This puzzle belongs to the classic family of hat problems, a tradition of logic where the decisive information is not always in what is seen. The third blind wise man variant has an especially elegant twist: whoever ends up solving the problem seems to have less information than anyone else. And yet, he is the only one who can answer with certainty.
The third blind wise man
Riddle statement
Three perfectly logical wise men are sitting in a circle. The king informs them that he has five hats: three white and two black. Next, place a hat on each wise man and hide the other two. The hats are visible to the others, although no one can see theirs.
Then the king asks them in turn, always in the same order, if they can deduce with certainty the color of their own hat. The first one answers no. The second also answers no. Then the third, who is blind, answers yes.
What color is your hat?
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Solution
Answer: The hat of the third wise man is white.
Explanation:
Let's call the three wise men A, B and C, according to the order in which they answer. C is the third wise man, the one who is blind.
A answers first. If A had seen that B and C were both wearing black hats, he would have known immediately that his own hat was white, because there are only two black hats available. But A replies that he doesn't know. Therefore, B and C cannot both wear black hats.
Then B answers. B has heard A's answer. Suppose C were wearing a black hat. In that case, B could reason like this: "If I also wore black, then A would have seen two black hats and would have known that his was white. But A didn't know. Therefore I can't wear black; I would have to wear white."
So, if C were wearing a black hat, B would have been able to deduce his own color. But B also answers that he doesn't know.
Then C concludes that his hat cannot be black. If it were, B would have known that his was white. Since B couldn't figure it out, C's hat has to be white.
The key is that each negative answer eliminates one possibility. The third wise man does not solve the riddle by looking at hats, but by understanding what the other two could have deduced.