four prisoners are arrested for a crime, but the jail is full and the jailer has nowhere to put them. He eventually comes up with the solution of giving them a puzzle so if they succeed they can go free but if they fail they are executed.
If any prisoner can figure out and say to the jailer what color hat he has on his head all four prisoners go free. If any prisoner suggests an incorrect answer, all four prisoners are executed. The puzzle is to find how the prisoners can escape, regardless of how the jailer distributes the hats.The jailer puts three of the men sitting in a line. The fourth man is put behind a screen (or in a separate room). He gives all four men party hats (as in diagram). The jailer explains that there are two red and two blue hats; that each prisoner is wearing one of the hats; and that each of the prisoners only see the hats in front of him but not on himself or behind him. The fourth man behind the screen can't see or be seen by any other prisoner. No communication between the prisoners is allowed.
Answer:
For the sake of explanation let's label the prisoners in line order A B and C. Thus B can see C (and C's hat color) and A can see B and C.
The prisoners know that there are only two hats of each color. So if A observes that B and C have hats of the same color, A would deduce that his own hat is the opposite color. However, if B and C have hats of different colors, then A can say nothing. The key is that prisoner B, after allowing an appropriate interval, and knowing what A would do, can deduce that if A says nothing the hats on B and C must be different. Being able to see C's hat he can deduce his own hat color.
In common with many puzzles of this type, the solution relies on the assumption that all participants are totally rational and are intelligent enough to make the appropriate deductions.
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