I'm not seeing why they need to turn the switch ON twice. Couldn't they flip the switch only once and have the counter wait for <number of prisoners> ONs (N - 1 + 1)?
Because the switches start in an unknown state. So the first time the counter enters the room and sees switch A on he has no way to know if that's because another prisoner turned it on, or because he's the first one in the room and it started in the on position.
