Monday, July 28, 2008

Riddle #22: The Hats

You are a prisoner and you are totally blind. The warden takes you and three others into a room and tells you that each of you will have the opportunity to be set free. He puts a blindfold on the other two prisoners (no need to put one on you), and makes this announcement:

"I have three white hats and two red hats. I will place a hat on each of your heads and remove the blindfold. If you can tell me what color hat is on YOUR OWN head, then you will be set free. However, if you are incorrect, then you will be executed."

He places a hat on each of your heads and removes the blindfolds. The first prisoner looks at you and the second prisoner and says, "I cannot tell what color hat is on my head." The second prisoner looks at you and the first prisoner and says, "I cannot tell what color hat is on my head."

You think about it for a minute and then, even though you are blind, you correctly deduce what color hat is on your own head. What color hat is it and how did you figure it out?

NOTE: The correct answer has been guessed. To read it, see the comments section.

Lateral Thinking Puzzlers by Paul Sloane has lots of classic riddles, clues and answers. Click here to order it.


  1. Okay, is it a white hat? There are
    three white hats to two red, so the chances of getting a white hat is greater.

  2. Well, you can't just guess, because if you are wrong, you die. So you have to be absolutely certain, and you have to tell me why you made that deduction.

  3. The other two have red hats. Each
    of them looked at you and the other prisoner and saw one of each
    so they could not tell. So your hat is different from the others, and the chance of getting white is
    greater than getting red, so your
    hat it white. Does that make any

  4. But it could also be that you have a red hat and the other two have white hats. They still would be unable to guess which one they had. I wouldn't be willing to risk my life on the law of probability. I know I'm missing something. I'll just have to keep pondering.

  5. Aaron, actually you don't figure out what the other prisoners are wearing. All you can figure out is what YOU are wearing, but you can know that with certainty, not just because of probability.

  6. Cathy, think carefully about the logic of your statement. Remember, the second prisoner knows that the first prisoner can't figure it out. Then he looks at you two and he still can't figure it out.

  7. Ok--I didn't get that P2 would know P1 was unable to figure it out. That means you have a white hat. P1 would have seen your hat. Whether P2 had a white hat or a red hat, he would not have been able to know his own hat. P2 cannot deduce from P1's answer what color he has and, for the same reasons, cannot guess by looking. If your hat had been red, P2 would have known his hat was white. So your hat must be white. Did I explain it clearly enough?

  8. Cathy, Yes, I think you've got it. Let me explain it another way. Prisoner 1 did NOT see two red hats, or else he would have known that his was white. He saw either two white hats OR a white and a red hat. Prisoner 2, knowing this, was still not able to tell what color hat he had. If your hat had been red, then he would have known he had a white hat. So therefore your hat has to be white.

  9. Alright, first, what color hats are on the other two prisoner's heads? That's an important piece of information.

  10. Anonymous, read all the comments. The interesting thing is, you do not need to know the colors of the other people's hats to tell what color your hat is. All you need to know is the information I gave you. And since the answer has been given, I'll explain it again, another way: First you know that prisoner 1 didn't know what color his own hat was; therefore he did not see two red hats. You also know that prisoner 2, knowing that prisoner 1 did not see two red hats, still didn't know what color his own hat was. That means that your hat could not have been red, because if it had been red, then prisoner 2 would have immediately known that his hat was white. Therefore you know you have a white hat. Make sense?

  11. But there is still a probability that prisoner 2, knowing the fact that prisoner 1 might be wearing a white hat and prisoner 3 wearing a red hat, might be actually wearing a white hat. It still confuses me.. Maybe my left brain would solve it :).

  12. The only way the first man knows which hat he has on is if he sees 2 red hats (because there are only 2 of those). So obviously he sees at least 1 white hat.

    The second man can know which hat he has on if he sees 2 red hats, OR if he sees a red hat on the 3rd prisoner, because if there were a red hat on him, his own hat must be white (or else the first prisoner could have guessed).

    Since the second man does not know, the 3rd man knows that his hat must be white.

  13. There seems to be several combinations of this.
    If 3rd man is wearing red,both of the other two would have to be wearing white, yet they would not be able to determine their own color. If the 3rd man is wearing white, both of the other two could be wearing either white or red, yet they still would not be able to determine their own color. So I dont see how the 3rd man could determine his color.