Help with the 'Three Hats Puzzle'

Factual Questions
21

Well The Ryan, as I said, if the solution turns out to be R-R-W, and Moe just “waited” (for no particular reason other than to mislead a potential puzzle-solver), then I will be severely disappointed. That would take all of the thought (and fun) out of solving the puzzle. “He saw two reds, so he knew he was wearing white.” Well, duh! What’s the point of the whole puzzle then?
We were told that they would sit around and try to figure out their own hat color, and that they were clever and truthful. If Moe knew it right off the bat and didn’t say anything, I’m sorry that is being untruthful.

As for seeing two reds being the only way to KNOW FOR SURE that one of them was wearing white - well, perhaps, but what happened to reasoning, thought, clues, etc? (all of which are mentioned in the setup as being things they will use to solve it.)

Nononono, R-R-W better not be the answer, with the weak explanation that Moe “just waited”.

22

In my “reverse engineering” of Randi’s solution, I misunderstood what Randi said. For some reason, I thought he said that RWW did work. Since he didn’t, much of what I said is bunk.

I can see how Moe would know if it was RRW, but I fail to see how we’re supposed to guess that. I’m curious to see the published solution.

23

The Ryan, the puzzle explicitly states that all three guys are astute and clever, and that they speak the truth. If Curly says he can’t tell what color hat he’s wearing, that means Curly does not see two red hats on Moe and Larry. If Moe sees a red hat on Larry, he knows his hat is not red, and so it must be white.

  • I’m presuming by him, you are referring to Moe.
24

Brad:

Randi did specifically tell me that R-W-W was incorrect, although it’s possible that he was being overly fussy about the order (although I don’t know why, because I never set up a M-L-C order, I was just talking about the distribution of the hats - whether Larry or Curly has the other white may matter, but not in the discussion we were having when he said it was wrong). He said the same about W-W-W.

Which only leaves R-R-W. Which will totally SUCK if it’s the answer. I too am very curious about his answer - and I’m more curious about his explanation of why W-W-W doens’t work.

25

Drop. I agree that the puzzle is poorly worded and misleading. However, if Randi is going to insist that R-R-W is the answer, then it also follows that he will insist that W-W-W (and some permutations of R-W-W) cannot be a reasonable answers.

If we take out the misleading parts of Randi’s text (esp. the 15-minute interval), Moe can solve the puzzle only if: at least one person is wearing a red hat and if (at least) one of the red-hat wearers is not Moe. Putting it another way, without the 15-min interval and the assumptions that go along with it, W-W-W or W-W-R (Moe in red):

(1) Each person will see either W-W or W-R and be equally stumped before any of them speaks.

(2) Moe person will know from the observations of the first two that R-R is not possible (since we know this would be immediately obvious to the person in W)…

(3) …but like us, Moe doesn’t know whether the others saw W-W or R-W.. Thus he cannot exclude the possibility that he is wearing a red hat; nor conclude with certainty that he is.

Thus two permutations are insoluble to Moe: W-W-W and W-W-R with Moe in the red hat.

My point is: the only reason W-W-W is solvable in Randi’s version is because of the misleading nature of some statements in that version. If Randi will not acknowledge that his version is misleading, then logically he will state that W-W-W is an insoluble possibility (to Moe and to us, the observers).

Likewise, he will state that W-W-R, with Moe in red, is an insoluble permutation.

Likewise, the permutation R-W-W (with Curly or Larry in red), is only partially solvable to us because of (3) above. However, if this permutation were the correct answer, Randi could not expect us to conclude what color hat each was wearing; we could conclude only that Moe was in white and one of the others was in red.

Therefore, setting aside the misleading statements in Randi’s version, the only possible solutions which could be completely solved by us are permutations of R-R-W.

… I think… somebody correct me if my logic doesn’t hold up.

Anyway, the idea of “truthfulness” isn’t the problem in Randi’s version, although it could be misleading. The real difficulty is this “15 minute interval” and insistence on the cleverness of each of them, which leads the observer to make certain assumptions about the thought processes of the three men.

26

RRW is the correct answer and the problem is stated correctly. It is just more devious than it appears on the surface.

We know that if a person sees two red hats he will know that his is white. Therefore if Moe speaks the truth, and does so as soon as he knows the answer, he cannot see two red hats.

This is true: Moe does not see two red hats, which if the solution is correct is what the others must be wearing. Moe does not see two red hats for Moe is blind.

If we go over what we have already established, we know that nobody can see two red hats without knowing the colour of their own. Likewise if someone sees a red hat and a white hat and hears no answer from the others, then he must know that his own hat is white. If that left only the possibility of W-W-W then both remaining players would know the answer when the first resigns - not just Moe.

If Moe is blind however, both other players see white and red, yet neither can know the colour of their own hat even when one player drops out. When both drop out, Moe knows that both must see one white and one red hat and therefore knows that he must have a red hat.

27

GRJ, how does Moe determine his hat is white “before the beer was on it’s way”, i.e. before Larry spoke? I don’t think your solution works.

28

I can, it is right here

29

with the puzzle being discussed at the bottom of this page

30

In thinking about this again (still!) last night, I realized that my logic yesterday was partly wrong. There is one solution which does not violate the conditions of Randi’s version of the puzzle. However, it is based on this not-unreasonable assumption: that Curly, Larry and Moe have to take turns, in that order, solving the puzzle. That is, Larry cannot speak before Curly; and Moe cannot speak before the other two.

Even though it is not specifically stated whether (a) each must wait his turn to solve the puzzle or (b) each must speak up as soon as he knows his own hat color, we have to assume it is (a); because Moe’s statement at the end clearly violates (b). But it is necessary to know which of these two “rules” applies, because each affects the logical thinking of the three men (and us, the observers) in a different way, as we have been discussing. Thus, one problem in Randi’s version is that this distinction is implied but not clearly made.

So, on to the solution. To keep clear on the order, I am going to refer to Curly, Larry and Moe as persons #1, #2 and #3, respectively – and unless otherwise stated, the hat colors are not discussed in any particular order:

Person #1 gives up, meaning he sees W-W or W-R, definitely not R-R. [Note: this should not have taken 15 minutes, and so is rather misleading; this is the second problem with Randi’s version.]

Person #2 knows that #1 sees either W-W or W-R because #1 couldn’t solve it. Likewise, if #2 sees R-R, then #2 could solve it instantly; so we know that #2 sees either W-W or W-R. Now, if #2 sees that #3 is wearing a red hat, then #2 knows that his own must be white – otherwise #1 would have seen R-R. If person #3 is wearing a white hat (regardless of #1’s color), then #2’s could be either color. Since #2 gives up, this means person #3 is wearing a white hat, and that #1 could be wearing either color.

Person #3 concludes that he is wearing a white hat. At first we might think he has arrived at his conclusion by working through the answers given by both #1 and #2. But #3 makes the statement (truthfully, we are told) that he knew his own hat color “before the beer was ordered” i.e. based only on the statement of #1. How is this possible?

It is possible only if #2 is wearing red. Hence when #1 gives up, #3 can conclude that #1 does not see two red hats. Therefore #3 knows his hat is white.

Additionally, #3’s statement tells us that #1 and #2 are not both wearing red. If they were, then #3 would have known his own hat color immediately – which would make his statement (re: when he knew his hat color) false, and we are told it is not.

Likewise, if #1 was in red and #2 in white, or both were in white, then #3 could not have been certain of his own hat color until #2 gave up – again, making his statement false. [Additionally, the solutions of W-W-W and R-W-W (1-2-3) could not be completely solved by us, the audience; Randi’s questions to us imply that we can solve it.]

Therefore, person #1 is in white, person #2 in red, and person #3 in white. This is the only solution which fits the information given, while making all statements “true”.

But again, Randi’s version is not as clear as it could be on the point of who speaks when. DropOfAHat Be sure to take him to task for this – as you said in the OP, this distinction is enormously important!

31

You can’t rule out #1 having a red hat. #3 (Moe) only says he knew his hat color “before the beer was ordered”. He never says he didn’t know it earlier than that, so his statement wouldn’t be false.

To your possibilities (a) each must wait his turn to solve the puzzle or (b) each must speak up as soon as he knows his own hat color I would add © Each can solve the puzzle whenever he feels like it. Nothing in the puzzle rules this out. I might do just that if I were Moe, and saw two red hats. And I’d laugh when they found out I was holding back.

32

As I’ve said a few times, if Moe saw two reds and held back for no particular reason, then that is a cop-out answer and it is not solvable as a logical extension of the premises as given. It is almost random, and no more correct than W-W-W (why is W-W-W wrong?? it fits the puzzle BETTER!) or W-R-W.

If Moe saw two reds and said nothing, then his deception sabotaged the others’ chances of figuring it out. Was this a contest to see who could figure it out first? If so, he had no reason to wait, he would have won by announcing the answer right away. We are not told if this was the case, which is yet another of the ambiguities of the puzzle as presented by Randi.

33

W-W-W can not be correct, because after Curly first speaks, Moe is able to figure out his hat color, but Larry is not. If Moe and Larry have the same color hats, then they see the same thing. If one of them can figure out their hat color, then so can the other. Therefore, one of them must have a red hat, and one must have a white hat.

If you assume that if one of them saw two red hats (and so knew his hat was white) that he would immediately say so, then Curly, who had to see one red and one white hat, could determine that his hat must be white, since the others didn’t say anything quickly. It’s only with the possibility of holding back when seeing two red hats that the puzzle is consistent. So holding back when seeing two red hats is not a cop-out, it’s a required possibility.

34

DropOfaHat, if you like puzzles, you might want to check out the Grey Labyrinth, and especially the Discussion Forums

35

Drop. As usual, I’m still agreeing with you. If you make the assumptions that you made originally, W-W-W is the only possible answer. Anything else should have been solved by one of them sooner. Since the “rules” are not clearly defined (for us, at least) by Randi, we can’t know whether Moe is allowed to “hold back” as long as he likes, or required speak as soon as he knows the answer. Depending on which way you, the observer, decide on this point, you will get a different answer.

In my solution, I was trying to determine what Randi intended but perhaps neglected to explain to us clearly. It’s an inductive puzzle on top of a deductive one!

*ZenBeam I see your point. “Before the beer was ordered” could mean “before person #1 (or anyone) said anything”. In which case #1 and #2 must both be wearing red hats. This is second ambiguity in Randi’s version which, depending on how you read it, leads you to a different answer. Boo, to Randi!

And yes, ZB, your third possibility is admissible: each can attempt to solve the puzzle at any time, but is not required to announce the solution as soon as he knows it. This is a re-statement of the first ambiguity, above; and depending on how you look at it, you get a different answer.

So to summarize:

Ambiguity One – (a) if you assume “before the beer” means after #1 spoke, then W-R-W (1-2-3) is the only possible solution. (b) if you assume “before the beer” means before #1 spoke, then R-R-W (1-2-3) is the only answer.

Ambiguity Two – (a) if you assume that each does not have to speak up as soon as he knows the answer (or that each must wait “his turn”), R-R-W and W-R-W (1-2-3) are both possible – which answer you get depends on how you decided on Ambiguity One. (b) if you assume that each must speak up as soon as he knows the answer (during that 15 minutes or at any point afterwards), W-W-W is the only solution.

gasp OK my friends, the logic centers of my brain are shutting down. I hope I’ve summarized things clearly. Let me know.

36

Zenbeam - thanks, yes I have been to The Grey Labyrinth often, I like it - and their solution to a similar puzzle agrees with mine and MJH2’s!!

Now, MJH2: your summation does boil down the ambiguities quite well - EXCEPT that I am now more convinced than ever that R-R-W cannot be the answer at all, regardless of how you interpret the ambiguities. Why? Well, Curly conceivably could do what he did in a R-R-W situation - seeing R&W, but hearing no proclamation forthcoming from Moe in White, he could still think that Moe is holding out (for whatever reason), and just give up. Once he does that, however, clever Larry cannot truthfully say that he does not know what color hat he has on - if Curly has truthfully said that he doesn’t know what he has on, then he did not see two reds, and Larry, seeing R&W, must know he has white on!! So R-R-W is out!!!
Am I admitting that C-W L-R M-W is a possibly viable answer? Well, I think that the 15-minute lapse indicates a greater degree of uncertainty/ambiguity than W-R-W allows for (i.e, it should have been solved earlier by 3 clever men), and therefore it is misleading. But if Moe (or Curly) were waiting for vocal confirmation of the other’s non-seeing-R-R status, then perhaps W-R-W could be the answer. COULD be. But three clever men should have figured it out much earlier, without waiting for each other to get bored…

37

{b]Drop.** Yes, fine. We are in agreeement. You are basically expanding on my arguments for Ambiguity Two(b). Let me re-state.

You can rule out red (in any number and permutation) only if each must speak up as soon as he knows the answer. When this does not happen immediately, each can conclude that no one sees two red hats, and each will assume the other two have arrived at the same conclusion. At some point after this, if anyone sees one red hat, he knows that is the only red hat and therefore knows the answer. Once again, no one speaks up. Therefore, no one sees any red hat(s) and all three are wearing white. This reasoning behind this solution depends on the assumption that they must speak as soon as they have the answer. You cannot conclude W-W-W without taking Ambiguity Two(b).

But if you take Ambiguity Two(a), then W-W-W is not possible because no one – not you, not any of the three men – can draw any conclusions about what the three men are seeing/thinking until one of them speaks.

I’m sorry, but I don’t know how to put it any more clearly than this. Perhaps ZenBeam can help out; the foregoing incorporates his ideas, too.

38

MJH2, a nit with your Ambiguity One. I assume “before the beer was on its way” means before #2 spoke, but not necessarily before or after #1 spoke. In that case, W-R-W (1-2-3) and R-R-W (1-2-3) are both possible.

DropOfaHat wrote:

I don’t see how this follows. Larry sees red on Curly, and white on Moe. He doesn’t know whether Curly saw RW or WW, so larry doesn’t know his own hat color.

Put yourself in Moe’s position, seeing two red hats. He knows he is wearing white, and coul immediately say so. How boring!! Much more interesting to wait and see if Curly or Larry incorrectly guesses white! If Curly and Larry admit this as a posibility, then deducing that they are wearing white because they see one red hat is no longer possible.

In the three hat puzzle on the Grey Labyrinth, the sages have a reason to answer the question quickly if they know the answer. That isn’t necessarily the case here, so the chain of reasoning is broken.

39

Building on the idea that each person would speak out the moment he knows his own hat color: NO possible arrangement of hats fits the problem, actually.

The cases for RRW and RWW (in any order) have already been made, each would be solved in a short amount of time. In the former, W would immmediately know he’s W. In the latter, each W would pause and watch the other W; in lack of any response, each would know he could not be wearing the second red hat, so each would then know he was W.

Finally, by the time 15 minutes have passed, certainly all would have figured out they were all wearing white, due to the lack of any responses (all other cases could be solved); therefore WWW can be solved given the absence of any other solution.

I just think it’s a very poorly written problem to begin with.

40

ZenBeam My wit’s with your nit. If one wants to decide between for/after, then you get one answer. If you don’t want to decide, then both are possible.

As for Drop.'s reasoning in his last post: he is operating under the assumption that “speak the truth” means “speak up as soon as you know the answer”. Which is one way of looking at it – yours is another. His line of reasoning follows from his assumption; it’s the assumption you don’t agree with. Fair enough.

Randi, joo got a lot of splainin’ to do!