Black and White Hats

Difficulty:
★★☆☆☆

Problem:

Alice, Bob, and Cathy have a box that contains three black hats and two white hats. They each close their eyes, take out a hat from a box and put it on. They close the box and open their eyes, so that they can see the color of the hats that the other two are wearing but they cannot see their own.

Alice says, "I don't know the color of my hat."

Bob says, "I don't know the color of my hat."

Cathy says, "I know the color of my hat."

What color is Cathy's hat?


Solution:

Let's think about what new information is revealed after each statement.

Alice doesn't know the color of her hat. Since there are only two white hats, if Bob and Cathy were both wearing white hats, Alice would figure out that her hat is black. But since she doesn't know, we (and Bob and Cathy) learn that Bob and Cathy are not both wearing white hats.

Now, if Cathy were wearing a white hat, Bob would know that he's wearing a black hat, since it can't be the case that he and Cathy are both wearing white hats. But since Bob doesn't know, Cathy must be wearing a black hat.