Three prisoners and colored hats in a line

Question

Three prisoners are told there are 5 hats: 3 white and 2 black. Three of the hats are placed on their heads; the other two are hidden. The prisoners stand in a line: the back prisoner sees the two in front, the middle sees only the front, and the front sees no one. None can see their own hat. Starting from the back, each is asked "Do you know the color of your hat?" The back prisoner says "No." The middle prisoner says "No." Then the front prisoner, who can see nothing, says "Yes, I know mine." What color is the front prisoner's hat, and how does he know?

Accepted Answer

The front prisoner's hat is white. He reasons from the two "No" answers. What the back prisoner's "No" reveals. The back prisoner sees the middle and front hats. There are only 2 black hats total. If the back prisoner saw two black hats in front, he'd know his own must be white (the two blacks are used up) and would say "Yes." Since he said "No," the middle and front hats are not both black. So at most one of {middle, front} is black. What the middle prisoner's "No" reveals. The middle prisoner sees only the front hat, and he has just heard the back prisoner's "No," so he too knows middle-and…

Three prisoners and colored hats in a line

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