… and Cabbage makes a good point, too. If they had to speak up as soon as they had an anwer, and were all so smart, all three of them should’ve known the answer was W-W-W after a short time.
I agreee 100% with this. Which is why that assumption can’t be correct.
We know that a person doesn’t have to speak up as soon as he knows the answer - Moe admits that he didn’t, and we’re told that he didn’t cheat.
Unless you choose to interpret the “before the beer” remark as somehow meaning after Larry spoke, which seems like a big stretch to me.
Well, James Randi has posted the solution on his website (see above links). It’s somewhat satisfying, although I still have have some problems with it. Here it is, verbatim.
[Did we get responses to the Three Caps puzzle! By far the most common was 2 red, one white, but that is not acceptable, nor is the white-white-white arrangement. This puzzle is a variation on an old theme. I’ve taken it and put a twist or two in, being the fiend that I am. In the illustration is a simple diagram showing the only four configurations possible.
We have to deal here with the fact that we already know that Moe’s right. We have an advantage that the players don’t have, and that is an important factor in working out our answers. Another thing is obvious from the start: If there are two players wearing red caps, the remaining guy knows right away what he’s wearing - white. Any of these three seeing two reds, would immediately respond, and that’s not what happened. The statement by a player that he does not know the answer, means that he’s not looking at two red caps - which would be the only circumstance under which he could say, before any other evidence comes in, that he knows. Moe’s the smartest of the three, because he sits and waits for evidence to develop.
Now for the distribution of caps in this giddy group. It goes Curly=white, Larry=red, and Moe=white, as shown in our illustration. You’ll also see there the four possibilities for the cap arrangements. If both Curly and Larry had been wearing white caps, Moe would know nothing about his own cap from that fact alone, nor from hearing Curly’s complaint, but after Larry didn’t solve the question, he would have known that he - Moe - was not wearing a red cap. How? If Moe had been wearing red, then Larry, hearing that Curly wasn’t looking at two red caps, would have known that he himself was wearing white, and he would have been the winner. He wasn’t the winner, as we know from our superior position. So, by elimination, Moe knows he’s wearing white. So neither white-white-white nor white-white-red (the first two combos shown) are the configurations we’re looking for. But read on.
Moe has to be looking at one red cap, and one white cap. If he sees Curly wearing a white cap, and Larry a red (combo #3), after Curly says that he can’t figure it out, Moe has total validation that he’s wearing a white cap, and he doesn’t need to hear Larry’s comment. Moe knows from the very first comment, that he’s wearing a white cap!
But wait a minute, I hear some of you saying, “But if Moe sees Curly wearing a red cap and Larry a white one (combo #4), after Curly’s comment, he’d know nothing useful in itself. But after Larry’s comment, he’d know that he - Moe - is wearing a white cap!”
Gotcha! You forgot that Moe said, “I’m absolutely certain, and I knew it before the beer was on its way!” (Italics added with glee.) Moe is revealing to us that he knew right after Curly had spoken, and didn’t need Larry’s observation!
Now, I got the argument that “I knew it before the beer was on its way” could also mean “I knew it before the very beginning,” and I admit that I could have written, “I knew it as soon as the beer was on its way.” You may have a point here, I confess. But that’s why I ask for your input, so that I, too, may be educated! And do I get it? Oh, yes.]
So when Randi told me that R-W-W was wrong, he WAS taking order into account, even though we hadn’t established a M-L-C order!
And he admits that the wording about the beer was a little ambiguous. Big woop.
More important for my orginal argument is that he admits that if any of the three of them had seen two reds, they would have spoken up rather quickly - and he points out that this didn’t happen. Well, I STILL think that three clever men would NOT have sat around for 15 minutes. If Moe saw Red on Larry & White on Curly, he should’ve known that he was in white because Curly didn’t jump up quickly to announce wearing red (which Randi admits he would’ve done) if Moe were in red as well as Larry.
And to my mind, the fact that Moe was helped by Curly’s information but not by Larry’s does NOT rule out W-W-W! Moe (& all of them) would have seen W-W on the other two. When Curly got bored and said he couldn’t figure it out, Moe could have reasoned that if he himself were wearing red, Larry or Curly would have been able to figure out that they were in white, because the other didn’t leap up immediately to annouce wearing white (which they admittedly would have done if they saw two reds) - so Curly’s resignation was all Moe would’ve needed to figure out he was in white. If Moe saw R&W, but still waited for Curly to announce that he didn’t know what color he was wearing (which meant that he DEFINITELY didn’t see two reds) - well, where’s the reasoning and thinking? Moe waited for affirmation that Curly didn’t see two reds, rather than reasoning it out. That may be prudent, but it doesn’t show much in the way of lateral thinking…
DropOfaHat
There are many ambiguities in this puzzle, but the definiton of “untruthful” is not one of them. This term has a very definite meaning, and simply remaining quiet does not fulfill it. “Untruthful” means saying something untrue. Seeing as how Moe didn’t say anything, he obviously didn’t say something untrue.
ZenBeam
Yes, the puzzle tells us that.
Yes, we know this.
We know this, but Moe doesn’t. When you quoted me, you dropped my emphasis on know. I put it in italics for a reason. Moe might think he has a white hat, but he doesn’t know it unless he sees two red hats. We’ve already established that no matter what the distribution of hats is, Moe must have been deceptive. And Moe knew that he was being deceptive. So, knowing that he was being deceptive, why should he asssume that no one else was being deceptive? Or just plain dumb? Huh?
The Ryan - if you look at Randi’s answer in my most recent post above, you’ll see that Moe did NOT see two red hats. Randi admits that if he (or any of the three of them) HAD seen two red hats, they would have spoken up about it rather quickly and the game would have been over. Which (he also admits) didn’t happen. According to Randi’s version, Moe saw W on C and R on L waited for Curly to say he didn’t know what was going before determining (correctly) that he (M) was wearing white, because Curly could not truthfully say he “didn’t know” if he saw two reds - then he waited for Larry to beg out too before revealing.
Randi’s only way of ruling out R-R-W is the same as the rest of us have been using - that someone seeing two reds would have spoken up quickly! Randi agrees with us.
However I still have a bit of a beef with Randi - I think Moe (or Curly, if he were as smart as Moe) could’ve used the absence of a quick declaration (which Randi admits WOULD HAVE HAPPENED if someone saw two reds - and he says “this didn’t happen”, which means no one saw two reds!) to determine that he was in white if he saw R & W, which is why in my heart of hearts I still think W-W-W is not totally incorrect.
Why so I still think W-W-W is plausible? Well, the way Randi eliminates it is by saying that Curly’s denial helped Moe but not Larry, which couldn’t be true if they saw and wore the same color hats, because they both could have solved it. Well, even in Randi’s version, Curly & Moe saw the same thing and wore the same thing, and Curly gave up and Moe solved it! Why? Because, according to Randi, “Moe is smarter”. Well, then he could’ve used Curly’s denial as a clue that LArry didn’t!
Anyway, this was an interesting exercise. Thanks all…
No, that is not Randi’s way of ruling it out. His/our way of ruling out R-R is: (a) the first two give up; and (b) Moe tells us he knows he’s wearing a white hat, and we are told that he is correct.
Randi really shouldn’t “agree” with your above statement. However, it doesn’t matter; the point is moot. Moe does not rule out R-R-W as a possibility because no one speaks up quickly. He rules it out on the basis of: (a) his own observations (one red hat on Larry); and then (b) what Curly says. These do not depend in any way on the idea that whoever solves the puzzle would speak up quickly.
Because they were not required to speak up as soon they had the answer, none could make the assumption that the others weren’t seeing R-R until someone said something. Internal evidence in the puzzle tells us that they didn’t have to speak up right away: even though Moe solved the puzzle after Curly’s statement, Moe didn’t announce the solution until Larry gave up, and we’re told that Moe did not cheat. It doesn’t matter why Moe waited; all we need to recognize is that Moe tells us when he solved it. And since Moe himself withheld the answer once he had it, he’d be the last one to expect that the others would speak up right away if they had the answer. So even he can’t conclude that Curly isn’t seeing R-R until Curly gives up.
Furthermore, Moe is not necessarily the smartest one; he’s just the most patient. Moe needed more information. Curly lost patience first, and in doing so gave Moe the information he needed to solve the puzzle. Perhaps he kept quiet and let Larry have a go, knowing that Larry was going to give up (since W-R-W is not solvable by Larry after Curly’s statement).
Well, Randi should not have said this because it implies that each of them had to speak up as soon as they had the answer, which we already know, explicitly, is not so. And we’ve already determined that Randi isn’t too careful about how he words things.
Aside from the minor quibble over what was meant by “before the beer”, which Randi explained, I am satisfied with his solution. Nothing else works.
And is Randi’s word infallibe?
[quote]
because Curly could not truthfully say he “didn’t know” if he saw two reds - then he waited for Larry to beg out too before revealing.[/qoute]
- Curly could have truthfully said that he didn’t know, if he wasn’t paying attention.
- How does Moe know that Curly is telling the truth? He doesn’t. Unless he (Moe) sees two red hats.
The fact is, the arrangement RRW does not violate any of the stated terms of the puzzle. Can anyone dispute this? The reason Randi gives for eliminating it is that if it had happened, Moe would have spoken up immediately. Says who? The puzzle nowhere says that people must speak up immediately upon figuring it out. Randi just added a new condition to the puzzle. The solution that Randi gives requires that the participants have knowledge which the puzzle never mentions: they must know that the other participants are smart and truthful. Sure, we know this, but do the players? Yet again, Randi is adding something new to the puzzle. My answer is the only one that fulfills the conditions of the puzzle as stated. If you want to get into what the solution is for some alternate puzzle which Randi has devised, thatis a completely different issue.
I’m disapointed with the solution. I don’t see how Randi can argue on the one hand
and on the other hand have Curly not know his hat is white by this exact reasoning.
MJH2, I don’t see how you rule out Curly and Larry both wearing red. If you assume this, and read through the puzzle, you never come across an inconsistancy.
The Ryan, I see what you’re saying now. I don’t think it’s unreasonable to take “Assume that all three guys … speak the truth” to mean they believe the others won’t lie, or that Moe’s statement implicitly assumes Curly isn’t lying. One red hat, on Larry, is certainly consistant with the puzzle.
Perhaps we should look at this as a variation of the Monty Hall type question… In other words, since Moe basically states that he didn’t need Larry’s information, we should just throw out what Larry says.
Imagine this. Curly says he can’t figure out what hat he has on. Then, Moe says (again, we are ignoring Larry), “I know what color hat I have.” It would be pretty obvious that the distribution has to be W-R-W, wouldn’t it?
Brian
Yep. So is the solution “They were all wearing white hats. After Curly spoke, Moe looked in the mirror and noticed that he was wearing a white hat.”
What you say is correct, Zenbeam. Randi contradicts himself here, and we know that he’s not careful about how he words things. (So it seems strange, doesn’t it, that he’s the one presenting the logic puzzles?) But as I said, his explanation (quoted above) is not really the way to rule out R-R-W anyhow.
I’m not ruling it out solely on the basis of the puzzle alone. It is also ruled out based on what Randi meant by “before the beer”, which he explains thusly:
[quote]
… Moe said, “I’m absolutely certain, and I knew it before the beer was on its way!” … Moe is revealing to us that he knew right after Curly had spoken …
[quote]
In other words, Moe needed to hear what Curly had to say before he could solve it; therefore Moe wasn’t seeing two red hats. That being so, the only other permutation Moe could solve right after Curly gave up is W-R-W. And to belabor the point, since Larry is wearing a red hat, and Curly gives up, Moe knows that Curly isn’t seeing two red hats; and since Moe sees one red hat on Larry, he knows Larry isn’t seeing two red hats. Ergo R-R-W is completely ruled out.
Granted, this depends on Randi’s clarification of the “before the beer” remark, which he admits was poorly worded. The original text, without Randi’s clarification, does not rule out R-R-W because “before the beer” could very well mean “as soon as we put on the hats and sat down”.
Before we had Randi’s answer, I made an assumption about the timing implied in “before the beer”, reasoning that the puzzle was intended to have only one solution but that Randi had erred somewhere in the wording. If “before the beer” meant “after Curly gave up but before Larry spoke”, then there is only one solution; therefore I assumed (correctly) that this is where Randi made the error.
An inductive reasoning problem on top of a deductive one.