The King calls in three wise men and tells them to all close their eyes. While their eyes are closed, he goes around and puts a hat on each of them.
“I put a blue or white hat on each of you,” the King says. “I won’t tell you what color each hat is, but I promise you that at least one of you has a blue hat.”
“Now open your eyes,” he continues. “You may not communicate with each other at all. Within one hour, one of you must call out the color of your own hat. If you aren’t able to do this, or if anyone calls out the wrong color, I will have you all exiled from the kingdom.”
The wise men open their eyes and look at the other mens’ hats. They stand there for almost the whole hour in silence, thinking. Just as time is about to run out, all three men figure out the color of their own hats and yell the colors out at the same time.
You can assume that all three men are perfect logicians, that they know that the others are perfect logicians, and that they all think at the same speed.
What colors are the three men’s hats?
Their hats are all blue in color.
There are 3 possible hat color combinations:
- [A] 1 blue, 2 white
- [B] 2 blue, 1 white
- [C] 3 blue
The color combination of 3 white hats is not possible since the king has already said that at least one of the wise men has a blue hat.
So, let’s start our analysis.
What if there were one blue hat and two white hats? Then the wise man with the blue hat would have seen two white hats and immediately called out that his own hat was blue, since he knew there is at least one blue hat. This didn’t happen, and thus the hat color combination [A] is ruled out.
Now, the wise men knew that only 2 hat color combinations are possible – combination [B] or [C].
What if there were two blue hats and one white hat? The men with the blue hats will see one white hat and one blue hat. They will conclude that color combination [B] is the case and would call out blue as their hat color. This also did not happen, and thus the hat color combination [B] is also ruled out.
After some time, when none of the wise men are able to identify the color of their own hats, combination [C] (of 3 blue hats) becomes the only possible option.