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What Colour is your Hat?

  1. Apr 30, 2007 #1
    A professor wished to figure out who was the smartest of his graduate students. He brought them into his office, blindfolded them, covered all the mirrors in the room, and sat them in chairs facing one another. He told them that he would put either a yellow or a green hat on their heads, but then placed, on all three students, a small green hat (representing Creativity from De Bono's Six Thinking Hat Colors) and no yellow ones! He then told them to remove their blindfolds and to look at one another, but not their own heads. He asked "raise your hand if you see a green hat" to
    which all three raised a hand. Next, he said "stand up if you know the color of the hat on your head." After a lengthy pause, only one student stood up and correctly stated the color of all three hats was green. How was this known? Outline your logic.

    This one took me about 15 minutes to figure out using a lot of "outside the box" thinking.
     
  2. jcsd
  3. May 1, 2007 #2
    because he could see two students hats, they both raised their hands so he knew his hat was green, there for all three hats were green.
     
  4. May 1, 2007 #3
    Not exactly. Think about it, if only two hats were green, then all three would STILL raise their hands. There are three people, person A, person B, and person C. Person A and Person B are both wearing green hats, while person C is wearing a yellow one. Person A would raise his hand seeing Person B's hat, person B would raise his hand seeing person A's hat. Finally person C would see both person A and person B's hat.

    One hint, the fact that all three are very intellegent is important to the answer.

    No one will be able to answer this one :)
     
  5. May 1, 2007 #4
    Gosh I'm so tempted to post the answer, but light bulb, you ARE on the right track.
     
  6. May 1, 2007 #5
    all you have to do is look at what direction 2 of the students are looking when they raise their hands so you don't raise your hand until you stand up which confused the other two students because they didn't know why you weren't raising you hand.

    or

    two of the student raised their hand when looking in only one direction while the student who stood up looked in both directions before raising his hand and standing up.
     
  7. May 1, 2007 #6
    Because this information is not presented in the question, the direction in which they looked is negligible in this situation.
     
  8. May 1, 2007 #7
    saw his reflection somewhere? took the hat off and looked at it?
     
  9. May 1, 2007 #8
    asked the other two students if he had a green hat?
     
  10. May 1, 2007 #9
    everyone asked each other what color the hat was and the student who stood up told one of the others his hat was yellow, while he is being told his hat is green the winner stood up. :P
     
  11. May 1, 2007 #10
    the room was lit with a blacklight, and the yellow would be glowing.
     
  12. May 1, 2007 #11
    He he he, sorry light bulb, it's not quite that easy, although when I was figuring it out before, I had WISHED it was. Anyways, I'm afraid I cannot give anymore hints or information until at least a few more people have looked at it and tried it. You were on the right track though.
     
  13. May 1, 2007 #12
    they were small beanies and only visible from the back so the stand up student made the finger motion to turn in a circle after he did one turn with a thumbs up from one of the other students. now if the student on the left turned counter-clockwise and the student on the right turned clockwise he would have been the first to see the colors.
     
  14. May 1, 2007 #13
    the stand up student stepped out for a sec to go to the bathroom, mirror.


    ok i'm done.
     
  15. May 1, 2007 #14
    You've got one creative mind light bulb, hence your name :D
    (and no, I intend that as being an insult or anything, you're answers were very amusing lol)
     
  16. May 1, 2007 #15
    thanks, i try :P
     
  17. May 1, 2007 #16
    Lets call them A, B & C, where C is the student who answered the question.

    Suppose C is yellow.
    If B was yellow, then A wouldnt see any green hat.
    But A saw a green hat. So B would have to be green.
    But B didnt answered anything, that is, B doesnt know his color.
    So, C cant be yellow, and all of them are green.

    :smile:
     
  18. May 1, 2007 #17
    Yes Rogerio, that is correct. Looks like my question isn't as hard as I thought, he he.

    My answer below:
    Given all of the information, we know that there are two green hats. Like Rogerio said, suppose C (ours) is yellow, and the other two, B and A, are green. A and B raise their hands, as do I. Now, if we (remember, "we" are in person C's head), if we go into the mind of person B, he would see a green hat (person A) and a yellow hat (person C [which is what we are]). At this point, he knows there are TWO green hats somewhere and he sees one yellow, and one green. He therefore, should assume that his is green, but he doesn't say anything, because in fact, he DOESN'T see a yellow hat on Person C (us), he sees two green hats! This means that we CANNOT assume that Person C (our) hat is yellow, it isn't possible!
     
  19. May 1, 2007 #18
    that doesn't work for me :grumpy: unless the field of vision from all 3 students is restricted. so i guess their not sitting as a triangle facing each other? are they sitting like a-b-c?


    sorry o:) too much ambiguity
     
    Last edited: May 1, 2007
  20. May 1, 2007 #19
    It doesn't matter how they're sitting, just try reading my answer again and again, and visualize being in person C's mind and then person B's mind. You'll get it, don't worry :)
     
  21. May 1, 2007 #20

    Imagine it this way:

    Let's pretend we're person C. We see that A and B each have green hats. So there's two possibilities. Either our hat is green, or yellow. First, let's suppose that our hat WAS yellow. What would happen?

    Well, if that were the case, person A would see a green and a yellow hat. He'd see a green hat on person B's head, and a yellow hat on person C's head. So in this case, person A could make a simple deduction. He would know that person B saw a green hat. And person B couldn't have seen the green hat on person C's head. Person B HAD to have seen the green hat on Person A's head. Therefore, person A would know that his own hat was green! So person A would stand up pretty quickly and say "I *know* I have a green hat!"

    But that didn't happen! Person A *didn't* make this conclusion. So obviously, since it DIDN'T happen, our hat (that is, person C's hat) is NOT yellow, because if it were, Person A (or person B for that matter) would be able to make a conclusion right away, and they didn't. Therefore, the hat on person C's head is green!

    More specifically, if all the hats are green, *nobody* can logically prove the color of their own hat if all they know is that each other person saw a green hat. Hence, everyone will sit around and say nothing. But that's the ONLY case where nobody can determine anything. For any OTHER combination of hat colors, at least one person can logically deduce the color of their own hat. Three green hats is the only way an impasse can be reached. So because we know that to be true, we actually *can* determine the color of our own hat, because if nobody is able to conclude anything, then we can conclude something!

    Of course, this relies on the intelligence of the other two people as well. If they're both 4-year old kids, they might have known enough to acknowledge that they saw a green hat, but not be smart enough to deduce their own hat color simply by the other people's actions. But for the sake of the problem, we're assuming that the other people are reasonably smart.


    DaveE
     
    Last edited: May 1, 2007
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