Wednesday, April 3, 2013

Puzzler - Prisoners and Lightbulbs

There are 100 prisoners in solitary confinement. There's a central living room with one light bulb that is  initially off. No prisoner can see the light bulb from his or her own cell. Everyday, the warden picks a prisoner at random to visit the living room. While there, the prisoner can toggle the bulb if he or she wishes. The prisoner also has the option of asserting that all 100 prisoners have been to the living room by now. If this assertion is false, all 100 prisoners are shot. However, if it is indeed true, all prisoners are set free. So the assertion should only be made if the prisoner is 100% certain of its validity. The prisoners are allowed to get together one night in the courtyard, to discuss a plan. What plan should they agree on, so that eventually, someone will make a correct assertion?

Solution: Here is one sub-optimal solution: Suppose each of the prisoners are assigned a different number from 0-99. Each day is taken modulo 100 and if the prisoner is assigned the same number as the day he turns the light on (or keeps it on) and otherwise he turns the light off (or keeps it off). Whenever a prisoner is called to the room and sees that the light is on, he makes note that the previous days' prisoner has been to the room. Once any of the prisoners has tallied every other prisoner being in the room, he can declare so and they will be set free. For example, say prisoner 50 is called to the room on day 134=34 (mod 100) and sees that the light is on. He concludes that prisoner 33 was in the room the previous day and then proceeds to turn the light off.

There is a simple way to alter this solution to make it way better -- the expected wait goes from ~10,000 to ~1,500 days. Can you think of it?


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