A logician invites 6 of his logician friends to help him celebrate his birthday. Each of the 6 guests is wearing a hat which is either red, yellow, or blue, and the logician host informs them all that there is at least 1 of each color. After they eat the cake, the host stands up and exclaims there is a special party prize for the first person who can deduce what color hat is on their head. The party guests all looked around the room at each other but no one claimed the prize immediately. Suddenly all 6 guests stood up and correctly identify what color hat was on their head.

What were the colors of their hats and how did they know?

(In reply to how about this? by jmnw)

I hope this explanation helps.

The 6 logicians do not deduce the color of their hats by looking at the hats of the others.  Each one sees 5 other hats, and yet cannot deduce the color of their own hat.  How can this happen?

You say, "As long as each person sees at least one of the 3 colors, there is no way to infer which color hat they themselves are wearing."  Exactly!  For each person to see all three colors, there must be at least two hats of each color; because if, say, only one hat is blue, then its wearer will see only red and yellow.  Therefore, there must be two red, two blue, and two yellow.

Reading further, however, we see that everyone deduced the color of their hat at the same time after a few moments.  How did they do this?  The same way we did.

Of course, we are assuming that they did not remove their hats, and that they all think at the exact same speed and in the exact same way.  Admittedly, these assumptions don't sound very realistic, but that is just the way that some puzzles go.

Posted by Tristan on 2004-11-19 02:51:27