# The Five Hats

Three men are being held prisoner. Let's call them Abel, Baker and Charlie. One day, a guard walks into their cell carrying five hats. Three of the hats are red, and two of the hats are white. The guard puts one hat on each prisoner's head. They cannot see their own hat, or the two hats the guard has left.

The guard says, "If any one of you can tell me what colour hat you have on your head, you can all go free."

Abel looks at the other two and says, "I don't know."

Baker looks at the other two and says, "I don't know."

Charlie, who is blind, tells the guard what colour hat he has on his head, and explains how he knows.

What colour hat did Charlie have on his head, and how did he know?

# Answer to The Five Hats

1. Charlie knows that if Abel saw two white hats, he would know that he had a red hat.

2. Charlie also knows that Baker did not see two white hats, or he would have known he had a red hat.

3. If Baker had seen a white hat on Charlie's head, he would have known he had a red hat, because he (Baker) knows Abel did not see two white hats.

Since Baker did not know what colour his hat was, Charlie knew he must have a red hat.

The easiest way to solve this problem is with a matrix or truth table showing all possibilities:

 Abel Baker Charlie 1. Charlie knows that if Abel saw two white hats, he would know that he had a red hat. 2. Charlie also knows that Baker did not see two white hats, or he would have known he had a red hat. 3. If Baker had seen a white hat on Charlie's head, he would have known he had a red hat, because he (Baker) knows Abel did not see two white hats.
Notice that we still don't know what colour hats Abel and Baker were wearing, but we have eliminated every possibilty that Charlie was wearing white.