Hard logic problem

  1. Suppose there are 4 people, two of which are wearing red hats, and two of which are wearing blue hats. one person is behind a wall, and the other three are standing in a uniform line, only able to see the person directly in front of them. Which person knows exactly which color hat he's wearing, and why?
  2. jcsd
  3. My initial response is the one behind the wall since there seem to be no restrictions as to if he can see the others in the line, or even his own hat for that matter.
  4. Does each person himself know what hat he is wearing.........or is that what you're asking? It's just not clear.
  5. If you're asking which person knows what hat he himself is wearing.......maybe the one behind the wall b/c he can take it off of his head and look at it without anyone knowing.
  6. that is what he is asking, yes. which one will get to know what colour his hat is.

    I would answer this one, but I've come to it many times, and I know the answer. Just a hint for everybody: you have to think in why he is the only one that knows his hat, and what does the fact that no other one knows which hat he is wearing means and impplies.
  7. If they can see their own hat, then everyone knows what color hat they are wearing.

    If they can't see their own hat, then no one can see more than one hat, and some can't even see that. So none of them knows what color hat they are wearing.
  8. no: each of the three in a row can see the other two people and thus, their hats.

    I'm not saying more because I want to see what is the answer of each, and how s/he got it.

    This is actually an easier problem than the normal one (with three people: each whashing the other two's hats) because altohught there are more people, it has easier solution.
  9. AKG

    AKG 2,576
    Science Advisor
    Homework Helper

    So, do we have people like this:

    x x x | x

    Where x is a person, and | is a wall? Also, would the left most person only be able to see the second person, and the second person only be able to see the third person? Would the third person (the one just to the left of the wall) and the fourth person (just to the right of the wall) not be able to see anyone? Given this, I don't see how it's possible for anyone to know which hat they have, unless they are allowed to communicate somehow.
  10. the above question isn't descriptive enough

    are they standing x-> x-> x-> | x-> OR x-> x-> x->x->| the question above doesn't say that they are separated by teh wall.
  11. Each person in the line can only see one of the other two people, except for the first in the line, who can't see anybody.
  12. It has to be some kind of trick question. The person behind the wall has 1-way glass for example so he can see all the people in the line.
  13. What if the two people in front of the person are wearing a blue and a red hat; then he would not know if he was wearing a blue or a red hat himself.
  14. If the third person could see both hats in front, and saw one red and one blue hat, then as you say, they wouldn't know what color their own hat was. The person in the middle would 'hear' the silence of the third person and realize that their own hat was a different color from the first person's hat. Since they can see the first person's hat, they know the color of their own hat.

    This is idle speculation. According to the wording of the puzzle, no one can see two other people. I continue to believe my solution is correct.

    However, that phrase "behind a wall" bothers me. Can the person behind the wall see any hats?

    The phrase: "only able to see the person directly in front of them". bothers me too. Can anybody see any hats?
  15. Oops.

    Actually, I was confusing this hat problem with a different one. There is no obligation or incentive for anyone in the puzzle to say anything so the person in the middle would have no way of interpreting the silence of the third person.
  16. If I understand it correctly that three persons are standing one behind another, the second person(or third) can tell which hat he is wearing. If the hat colors of first and second persons are same then the third person(so the third person) can easily tell which hat he is wearing. If the third person can't say what color hat he is wearing(because he is looking at two different colored hats) then second person's hat color is other first person's hat color.
  17. Jimmy,

    It seems we both are right(actually I posted before reading your post).

    But the other problem given in the first thread(which was initiated by the forum adminstrator) of this forum is much tedious and can't be solved if all the three persons are of not equal intelligence. For ex. if first person can't think what the second person thinks about the third person.
  18. Quark, it is difficult for me to imagine which part of your answer you consider to be correct. Your solution requires that the third person be able to see two other people. The problem specifically disallows the third person from being able to see more than one other person. I quote:

    the other three are standing in a uniform line, only able to see the person directly in front of them.

    This puzzle is getting tedious sjsustudent. What is the answer?
  19. Isn't just the guy at the back f the line? Nobody can see if they look, because they're all facing forwards...or have i missed something?
  20. lisab

    lisab 3,188
    Staff Emeritus
    Science Advisor
    Gold Member

    "Pssst...hey, buddy...yeah, you standing behind me. Can you tell me what color hat is on my head?"
  21. Logical analization

    Ok there are only a fixed amount of logical outcome's from this...

    1."behind a wall" dosnt state that the wall is higher than his field of vision.
    2.The " a wall" is not stated to block his vision/or be no transparent.
    3. Dosnt state that the wall cannot be a mirror.
    4.Dirctly in front of him dosnt state that person in the middle of the uniform line has perif-vision of the two people on each side.
    5. i can go on and on about there is a lack of guidline's what is allowed to be done and not, due to lack of pre-set information discribing each thing.
    1. 2 people have blue hats=true
    2. 2 people have red hats=true
    3. there is one wall=true
    4. there is one person behind the wall=true
    5. there are 4 people in total=true
    6. 3 of the 4 people are standing in a uniform line
    only able to see directly in front of them=true
    7. dose any person knows exactly what color hat they have on=unknown

    So from these 6 true's and 1 unknown factor, there are only 4 logical correct outcome's each one having a equal % of being correct due to the lack of information, but one has a higher % of being correct with the flawed information given...

    8. the person behind the wall know's because the wall is not in between him and the 3 uniform lined up people, so he can see all three of them and which hat they have on, and therefore he knows which hat he has on. because 1-6 are true and 7 is unknown 8 must be true.. for a person to exactly know which hat he has on..but 8 can only be true if 9 is true
    9.(he's behind a wall but he can turn around and other 3 uniform lined up people have there back's turned when the person behind the wall turns around)=true

    So the person behind the wall has the highest logical % of being the person to exactly know which hat he has on because 1,2,3,4,5,6,9 are true and 7 is unknown and thats

    he know's why because he can turn around and see the three people in uniform line facing the other way but conflict #10 is about there all infront of a wall but not, but are but not..... this question is very silly and lack's detail's... i hope this question isnt used in any test in any schools ever...
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