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Three hats

Q:

The three wisest sages in the land were brought before the king to see which of them were worthy to become the king’s advisor. After passing many tests of cunning and invention, they were pitted against each other in a final battle of the wits.

 

Led blind-folded into a small room, the sages were seated around a small wooden table as the king described the test for them.

 

“Upon each of your heads I have placed a hat. Now you are either wearing a blue hat or a white hat. All I will tell you is this- at least one of you is wearing a blue hat. There may be only one blue hat and two white hats, there may be two blue hats and one white hat, or there may be three blue hats. But you may be certain that there are not three white hats.”

 

“I will shortly remove your blind folds, and the test will begin. The first to correctly announce the color of his hat shall be my adviser.  Be warned however, he who guesses wrongly shall be beheaded. If not one of you answers within the hour, you will be sent home and I will seek elsewhere for wisdom.”

 

With that, the king uncovered the sages’ eyes and sat in the corner and waited. One sage looked around and saw that his competitors each were wearing blue hats. From the look in their eyes he could see their thoughts were the same as his, “What is the color of my hat?”

 

For what seemed like hours no one spoke. Finally he stood up and said, “The color of the hat I am wearing is…”

A:
..."Blue." At first glance, this problem appears to be impossible to solve. Contributing to this is the feeling that the King's only real clue- that there is at least one blue hat- is useless since the sage can clearly see that there are at least two blue hats.

Don't feel bad if you sat stuck on this one for a while: as the puzzle clearly states, so did the three wisest sages in the kingdom. It is this fact that allowed our sage to give his answer. In truth, any one of them would have come up with it, given enough time. Why?

Consider a situation which we knew was not the case- that there was exactly one blue hat. What would happen? There would be a split second of pondering by the person wearing that hat, and he would say "I am wearing a blue hat." No real puzzle there, but of course there wasn't just one blue hat. The important fact is that everyone knew there was not one blue hat. But more importantly than that, everyone knew, or could quickly figure out that everyone else knew this (by the fact that answer was did not come out in the first few seconds.)

This leaves everyone wondering, "Are there two or three blue hats?"

Consider this less obvious situation- that there were exactly two blue hats. This seems a very real possibility at first, after all, we can see exactly two blue hats. So everyone sits and thinks- for a little while. But if there are only two hats, then two people see one blue and one white hat. These two people will very quickly, by virtue of the other's silence, rule out the possibility that there is only one blue hat. One of these two lucky sages would cry blue within a few short minutes, if that long.

There is only one case which forces the three sages to sit in silence- three blue hats. Our sage, through his sharp wits was the first to reach this conclusion.
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