Question

Three perfectly logical men are told to stand in a straight line, one in front of the other. A hat is put on each of their heads. Each of these hats was selected from a group of five hats: two identical black hats and three identical white hats. None of the men can see the hat on his own head, and they can only see the person's hat in front of him. In how many distributions of the hats can the person in front deduce his own hat color?

Answer 1

If the first two men in the line wear black hats, the last person will identify that he is wearing a white hat (as there are only 2 black hats). So, the person in front will not able to deduce his own hat color. For, the person in front to deduce his own hat color, the first two person are either wearing at least one white hat. The possible distribution of hats of first 2 persons so that the last person is not able to deduce his own hat color is WW, WB and BW.

Now, for the BW combination, the middle person will see the first person wearing a black hat and he deduce that he is wearing a white hat because if he is wearing a black hat then as shown above, the last person is able to deduce his own hat color (BB combination). So, possible distribution of hat of front person so that the middle person is not able to deduce his own hat color is W.

Hence the front person should wear white then last 2 persons will not able to recognize their hat color. If the last 2 person are not able to deduce there own hat color, then the front person will definitely sure that his hat color is white.

So, the possible distributions of the hats so that the person in front deduce his own hat color are

W, B, B

W, B, W

W, W, B

W, W, W

There are total 4 distributions.