A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test. The princes are gathered into a room and seated, facing one another, 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 prince's are unable to deduce the color of their hat, or are unwilling to guess. What color is your hat? Note: You know that your competitors are very intelligent and want nothing more than to marry the princess. You also know that the king is a man of his word.

Guest Jul 10, 2017

#1**+1 **

The wording of this question is not complete.

The King has devised a FAIR test. Where each Prince has equal chance of knowing or guessing the correct answer.

The only way that the chance of guessing correctly can be the EQUAL is if all three of them have a white hat on.

Melody
Jul 12, 2017

#2**+1 **

You're right melody, the only fair test is the test where everybody gets the same hat color (not that trivial, but true). But i think this is not the way you were supposed to answer the question. Here's how i think we are supposed to answer it:

Suppose you have a black hat. That means each of the other competitors see one black hat and one white hat. And because the other competitors are known to be very intelligent, they can infer from that situation they have a white hat and say that color.

Why? Each of the other competitors thinks for himself "suppose i had a black hat. That means the othee competitor WITH THE WHITE HAT (not you, we assumed you have a black hat) sees 2 black hats, and because there are only 2 black hat IMMEDIATELY realizes he has a white hat. But that didnt happen, so i have a white hat.

But that didnt happen either, so you have a white hat.

Guest Jul 13, 2017

#3**0 **

Hmmmmm.........the other two princes could not immediately deduce the color of his own hat...

But.... if you have a white hat, the other two princes see the same scenario you see. Each prince is looking at 2 princes with white hats who, at first, could not deduce his own hat color.

So why can you deduce your hat color, but not the other two?

.....After I thought about that question I realized this.....

Well........when each prince realizes that nobody could deduce his own hat color, each will simultaneously know that his hat is white.

Because.....

If *anyone* is wearing a black hat, then someone will be able to deduce his own hat color....

If 2 guys have black hats, the white hat will know.

If 1 guy has a black hat, the 2 white hats will know when each sees the other doesn't know. and so

If 0 guys have black hats, all 3 white hats will know when each sees the others don't know.

hectictar
Jul 13, 2017

#4**0 **

This would make it more precise:

Each prince has a little whiteboard. After they all get their hats and look at each other, they write down whether or not they know their own hat color. Then, all at once, they show their whiteboards to each other. If nobody yet knows, they turn it back around and do it again.

Because, as it is....

If only you have a black hat, you see 2 white hats who don't immediately know their own hat color.

If nobody has a black hat, you see 2 white hats who don't immediately know their own hat color.

But in the first situation, the 2 white hats will know their own hat color sooner.

( I added this in a separate answer because I didn't want my other answer to look really long... )

hectictar
Jul 13, 2017