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 re: Another Aproach: by jmnw)

Hi Jen,

I know Tristan already explained the problem as well, but I thought this might help too.

The important clue is that there was a pause.  "No one claimed the prize immediately."

You gave the example: "If I see 2 yellow hats, 2 red hats, and 1 blue hat, why does this mean I am wearing blue also? I could just as well have on yellow or red, as each color has already been represented at least once."

If you were wearing a yellow hat (so there are 3 yellow, 2 red and 1 blue) what does the person wearing Blue see?  3 yellow and 2 red, right?  So IF you were wearing yellow, the person wearing the blue hat would have known their own hat color immediately and said something - no pause.

But since no one said anything immediately, you know that this is not true, so you must not be wearing a yellow hat.  (Same argument goes for you wearing a red hat).

The basic point is that if there were only one hat in a particular color, that person would have known so immediately because they'd only see the other two colors and think "but the host said there was at least one of each color, so I must have the color I don't see" and then they would say their hat color immediately.

Since no one says anything immediately we know that there isn't just one hat in a particular color.  In other words there must be at least two hats in each color.  Since there are 3 colors and 6 hats it just so happens that there are exactly two hats in each color.

Sorry for the wordy explanation, but i hoped that helped!

Posted by nikki on 2004-11-19 13:15:01