What Color is Chuck's Hat?

:date: 2014-02-01 10:56

:summary: A solution to the Three Hats puzzle.

I recently ran across a cool puzzle site. Here is one of the puzzles and a solution:

There are 3 black hats and 2 white hats in a box. Three men (we will call them A, B, & C) each reach into the box and place one of the hats on his own head. They cannot see what color hat they have chosen. The men are situated in a way that A can see the hats on B & C's heads, B can only see the hat on C's head and C cannot see any hats. When A is asked if he knows the color of the hat he is wearing, he says no. When B is asked if he knows the color of the hat he is wearing he says no. When C is asked if he knows the color of the hat he is wearing he says yes and he is correct. What color hat and how can this be?

Solution

I'm going to call the three men Alan, Bob, and Chuck.

First, let's imagine what Alan might be able to see. There are four possibilities.

===========  =============
 Bob's Hat    Chuck's Hat
===========  =============
 White        White
 Black        White
 White        Black
 Black        Black
===========  =============

Since there are only two white hats, if Alan sees that both Bob's and Chuck's hats are white his own hat would have to be black. In other words, by admitting that he can't tell which hat he is wearing Alan is saying that either or both of Bob's and Chuck's hats are black. If we eliminate the case of both hats being white we are left with three possibilities.

===========  =============
 Bob's Hat    Chuck's Hat
===========  =============
 Black        White
 White        Black
 Black        Black
===========  =============

From this it should be easy to see that if Bob sees that Chuck's hat is white his own hat would have to be black. Bob would be uncertain of the color of his own hat only if Chuck's hat is black. So Chuck, being no dummy, can conclude that his own hat is black.

It's an elegant puzzle with a very simple and satisfying solution.