Help with the 'Three Hats Puzzle'

Factual Questions
1

Some of you may have seen this puzzle presented in various forms - I saw it recently on James Randi’s website. Below is the puzzle as he presented it, followed by my answer (which I SWEAR has to be right!). Randi told me I was wrong - I am hoping the Teeming Millions will either back up my answer, or help me to see the light. Either way, at least I’ll be able to sleep nights. Here goes…
{Curly, Larry, and Moe, having nothing much better to do one evening, agree to play a strange game. There are two red and three white caps in a drawer. They turn out the lights, each reaches into the drawer in the dark, removes one cap, and places it on his head. Then the drawer is closed, the lights are switched on, and the players sit down, each to try to guess which color of cap he himself is wearing. Fifteen minutes goes by.
Perhaps unlike others with the same names we might know about, these three chaps are pretty smart. Each knows that there might be ways, by observing, listening, and reasoning, of knowing what color of cap he’s wearing.
Curly is the first to speak. “This is a stupid game! Whose idea was this, anyway? And whose caps are these? I don’t know what color I’m wearing, and I don’t care! I want a cold beer!” And he calls the delicatessen downstairs to order a cold six-pack. Moe smiles, but stays silent.
Larry is next to break the silence. “I don’t know what color I’m wearing, either! I agree this is a dumb game! I might have a green cap, but I’d never know it! Let’s order in some pizza and play poker!” Moe smiles even more broadly. Then he speaks up.
“Well, I’m wearing a white cap! I’m absolutely certain, and I knew it before the beer was on its way! So pass me over a cold brew! And I’ll buy the pizza!”
And he’s right. He is wearing a white cap. He didn’t cheat. He figured it out.
Assume that all three guys are astute and clever, and that they speak the truth. Two questions: How did Moe know? And what was the distribution of caps?"}
Here was my answer, which I STILL think is right:

[They were ALL wearing white hats. Moe figured this out based on what he saw AND his friends’ behavior, but we can determine it without even knowing what he saw:
There’s no way the distribution was two red & one white, because if any one of them had seen two red hats on the others, he would have known immediately (they are all clever) that he himself must have a white hat on; yet no one spoke up right away - so 2R/1W is out.
Similarly, 1R/2W is eliminated - if anyone saw 1R/1W on his companions, he would realize that he was wearing white when the one he saw in white didn’t stand up to announce wearing white (which someone who had seen 2R would have done quickly).
So NO ONE was wearing a red hat, which is the only way three clever men would stay silent for so long.
They all saw two white hats on the others, but only Moe had the patience to wait for the others’ reaction to base his answer on.]

Randi e-mailed me back to tell me: “Nope. Try R-W-W. Would that work?” I replied that it WOULDN’T work, for reasons outlined above - if there was a red hat on anyone, they would have solved it rather quickly. He then told me that neither R-W-W nor W-W-W were correct! All that leaves is R-R-W (unless order matters, but we never established a M-L-C order to begin with!), which should have been solved instantaneously, not after 15 minutes!!! (Remember, he specifically said they were “truthful”, so if they knew an answer and sat on it, they violated that premise.)

He won’t post his answer until next Monday, 10/1 - I can’t wait until then! HHEEELLLLPP!!!

2

I don’t know what happened - above was the title I submitted for this thread…

[note: VBulletin hates quotation marks in thread titles. There is a way to do it somewhere in ATMB if you are interested. At any rate, I fixed it for you. -manhattan]

[Edited by manhattan on 09-27-2000 at 05:01 PM]

3

If Moe saw his friends with two white hats, he couldn’t be sure whether he had on a red hat or a white hat, as his friends’ silence would still occur.

However, suppose Larry was wearing a red cap. Before anyone speaks, Curly doesn’t have enough information to know he’s wearing a white cap. Once he declares that he doesn’t know, Moe now knows that Curly didn’t see two red caps, and only then knows for sure that his own cap is white.

The hint is that Moe “knew before the beer was on its way”. Curly orders the beer after he gives up.

4

Well… given the way the question is stated, I agree with you: the only correct answer is W-W-W.

Here is my logic (probably very much like yours):

(1) No one spoke up right away; so none of them could tell immediately what color he was wearing.

Thus R-R-W was out; otherwise one of them would have known immediately what color his hat was, and said so (as the story implies).

(2) Curly’s observation did not help Larry; Larry did not have enough information either.

This means Curly saw either W-W or R-W. Larry’s possibilities are similar. But if either of them saw R-W, then he would have known the answer was R-W-W (with himself in the white hat), since he could assume from (1) that R-R-W was out.

This suggests that both Curly and Larry saw W-W.

(3) Moe knew he was wearing a white hat.

Based on (1) and (2), Moe could assume that no red hats were involved.

If Randi is going to insist the answer is R-R-W, then the story is misleading. The “trick” I see here is that Moe said he knew “before the beer was on its way”, i.e. before Larry’s observation and possibly before Curly’s. In other words, before any of them spoke up – meaning Moe had to see R-R. However, as you point out: we are told that they sat for 15 minutes, that none of them said a word, and that each is clever enough to figure this out on his own (given sufficient information). If Moe saw R-R, why did he wait? If one/any of them saw R-W, then after “15 minutes of silence” he could assume that there was only one red hat in play and solve the puzzle accordingly. The story implies that Moe needed the observations of the other two to figure it out; and then the story contradicts itself by saying that Moe figured it out right away but said nothing for 15 minutes. That’s misleading.

So yes, based on the way the information is presented, a solution of R-R-W or R-W-W is not possible.

Now, if we had been told they simply took turns speaking right after they put on the hats, R-R-W becomes the only solvable scenario. W-W-W and R-W-W would have impossible for any of them to figure out, because each would see only W-W or R-W, which offers no help. The “15 minutes of silence” really throws a wrench into the works…

5

MJH2, many of your points are ones I made in my exchanges with Randi!! (I left them out of this post to save time.)

He clearly states that they are being TRUTHFUL - so if it was W-R-R and Moe waited that long, then he (and Randi) violated that premise.

And if it was W-W-R, then they can’t be as bright as Randi suggests - whoever had on white (and saw R-W) should have realized they had on White when the other white-wearer didn’t announce that he had on white (which he would have done if he saw R-R)! And that would NOT have taken 15 minutes…

Punoqllads - If Larry is in Red, and Moe is in White, then Curly certainly DOES have enough info to determine that he’s wearing white, and pretty quickly! Remember, they are all smart - if Moe doesn’t immediately stand up and say he’s wearing white (which he would do if he saw R-R), that means that Curly must be in white. Similarly, Moe would’ve made the same observation much quicker than he did.

And don’t forget the worst thing of all - in Randi’s e-mails he said that W-W-R and W-W-W were wrong!! (Which means you’re wrong, Punoqllads - unless he’s picky about the order of W-W-R, which is ridiculous…) That only leaves R-R-W, which Moe should have gotten right away!! (Again, unless the setup is inaccurately expressed.)

I am undeterred - W-W-W is the only arrangement that would keep three clever men puzzled for 15 minutes.

6

I think the trick here is that you know each of the three men is clever, but they don’t know that each other is clever.

For example, if I’m playing the game with a couple of others who I happen to think are complete idiots, and I see one wearing red, the other wearing white, I cannot conclude that I’m wearing white. Maybe the guy with the white hat sees two red hats and is too stupid to realize anything. Anyway, that’s the only sense I can make out of it.

Considering that, the only possibility left is WRW, different from RWW (order being important), the order respective to Curly, Larry, and Moe.

7

RRW is entirely possible. Why does everyone presume that Moe would have spoken up as soon as he saw the other two red hats? Perhaps he was so sure of himself, that he just wanted to see what the other guys would say before speaking up himself?

What was the big rush? Was there a prize for being the first correct answer and you think he should have been afraid one of the others would just have a lucky guess?

8

OK, but think about this:

If Moe were to speak up right away, saying “I know I have a white hat on”, the immediate conclusion of the other two would be that Moe saw two red hats. They would therefore also be able to tell immediately that they were each wearing a red hat.

Therefore, Moe has to wait until both of them give up before making his announcement.
BTW, I came to the same conclusion you did, completely overlooking the “before the beer” comment. Oops.

9

DropOfaHat,

Where is this problem defined? In the OP it seems to me that RRW is a given, and yet you seem to want to argue that that is not the case. I can only assume that your wording of the problem is wrong. Please link me to the place where you found the problem.

10

For RWW to work, like Randi says, it seems to me that it cannot be a requirement for someone to speak up as soon as the figure out what color they’re wearing. There are 2[sup]3[/sup]=8 possible combinations, but one of those (RRR) is actually impossible, so we’re left with 7 (apologies for the crappy formatting):

L M C (Larry, Moe, Curly)

R R W (a)
R W W (b)
R W R (c)
W R W (d)
W R R (e)
W W W (f)
W W R (g)

Moe says that he knew he was white “before the beer was on its way.” This means (to me) that he either knew immediately, or that Curly’s statement was enough.

Curly’s statement eliminated (a) above - RRW. Moe could already see what Larry and Curly had on, so what must things have been for that statement to have helped him? If it was (c) - RWR, Moe would have known immediately, for reasons already discussed. If he saw Larry in white and Curly in either color, there is still ambiguity (d-f and e-g above), so Curly’s statement would not have helped.

If Curly’s statement is what did it for Moe, then the only possible answer is (b) - RWW (Larry in red). Moe sees Larry in red, and Curly in white. When Curly speaks up, Moe knows he’s not in red (Curly would have seen 2 reds and known), so he must be in white. He could have spoken up then, but apparently was not required to.

What does Larry see in that case? He sees Moe and Curly both in white, leaving (b) and (f) as possibilities. Curly’s statement did nothing to resolve that, so he still didn’t know what color he wore when he spoke.

I think the rules and some of the statements haven’t been done all that clearly, but that what we’re supposed to deduce is based on these two things (reverse-engineering from Randi’s solution):

1 - A person does not have to speak up as soon as he figures out what he’s wearing

2 - Moe’s cryptic remark is taken to mean that he figured it out after Curly spoke. If we don’t assume that, then I don’t think we can rule out (c).

Given those, ah, assumptions, I think that Larry in red, Moe and Curly in white is the correct answer.

My $0.02 US :slight_smile:

11

Additionally there are two versions of the hat puzzle discussed in this thread. Neither one matches the version in your OP. The second version is discussed in the last two posts in the thread and is close to yours.

12

Assuming that the OP has correctly transcribed the puzzle as presented by Randi, I think the problem is in the presentation. The way the story is told implies that all three are smart enough to figure it out – and each of them knows that the others are smart enough to know: (a) that seeing R-R gives the answer immediately; and (b) after 15 minutes and (a), seeing R-W gives the answer away too.

The basic puzzle (presented in several forms in the SDMB thread to which Lance Turbo has given us the link, above) was, in my opinion, embellished a bit too much by Randi, thereby confusing the issue.

Sometimes it doesn’t pay to be too clever.

13

I don’t think there’s a way to determine what color hat Curly has. When Curly says he can’t tell his hat color, that means at least one of Moe and Larry have a white hat. Since Moe knows his hat color at this point (before the beer is on the way), he must see red on Larry, and know he has white. Moe doesn’t speak up immediately even though he knows his hat color, so I don’t think we can make the assumption he would have spoke up if he saw two red hats. He may just be playing mind games.

Thus the distribution is Moe:white, Larry:red, Curly:unknown. Does James Randi ever say specifically that you know precisely which color each has?

14

DropOfaHat, could you provide a link to James Randi’s website?

15

First of all, the link to the puzzle is [www.randi.org/jr/archive.html] - click on the Sept. 24th article, and scroll most of the way down.

No, Randi doesn’t tell the distribution - he wants us to figure it out.

I copied the text exactly off the website, so if it’s unclear, it’s Randi’s fault.

Now, the reason I’ve been assuming that all three men are clever (and that the three friends know that each other are clever), and that they are honest is that THOSE WERE IN THE PREMISES OF THE PUZZLE AS POSED BY MR. RANDI! We have nothing else to go by…

If it turns out that R-R-W is the answer, I will be VERY disappointed. Moe would have known immediately that he was wearing white, and him “holding out” is extremely misleading (which is fine - it is after all, a riddle) but more to the point it’s dishonest - it violates the premise that they’re all being truthful. Some of the suggestions above about why he would hold out (didn’t want to reveal to others that they were wearing red; wanted to see what others would say; mind games) are weak, and don’t derive from the premises as given. These three were simply trying to figure out what color they themselves were wearing; they are all clever and truthful. If R-R-W is the answer, then it’s fairly random, and arguably any of our solutions could just as well (better!) fit the premises.

If W-R-W (Say, Curly in Red) is the solution (remember, Randi said it wasn’t, unless he’s picky about the order, but we never set up a M-L-C order to begin with!), there is still a problem. Larry would have seen R-W, and when Moe didn’t immediately announce wearing white (which Moe would have done if he saw two reds, being clever and honest), Larry should have realized, fairly quickly, that he had to be in white.

To my mind, W-W-W is the only arrangement that would have stumped three clever, honest men for so long.
The “beer” timing is (I’m sure, purposefully) ambiguous - “before the beer was on its way” could have been any time from when the lights went on to even the end of Curly’s phone call (at which point the beer would arguably be on its way). So Curly’s information may or may not have been useful. If these guys had been characterized as being of only moderate intelligence, and Moe saw R-W, then he may have needed Curly’s (in white) information to know for sure that he was wearing white and not red. But they were all characterized as being clever and able to pick up on certain information. If they were stupid or deceptive, maybe they’d sit there for fifteen minutes before revealing anything, but they’re nither!
Thanks for all the help. I’m still not sure what answer Randi has in mind, and I’m also not sure why W-W-W is ruiled out!

16

DropOfaHat, your answer is basically correct, but it assumes all three are smart enough to figure out the puzzle.

The three possibilities are of course Red-Red-White(RRW), Red-White-White(RWW), and White-White-White(WWW).

Restating your solution:

  1. If the case was RRW, the person wearing the white hat would have looked at the other two, noticed that the other two were wearing red hats, and could immediately speak up and say that he was wearing a white hat. Since no one spoke up, everyone knows that RRW cannot be the case.

  2. If the case was RWW, the two people wearing the white hats would have noticed one red hat and one white hat. Since no one spoke up immediately, these two people know that RRW cannot be the case, so that these two individuals know they must be wearing a white hat. They would simply have to wait for a bit to make sure no one immediately announced white (indicating a RRW case), and then either one of them could announnce that they were wearing a white hat, confident that they were correct.

  3. Thus, the only situation that causes all three to sit in silence is http://www.

BTW, as an interesting aside, anyone who is wearing a red hat in the puzzle cannot determine that they are in fact wearing a red hat. So again, white is still the only viable answer if you assume that no one was guessing.

17

Darn URL parsing. The line should read:

  1. Thus, the only situation that causes all three to sit in silence is White-White-White.
18

Thanks, Caldazar. Now I know I’m at least making some sense. And yes, I do assume all three are smart enough to solve the puzzle, because Randi said they were!!

What’s killing me is that Randi (not on the website, just in a private e-mail to me) said the neither W-W-W nor R-W-W was correct. I can only assume he isn’t bickering about the order of R-W-W - it may matter which (Curly or Larry) has a red hat (if one of them even does, which I still don’t buy) but we never established an order of M-L-C, so if his answer turns out to be something like W-R-W, then saying R-W-W is incorrect is being a bit too literal-minded for my taste.

And if the answer turns out to be R-R-W, then the whole thing is a wasted exercize, because that’s the FIRST possibility thrown out by all of us, and for good reason!

Yes, I agree, a red-wearer is not in a good position to solve the puzzle - which is why I insist it has to be 3 whites, because they’d all be looking at two whites, thinking they might be in red, until Moe figures out that if he were in red someone would have known they were wearing white!

sigh

I’ll let everyone know what Randi’s answer is.

Thanks for the discussion.

19

Martin Gardner devotes a chapter to this problem in “Penrose Tiles to Trapdoor Ciphers and the Return of Dr. Matrix” (sadly now out of print).

IIRC, he first removes the fuzzy time conditions and reformulates the problem by arranging the participants in a single file of chairs, where the nth participant can see the (n-1)th prior participants. Once you do that, it becomes fairly easy to understand the problem.

20

No, the only way for him to know that he had a white hat would be to see two red hats. As for him not speaking, I don’t see how this violates the terms of the puzzle. While this may have been “dishonest by ommision” in a sense, the puzzle never said that they were honest. It just says “and that they speak the truth.” Moe did speak the truth; he just waited a long time to do so.