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?

All three colors are present.

The hats are {red, yellow, blue, ?, ?, ?}. 

If {red, yellow, blue, red, red, red}, then the logician in the yellow hat would have spoken up, since he would have known that his hat couldn't possibly be blue or red, since all colors are present. (The one in the blue hat would have spoken up too.)

If {red, yellow, blue, red, red, yellow}, then the logician in the blue hat would see only yellow and red and would have spoken up.

The same reasoning applies to any case where a color is represented more than once in {?,?,?}.

So the only arrangement consistent with the prolonged silence of the logicians is {red, yellow, blue, red, yellow, blue}. Then each logician identified his color by looking at the other five; he knew his color was the singular one among those he observed: e.g. if he saw 2 red hats, 2 blue hats and 1 yellow hat, his hat must be yellow. 

Someone once made a "symmetry argument" that also leads to this conclusion.    

 

 

 

 

Edited on November 17, 2004, 7:43 am

Posted by Penny on 2004-11-17 07:10:21