Listening to the latest Car Talk, I was happy to hear a variant of the muddy children puzzle as this week’s “puzzelah.” Here it goes, straight from Ray:
The warden admits three prisoners into his chambers. He tells them, “One of you fellas is going to have a chance to get out. Here’s the deal.
“I’m going to blindfold all of you, then I’m going to put hats on your heads. I have three white hats and two black hats. Each of you is going to get a hat. You have to figure out which color hat you have to get released.”
He blindfolds them and puts a hat on each prisoner. They’re led out of the room in single file. When the blindfolds are removed, the guy in the back can see the two people in front of him, the guy in the middle can see the one guy in front of him, and the guy in front can see nobody.
They walk around the prison, stopping outside the warden’s office. The warden says to the fellow in back, who can see the two people in front of him, and their hats, “Can you tell me what color your hat is?”
Don’t forget, there are three white hats and two black hats available. The fellow in back says nothing. He doesn’t know.
The fellow in the middle is asked the same question. He is unable to answer.
The guy in the very front, who can see no hats, knows. He says, “I can identify the color of my hat.”
How does he know?
Those of you who have taken Advanced Game Theory will be done with this before I can finish this sentence. And if you haven’t taken that course yet and enjoy this sort of thing, you should definitely take Advanced Game Theory (Econ 410) this coming Spring, where we ponder some similar puzzles (and, yes, I do know that the course title has “& Applications” in it). All this goes under the heading Interactive Epistemology, and it is generally as complicated as it sounds. But also very important, as the surprising answer to this puzzle will no doubt show you.
Prof. Gerard,
The answer does generalize.
The answer (specific, not general):
If the front two hats were black, then the back person would know his hat was white. Therefore, at least one of the two front hats must be white. If the front hat is black, the middle person would know that his had to be the white one. Since he doesn’t know, the front person must be wearing a white hat.
If we think we know the answer, do you want us to post our answer here?
I hope it’s not too surprising.
It seems like it would be fun to generalize this to n players with n -1 white hats, no?