100 prisoners are put into solitary cells. There's a central living room with one light bulb; the bulb is initially off. No prisoner can see the light bulb from his or her own cell. Every day, the warden picks a prisoner at random, and that prisoner goes to the central living room. While there, the prisoner can toggle the bulb if he or she wishes. Also, the prisoner has the option of asserting the claim that all 100 prisoners have been to the living room. If this assertion is false (that is, some prisoners still haven't been to the living room), all 100 prisoners will be shot for their stupidity. However, if it is indeed true, all prisoners are set free and inducted into MENSA, since the world can always use more smart people. Thus, the assertion should only be made if the prisoner is 100% certain of its validity.
The prisoners are allowed to get together one night, to discuss a plan. What plan should they agree on, so that eventually, someone will make a correct assertion?
This would work if the prisoners will live for a REALLY long time:
Designate one prisoner as a counter who counts the number of times it's certain that someone new has been in the room. Every other prisoner essentially signals the counter when they have been in the room.
For every prisoner except the counter, the strategy is this: if the light bulb is off and they have never turned the light bulb on before, turn it on now. Otherwise, leave it.
If the counter finds that the light bulb is on, he or she turns it off. After doing this 99 times it's certain that every prisoner has been in the room.