Note: Read this problem carefully, because it's completely different from the original.

As before, 100 prisoners are put into solitary cells, and there's a room with a light bulb. (No prisoner can see the light bulb from his or her own cell.) Every night, the warden picks a prisoner at random, and that prisoner goes to the living room. While there, the prisoner can toggle the bulb if he or she wishes. but this time, the prisoner needs to assert that he knows, which prisoner was in the living room before him. If the assertion is false, all 100 prisoners will be shot. However, if it is indeed true, all prisoners are set free. 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.

But, the prisoners know that after that night, when they will go back to their solitary cells. the warden will choose one prisoner secretly (and this time, not randomly) and will kill him.

What plan should they agree on, so that eventually, someone will make a correct assertion?

Designate 7 prisoners as Sunday, Monday, ... , Saturday. Anyone going in the first day will turn off the light (or leave it off) if he's not the designee for that day of the week, but turn it (or leave it) on if he is the designee. From then on, if one of the 7 designees is called in on his day of the week he turns on the light. If anyone finds the light on (after the first time) he announces the name of the designee for the previous day of the week.

This is a particular example of a count mod some number, in this case 7.  Other numbers might be more optimal in terms of expected speed of release.

Posted by Charlie on 2007-03-07 11:52:17