Puzzle 65 | Hats Off

A king wants his daughter to marry one of 3 young princes, so he devises an intelligence test. The princes are gathered into a room and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining hats hidden in a different room. The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter. A wrong guess will mean death. The blindfolds are then removed. You are one of the princes. You see 2 white hats on the other prince’s heads. After some time you realize that the other princes are unable to guess the color of their hat. What is the color of your hat?

Solution:

There are two possibilities – either the hat on your head could be black or white. There are 3 cases :

1) either the king could have chosen 2 black hats and 1 white hat,

2) the king could have chosen 1 black and 2 white hats, and

3) the king could have chosen 3 white hats.

Let us examine these cases 1 by 1.

Case 1: There are 2 black hats and 1 white hat on the princes’ heads. In this case, the prince wearing the white hat can see that the other two princes have black caps on their heads and can immediately guess that he is wearing a white cap. This case is obviously not true.

Case 2: There is 1 black hat and 2 white hats on the princes’ heads. In this case, any prince wearing the white hat (Prince A) can see that one other prince is wearing a white hat (Prince B), while the third is wearing a black hat (Prince C). Now, if he was wearing a black hat (as in case 1), then the prince with the white hat (Prince B) would immediately have guessed the color of his hat. However, this is not true because the other prince was unable to guess the color of his hat. Hence, it is easy for this prince (Prince A) to guess that he is wearing a white hat. This does not happen, as we know that the princes are unable to guess the color of their hats. Therefore, this case is also not true.

Case 3: This only leaves us with the case where the king choose 3 white hats. Therefore, any of the three princes can, after waiting for a while, safely guess that they are wearing a white hat.