Problem #2 The Five Hats Problem Three men are condemned to die, one of whom is

Question

Problem #2 The Five Hats Problem Three men are condemned to die, one of whom is blind. The king decides he will offer them an opportunity to be set free. The three men are arranged in a circle facing one another. The king produces 5 hats: 2 black and 3 white. The king places a hat on the head of each person and then destroys the two remaining hats. The men have no idea which hats have been destroyed. The king instructs them, "The first one of you whocan tell me the color of his hat will be set free." A period of time passes in silence and then finally the blind man tells the king the color of his hat and is set free. What color hat was the blind man wearing and how did he know? This does not have a "trick" answer-your answer should be very logical and well thought out. Be able to explain your answer from the viewpoint of each of the three prisoners. Hint: Each sighted man can see the blind man's hat as well as that of · · . · 8 . · · the other sighted man. What does the pause in time infer? What are the possible states?

Explanation / Answer

ANSWER: The blind prisoner’s hat is WHITE.

PROOF: There are seven possible permutations of hats:

1 W B B
2 B W B
3 W W B
4 W B W
5 B W W
6 B B W
7 W W W

(1) If the permutation 1 were correct, then the 1st prisoner would see two BLACK hats and know his hat was WHITE. But the 1st prisoner doesn’t know what color his hat is, so permutation 1 is ruled out.

(2) The 2nd prisoner looks at the 3rd prisoner. If he sees a BLACK hat on the 3rd prisoner, then he will know that his hat must be WHITE. That is, if the 3rd prisoner has a BLACK hat, the 2nd prisoner would know that either permutation 2 or permutation 3 is correct, and either way, his hat must be WHITE. The only permutation where the 3rd prisoner could have a BLACK hat and the 2nd prisoner NOT have a WHITE hat is permutation 1, which is already ruled out already in (1). So if the 2nd prisoner sees a BLACK had on the 3rd prisoner, we would know his hat is WHITE. But he does not know what color his hat is. Therefore, he does not see a BLACK hat on the 3rd prisoner. Therefore, the 3rd prisoner’s hat must be WHITE.

(3) To sum up, the 1st prisoner’s ignorance rules out permutation 1. The 2nd prisoner’s ignorance rules out permutation 2 and permutation 3. Since permutations 1-3 are ruled out, the correct answer must be 4, 5, 6 or 7. And it doesn’t matter which one is correct, because in all of these, the 3rd prisoner’s hat is WHITE, which is all he needs to know.

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