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 The Logician's Birthday (Posted on 2004-11-17)

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?

 Submitted by Erik O. Rating: 3.5417 (24 votes) Solution: (Hide) Since we know there is at least 1 of each color, the distribution of hats can be one of the following: 1, 1, 4 1, 2, 3 2, 2, 2 (it doesn't matter which column represents which color) In the first case, all 6 guests would look around to check out the colors. The two logicians who were singly represented would know immediately that they only saw two colors instead of three and know that the hat on their head was the third color. In this case, two guests would stand up immediately and claim their special prize. In the second case, only 1 logician would see only two colors represented and know immediately that their hat was the third color. Since no one stood up immediately to claim their prize, they each must have seen at least 1 of each color hat. Knowing that the person they assumed was wearing the singular hat would know right away that their hat was singular they would expect that person to claim the prize. When no one got up immeidately they each realized that they, too, were wearing that color. All 6 guests, being logicians, recognized this at the same time and all claimed their prizes.

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