photo credit: Toni Blay via photopin cc
In a "Lights Out" puzzle, the goal is to turn everything off, by hitting switches that toggle certain combinations on the board.
For instance, I might encounter a door guarded by laser beams. There are 4 laser beams bouncing around and if they are all turned off, I can make it through the door safely. If even one of them is on, I will get disintegrated. We'll call these lasers 1, 2, 3, and 4.
O O O O
In front of the laser-guarded door are 5 levers, which we'll call A, B, C, D, and E. Each lever can toggle one or more of the laser beams. So maybe if I hit A, it toggles 1 and 2 (turning them both off):
x x O O (x means off, O means on)
but if I then hit B and it toggles 2 and 3, then 2 gets turned back on while 3 gets turned off:
x O x O
So, the puzzle is to find the right sequence of levers that toggles all of the laser beams into an "off" position.
This is made more fun/crazy/insane/mean if the GM applies a time limit, like a crushing wall moving in from the opposite side of the room.