game with a 5 × 5 grid of lighted buttons. Its forerunner was Magic Square, a 3 × 3 game on Parker Brothers' Merlin in 1978.
Klaus Sutner analyzed n × n boards in 1989. In 1998 Marlow Anderson and Todd Feil used linear algebra to show that only one 5 × 5 board in four can be solved, and each of those in exactly four ways.
The arithmetic underneath has 1 + 1 = 0, since a light pressed twice is back where it started.