Hat Confusion!
Alice, Barry, and Celia are each assigned a blue or red hat. They can see each other's hats, but not their own, and they know there is at least one blue hat.
Alice says, "I don't know the color of my hat."
Then, Barry says, "I don't know the color of my hat."
What color is Celia's hat?
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9 solutions
But why cant be there 3 BLUE hats ?
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Each person dosent know there colour of there hat they dont know that there are only 2 colour hats for instense alice sees celia with a blue hat she knows that thats the blue hat elimenated she sees barry with a red hat but does not know her own colour as she was not instructed that there is only red and blue colour hats
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The prompt indicates "at least one" which is not an exclusive statement. Meaning there can be more that one blue hat but there must be "at least one".
Actually the question says that they are each assigned either a red hat or a blue hat, it is true that it doesn't say anything about other colors, but you should know when to assume what and what assumptions will most likely be right or wrong. Otherwise you should probably go delete your account or something like that. Also learn some grammar, and learn to spell to.
I'm with you on that one, nowhere does it say that there are definitely any red hats at all, only that there is definitely at least one blue.
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There can be 3 blue hats.
Our only rule is there is atleast one blue hat. A says "I don't know the color of my hat" meaning she sees at least one other blue hat (if she only saw red hat's she would know she has the only blue since one blue is required. )
* Conclusion: We know that there must be a blue hat on B or C, or both B and C.
B then says "I don't know the color of my hat." If C were wearing red he would know his must be blue to satisfy our * conclusion above. But since he doesn't know, C's hat must be blue.
This doesn't necessarily answer what color the other two hat's are. They could not be red, both be blue, or a mix. But C's must be red.
How can Celia will eliminate this combination {Alice, Berry, Celia} --> {Blue, Blue, Red}
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This configurant couldn't exist, since Barry would have known his hat's colour
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This is the answer I have been looking for. For Celia to not know her hat, it is not enough that BBB is possible, but BBR has to be possible as well. However, if Celia had R then Barry would know that his hat was Blue because, if he had Red, Alice would know her hat. Therefore, BBR isn't an option.
If BBR was possible then Barry would know that his hat is blue as a conclusion from what Alice said. He would have seen that the others have 1 blue (Alice) and 1 red (Celia), at the same time Alice said she doesn’t know. That means Alice is seeing 1 red & 1 blue, which makes him the blue.
Can it be there were three blue hats?
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Yes, there can be 3 blues and C still would have guessed the hat. Let's assume they are all wearing blue hats only, but neither of them know. A sees that B and C are wearing blue hats, but A's not sure if A's wearing a red or blue. B sees A and C wearing blue hats, B understands that the only reason A would say I don't know is if A saw 2 blue hats or 1 blue and 1 red (Both can't be red because then A would have guessed that A's wearing blue, since we know at least 1 is wearing blue). If C was wearing red, then B would have know that B's wearing Blue. Since C's wearing Blue , B is not sure if B has red or blue, so B say I don't know. Hence C knows C has blue and C can even see other's are wearing blue.
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But A can also think like this: A sees two blue hats and A will deduce if she's wearing a red hat, then the other two would have guessed out their own colour. The reason is if two red hats is present would make an instant answer, and since no instant answer means that B and C can guess their own hat too! B would know his hat is blue since there aren't two red hats, vice versa. (THIS IS ALL FROM A's THOUGHT)
In fact, A is wearing a blue hat now and the scenario will not happen(logically). A would know his own hat colour because B and C have no response.
B and C can also think like A and would make a paradox and thus making this question invalid.
This is my personal thinking and only would make sense when not following the sentence up there, hope there's no problem ;)
but what if there are two REDS and one BLUE
How can Celia will eliminate this combination {Alice, Berry, Celia} --> {Blue, Blue, Red}? From the scenario Celia can answer correctly about the color of the hat on her head but it can be red also!!!
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In this particular case Berry can see Celia wearing red cap and a blue one on Alice. Also he learns from Alice's response that Alice can't determine his cap colour which means that Alice can see atleast one blue cap. So Berry will know that he must be wearing a blue cap since he can see Celia has a red one. Thus he would have rightly guessed the colour. But thats not the case instead Berry said he doesn't know his cap's colour which aids us to safely nullify this combination.
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Thanks :) I figured out the logic just after posting my comments. However thanks for your reply.
Your comment makes a sense. The only thing I don't understand is that we don't know what C can see. How can we determine that?
But the problem is it we have know way of knowing what A and B's hat colours are because they are not given to us. Therefore we lack necessary information to make a decision.
It never said that there has to be at least one red hat.
We can also have A and B with both blue hats and C with a red hat. Therefore, contradiction. There is simply not enough information to solve this problem.
Assuming that there are at least 2 blue hats is inconsistent because it is possible that Celia hasn't spoken.
I am of the opinion that this cannot be determined in every case. Why? Because the rules state that there is at least one blue hat; it does not state that there is at least one red. If all three are wearing blue; no single person can determine if they are wearing blue or red themselves.
I agree with the other commenters in the case of three blue hats. There is no assurance there is a red hat so no matter who is asked (or what the former responses were) there would be no way for any one person to determine the color of his/her own hat given the information provided. In this case (three blue hats) it goes like this: Alice: "I don't know what color my hat is." (Barry and Celia both see two blue hats so they think, "right, she wouldn't know") Barry: "I don't know what color MY hat is." (Alice sees that Celia's hat is blue, but still doesn't know what color her own is because either way the criteria would be satisfied.) Celia: (she knows Alice saw a blue hat on either her or Barry and Barry saw a blue hat on either her or Alice, but she sees blue hats on both of them so her hat could still be red...or blue.) Conclusion: Celia does NOT have enough information to know her hat is blue.
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Thank you! That’s exactly what I thought. They have a mistake in this question.. There is no enough information to determine. Perfectly described in the comment of Joshua Ellington where all Hats are blue. And there is no doubt about that.
noooo why which VC first fyi duct
Never said there had to be at least one red. There certainly be three blue hats.
But you said AT LEAST 1. From my frame of reference I thought that Celia could have a red hat, and Barry and Alice both have blue hats. Or Celia could have a blue hat, and Barry and Alice have red hats. Therefore you were not specific enough.
Alice doesn't know -> The other 2 people cannot wear 2 reds -> They can only be either: 2 blues or, 1 blue 1 red (1) Then Barry doesn't know: - If Celia's hat is red, Barry should have known immediately that his hat should be blue, according to (1) - But because Barry does not -> Celia's hat can't be red -> So it's blue
Start with A.
If A saw two red hats, he would know he has a blue hat because there are more than 0 blue hats. This can't be the case, since A started off by saying he doesn't know which hat he has.
Therefore, either:
- Both have blue (1).
- B has a blue and C has a red (2)
- B has a red and C has a blue (3)
Case (1):
B sees:
- Red on A, blue on C: he can't know his own color.
- Blue on A, blue on C: he still can't know his own color because they weren't told that there were blue hats and red hats. They were just told that there are more than 0 blue hats. He could have a blue hat for all he knows.
C sees:
- Blue on B, red on A : doesn't know
- Blue on B, blue on A: doesn't know (following the same logic as above).
Since C does know his own color, this case is invalid for C.
Case (2):
B sees:
- Red on C, red on A: he would know his own color, since there has to be more than 0 blue hats. This is therefore not the case.
- Red on C, blue on A: doesn't know.
C sees:
- Blue on B, red on A: doesn't know.
- Blue on B, blue on A: doesn't know.
This case is also invalid for both B and C.
Case (3):
B sees:
- Blue on C, red on A: doesn't know
- Blue on C, blue on A: doesn't know
C sees:
- Red on B, blue on A: doesn't know
- Red on B, red on A: knows his own color because > 0 red hats.
Since the only case which fits the conversation in Case 3, C has to have the blue hat. :D
But look at this case:
Claire is wearing a Red hat and Barry and Alice are wearing Blue hats.
So, Claire looks at Barry and Alice and sees two Blue Hats, can't determine from that info alone.
Barry looks at Claire and Alice and sees a Red Hat and a Blue Hat, he can't determine.
Alice looks at Claire and Barry and sees a Red Hat and a Blue Hat, she can't determine.
To me, it seems as if in this case the conversation could apply and Claire could be wearing a red hat.
I also thought that it was blue originally by your argument.
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From your case, what Alice sees is a Blue and a Red, since she does not see a Red and a Red, she does not know what her hat is.
Now, Barry, upon hearing that Alice does not know what her hat is, he knows that Alice either sees 2 blues or a blue and a red.
However, Barry sees a Red (Celia) and a blue (Alice).
So Barry knows that he is the Blue guy since if he is red, Alice would see 2 red and immediately know that she is blue.
Thus, Barry would have said that he knows his hat colour.
Hope that helped.
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Ah, I was under the impression that the two declarations of undecidedness were at the same time. i.e. not being able to use each other.
That clears it up. :)
Your solution is based on the idea that C has determined what color they're wearing, but the problem doesn't state that. The problem asks if it is possible for C to determine their hat color based on the previous statements.
I think your cases, as outlined, prove that it is not determinable, and I believe the problem is incorrect.
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I agree. The solution is incorrect...if all that is known is that there is at least one blue hat (not necessarily ANY red) then all three could be blue. If that were the case, each person would see two blue hats and not knowing any more than that there is at least one blue, no one could definitively say the color of his/her own hat. This is only one specific scenario that refutes the stated answer to the question. There are more.
It would fit that Celia could have the red hat, and the other two would have blue hats. Both of them would see 1 red and 1 blue hat, and therefore could not determine which color. The only way to determine the color is if you see two reds. This riddle is invalid
Your argument is missing some information from the riddle, as presented at this time. That is, A said that she doesn’t know what her hat is, then B said he doesn’t know what his hat is. B knows that A didn’t know her hat color, which could only happen if either or both B and C had a blue hat. With the information given by A’s statement, B can only be undecided about his hat color if C had a blue hat. If C had a red hat, B could say he had a blue hat.
And hence is our required solution.
Celia has BLUE COLOR HAT.
If there's at least one blue hat, there's anywhere from one to three blue hats, and anywhere from two to zero red hats. Person A sees a red hat and a blue hat on B and C. He only knows that there's a non-zero number of blues, so can't make a call. Person B sees a red on A and a blue on C and has the same situation as A.
C sees two red hats, knows there's at least a blue, and therefore knows he's blue. If either A or B has a Blue hat, C can't know. Ergo, must be blue with the other two red.
The hint: all of them being told that there are more than 0 blue hat, breaks the ice.
Since A and B both are confused because of the statement above, which means that :
A is seeing one blue and one red hat. B is also seeing the same sight too as both A and B are wearing a red Hat. The blue hat wearer is C as he can judge from the confusion of both of them and keeping the statement of blue hat more than zero in his mind, he guessed his hat color. Since A and B both of them seeing a different colored hatand not able to make out the colour of their hat and remain confused.
The deleted statement: And C walks away, leaving A and B, who still do not know the colour of their hats.
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Now it looks better, you made C a better person, who was before a bit selfish. :p
Also , C gives each of A,B an "apple" , since they are poor enough to not know the colour of their hats.*
*Terms and Conditions: @Julian Poon
You know what, I'm removing that statement... :P. Good one!
Wait... A could be seeing 2 blues and he will still be confused
If C is seeing two red hats, then he can guess his hat colour without all this conversation. But he said that he knows the colour only after he heard the other two statements.
Alice, by stating ""I don't know the color of my hat.". informs the others that she sees at least one blue hat. This is because if she saw a red hat on both Barry and Celia she would then know her hat is blue.
Barry now knows that either he or Celia (or both) must have a blue hat. If Celia had a red hat, he would know it must be he that has the blue hat. But since he states "I don't know the color of my hat.", the only conclusion is that he sees a blue hat on Celia.
If A saw both B and C wear a red hat, she will know that she wears a blue hat. Since A doesn't know the color of her hat, it means that at least one of B and C wears a blue hat. Knowing this, if B saw C wears a red hat, he will know that he wears a blue hat. So B doesn't know the color of his hat because he saw C wears a blue hat.
Because Alice didn't know her hat color, she either saw two blue hats or one red hat. Because Berry also does not know his hat color, Celia must have a blue hat, because otherwise Berry would know his hat color after Alice had said that she doesn't know her hat color. Because in this case, Alice had to see a red and a blue hat or two blue hats. If Celia had had a red hat, Berry would have known that he has to have a blue hat, because otherwise Alice would see two red hats and therefore knows that she has a blue hat, due to the fact that there must be at least one blue hat.
So everyone, including Celia, have a blue hat.
If only one person had a blue hat, that person would know immediately as they could see that no one else has a blue hat and there is at least one blue hat. Therefore there must be at least two blue hats.
If A and B have blue hats but not C, then B would know his hat colour after A said that he didn't know, because he would realize that the only way A wouldn't know is if there was another blue hat, and if C had a red hat then B would know his was blue.
By the same logic, if A and C had blue hats but B had a red hat then C would know his hat colour without B saying that he didn't know his. So either A had a red hat and the other two have blue hats, or all three have blue hats, making C's hat blue .