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?

A few people put down the correct answer but failed to explain the reasoning behind the explanation.

First to consider is the total number of possible hat arrangements that can be made using the three colors:

R=Red
B=Blue
Y=Yellow

1. R R R R Y B
2. R R R Y Y B
3. R R R Y B B
4. Y Y Y Y R B
5. Y Y Y R R B
6. Y Y Y R B B
7. B B B B R Y
8. B B B R R Y
9. B B B R Y Y
10. R R Y Y B B

Then we have to consider what the people wearing each sequence would see seated around them. We automatically can removed numbers 1-9 because in ever instance there is ONE or TWO hats of one color. In all these instances the person wearing the singularly colored hat would easily stand up before anyone else, instead as a group as described, or in some cases (#1, 4 & 7) two people would stand together and declare their colors.

To demonstrate this, take #4: In this instance, 4 people were wearing YELLOW hats, 1 person is wearing RED, and the last is wearing BLUE.

YELLOW wearing people would see 3 other YELLOW hats, 1 BLUE hat, and 1 RED hat. They obviously could not assume they were wearing any color over the other, since there is one of each color apparent to them.

RED wearing people would see 4 YELLOW hats and 1 BLUE hats. Since there was no RED hats visible and each color was present at least once, they know they are wearing RED. This is also true in the instance of the BLUE hat wearing person. Both RED and BLUE would stand up. However the riddle states EVERYONE stands up together...

ONLY in the instance of number 10, would there be cause for pause and then the realization, with everyone rising together....

Consider the following.
Assume that you are wearing the YELLOW hat. You would see 2 RED, 2 BLUE and 1 YELLOW. When you look around and no one gets up immediately, you learn a few things. Mainly that your hat is neither RED nor BLUE, because if it WAS, then the person wearing YELLOW would have already stood up! Because they did not stand up you can only be wearing the same color as them.

Consider this to completely explain as demonstrated by each person and what they would see…

#1 & 4 – R R B B Y
#2 & 5 – R R B Y Y
#3 & 6 – R B B Y Y

Only in the case of two of each color would there be reason to pause, in every other situation, one or two people would stand up immediately as I stated above.

Posted by Gary on 2004-12-10 00:08:27