Coloured lenses :-)

What I don't get: if they already knew that there were only brown- and blue-eyed people, why weren't they extinct a long time yet? In this case, the anthropologist only learns them that there were at least two of every colour, but even that can not have been news and can't have changed the problem.
If they all knew that there were people with brown and people with blue eyes, and one person saw only brown-eyers, then he would have known he had blue eyes, and committed suicide. As no one had committed suicide, they must have known that there were at least two of every colour.

Basically, they can't have known beforehand that there were only two colours, or they would have been long gone.

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Knowledge is a curse, but ignorance is worse

Probably wouldn't work, since they remember what they saw before. It is usually sufficient to test any solution against the case where there is only 2 blue-eyed people, so each know the other person who has blue eyes.

Go back to the special case of two blue-eyed people. Everyone in the tribe could always see that the tribe has brown and blue eyes, which is all that the visitor said. So it seems there is no new information. That's why part (2) is so curious. But as montebianco correctly surmised, in the case where there are two people with blue eyes, each of those people knows there are blue-eyed people, but they do not know that the other blue-eyed person knows this! It is knowledge about the other person's knowledge that is the crucial aspect, and this chain continues if there are 5 blue-eyed people. montebianco was exactly correct in his explanation, and I was surprised he got it so quickly.

Nice icon, haven't seen that one before!

Ah, got it (I think): they did all know that there were blue- and brown-eyed people, but they didn't know that they did all know (i.e. that the other ones knew as well).
But this only goes when there is a 18-2 distribution, not with the initial given 15-5 distribution. If they knew upfront that there were only two colours, then no new info is given by the anthropologist in this case.

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Knowledge is a curse, but ignorance is worse

Are they allowed to lie? How about if one bluie and one brownie announce that they weren't paying attention during the meeting, what's all the fuss about?

Edit - although now that I think about it, this raises a whole bunch of new problems...

Can they bash their heads with rocks until they forget?

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Quaeso quousque humi defixa tua mens erit? Nonne aspicis, quae in templa veneris?

It works with all distributions. In the case of a 17-3 distribution, there would be three blue eyed people, A, B, and C.

Guy A would know that guy B has blue eyes.
Guy A would know that guy B knows that guy C has blue eyes.
But, Guy A can't know whether guy B knows whether guy C knows that anyone has blue eyes.

Guy A cannot conclude that guy B knows that guy C knows that anyone in the tribe has blue eyes. When the anthropologist arrives, guy A can make this conclusion. He now knows that guy B knows that guy C knows that at least one person has blue eyes.

It works the same with a 16-4 distribution, and all distributions.

But is this correct? If all know that that there are at least four blue-eyed people. They all know that everone can see at three blue-eyed and everyone knows the last statement is just a subset of knowing that everone knows there are at least three blue-eyed.

Perhaps the information revealed by the anthropologist is not important for its information content - but rather it becomes a "focal point" in time to actually consider the logical ramifications regarding distribution of eye color. Essentially, they already did have all the same information provided by the visitor - they just had managed to avoid thinking about it as a group!

Listen to ZZ Top and get some cheap sunglasses...

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The ether of general relativity therefore differs from that of classical mechanics or the special theory of relativity respectively, in so far as it is not 'absolute', but is determined in its locally variable properties by ponderable matter.

Albert Einstein, "On the Ether", 1924

Bong

But that isn't permanent. Massive brain trauma generally is.

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Quaeso quousque humi defixa tua mens erit? Nonne aspicis, quae in templa veneris?

The chief declares a brown-eyed person must go first.

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Lighten up! This is a stellar board!

Maybe one of the blue-eyed people can lie and say that they didn't hear the anthropologist.

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Quaeso quousque humi defixa tua mens erit? Nonne aspicis, quae in templa veneris?

The blue-eyed people don't know they're blue-eyed. Who tells one to lie?

Maybe I'm not understanding how the tribe progresses to mass suicide, then...

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Quaeso quousque humi defixa tua mens erit? Nonne aspicis, quae in templa veneris?

This raises an interesting point. Unless the villagers assume that everyone has heard and understood the anthropologist's remark, they can't draw conclusions from the other peoples' lack of action.

Now this person, call him W, would only "know" that he had blue eyes after the meeting on the fourth day. He would feel confident about his eye color as long as he believed that at least one other blue-eyed person, say X, had heard the anthropologist and that X believed that at least one other blue-eyed person, say Y had heard the anthropologist and that X also believed that Y believed that at least one other blue-eyed person, say Z, had heard the anthropologist. W would believe this because otherwise X would have committed suicide on the third day because of Y's lack of action on the second day which would have resulted from the lack of action by Z on the first day. W would still be morally compelled to commit suicide on the fifth day, however.

If W waits until he knows he has blue eyes, he cannot promulgate his lie until after the fourth day. This delay will certainly arouse the suspicion of the other members of the tribe. As the above argument indicates, to feel certain of his own eye color, another blue-eyed person does not have to believe that W heard the anthropologist, only that there exists a chain such as the one that W relied upon. The chain would not have to actually include W.

Furthermore, the others may not believe W's lie, even if they pass off his suicide as just the result of a bad hair day.

Of course, anyone could tell a similar lie at any time before knowing his own eye color. We'd call it a white lie because it couldn't do any harm in any case.

The idea of raising some kind of doubt is a good one. To invalidate an alleged proof, I don't need to disprove the proposition. I need merely show that we cannot be 100% certain of the conclusion. As long as there is the slightest doubt, the slghtest possible loophole, we can't really say we know. If a tribe member doesn't know his eye color, he doesn't have to commit suicide. Looked at this way, how could a tribesman really ever know that everyone had heard the anthropologist and made the appropriate deductions. There is always some modicum of doubt about what has actually transpired in the course of human affairs - who has heard, done, and thought what.

Well, eventually they learn, but all blue-eyed people learn at the same time.

I agree with this, but it goes even further. Not only do they all need to hear it, they all need to know everyone else heard it. But they also need to know that everyone else knows everyone else heard it, etc.

I think it must be something along these lines, but now we need Ken G to tell us if lying is allowed

No, I think you are wrong. 3 people with blue eyes: A, B and C.
A would know that B and C have blue eyes, and would know that B knows that C has blue eyes, and that C knows that B has blue eyes. So he knows that everyone knows that there are people with brown and people (at least one) with blue eyes. Specifically, this line is wrong:
So A knows that everyone knows that there are two colours, but he doesn't know if everyone knows that everyone knows. This is a new problem, not the one I described ("but they didn't know that they did all know (i.e. that the other ones knew as well"), but a level deeper.
Given the original problem (15-5), we know that everyone sees at least 4 blue eyed people. Now, the worst one of those might think is that everyone sees at least 3 blue eyed people, and no one in this tribe has the possibility to believe that anyone sees only two blue-eyes people or less. So everybody knows that everybody knows that there are brown- and blue-eyed people in the tribe, and the anthropologist gives no new info. So why was there still a tribe for him to visit?

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Knowledge is a curse, but ignorance is worse

Ok, I must be missing something here. He only told the tribe that their own eyes are either going to be blue or brown. But he never said how many people had brown and how many had blue. That information was given to us after the fact.

Lets say I am in a room with 18 other people. All I can see are brown eyes and blue eyes. Now I don't know that my eyes are brown or blue. They could be hazel, green, etc. But someone comes along and tells us that all of our eyes are either brown or blue, but they don't tell us how many of each they see. That right there tells me that they must be one or the other. I look around and see 9 of each color. How am I supposed to know what color my eyes are without asking everyone else how many of each color they see? And why would I want to start asking around at that point anyway if it meant my death?

This is basically what I was getting at too. All he did was say that your eyes are either brown or blue, as opposed to an unlimited number of options. Without giving an exact number, I don't see how it would work out in this example. he said "people" which is quite ambiguous. But it's safe to assume he means more than one person.

So if I am one with blue eyes, I would look around and see 4 sets of blues and 15 sets of browns. Without knowing exactly how many of each there are supposed to be, how could I use that new information to determine what I have? In order to find out that there are indeed 5 people with blue eyes, I would have to ask people how many sets of blues and how many sets of browns they see. But I am obviously not going to do that. If I was going to do it, then it would have happened long before and everyone would have been dead anyway.

5 have blue eyes rest 15 are having black/grey/greenish ? this is some what a great puzzle for me Ken ! you mean every 5 members are having different eyes of different colors? the answer is easy 20/5=4 means rest 3 groups of 5 members each having 1. black 2. greay, 3. greenish eyes, but this is not necessary at all that their eyes should not be "all blue", or majority of them are having blue eyes, or you tell me now.

sunil

The more blue-eyed people there are (up to 10), the more recursions have to be made. If there is only one, the new information is that everyone now knows that there is at least one blue-eyed person. If there are two, it is that everyone knows that everyone knows that there is at least one blue-eyed person. If there are three, it is that everyone knows that everyone knows that everyone knows that there is at least one blue-eyed person. And so on.

It does indeed raise some deep questions about what "knowing" means, from the point of view of logic. Probably, this is a case where there is difficulty intersecting pure logic with humanness, which is always an interesting paradox when trying to apply logic in the real world (witness the seemingly meaningful statement "I am not telling the truth"). But for the purposes of the puzzle, let's assert that everyone can see everyone else is in sound mind and body, and all have a graven look on their faces a few seconds after the visitor speaks. Note that if anyone happened to not be at the meeting that day (which is not allowed by their religious commitment, it is just hypothetical), then that person would simply be "out of loop", there would be no effect on the rest of the tribe's fate.

There are two phases of information dissemination. The first phase is when the visitor speaks, and causes people to know what others know as in montebianco's solution to part (2). The second phase is each morning, as a new meeting goes by and nobody commits suicide, this brings in the final bits of info that doom the tribe. There is no issue of "starting the clock", or conveying that the only possibilities are brown and blue. For example, the visitor could have met with each tribesperson individually and made the same exact statement one-on-one, and it would have caused no problems. He could even tell each person, in this case, that he was going to meet with each person and make that one statement. What would doom the tribe is if he also said he was going to tell each person that we was telling each person that he was telling each person.... 5 times. So what is the tribe going to do to save itself, that doesn't require killing anyone, or pretending that there is doubt where there is none?

Yes, but that doesn't matter, does it? If you see four people with blue eyes, then you know that everyone can see at least three people with blue eyes, and everyone knows that everyone else can see at least two people with blue eyes. So everyone knows that everyone knows that there are people with blue eyes and people with brown eyes. You know that when there is a 15-5 distribution (or a 16-4 and you are one of the 16), so the anthropologist has told nothing new, unless they didn't know that there were only two colours . This is the crucial piece of information that starts the tribicide, not the "and everyone can see it".

Let's do the game again but without the anthropologist. Let's say that the tribe meets every evening, and a person is supposed to commit suicide at midnight.
Case A: they don't know that there are only two colours. Everyone lives happily ever after.
Case B: they know that there are only two colours.
B1. Imagine person A seeing nobody with blue eyes. He kills himself on day 1 at midnight, and the next evening, all the other persons notice A missing, deduce that they have brown eyes, and kill themselves at midnight.
B2. A sees one person with blue eyes. The next day, that person is still around. They both know what they have to do that night, and the tribe the night after.
B3. A sees two persons with blue eyes. He knows they will not kill themselves tonight, but if they don't kill themselves tomorrow night, he has blue eyes as well.
And so on, ad infinitum.
Case C: the tribe knows that there are max. 2 colours. In this case, the tribe lives happily ever after as well, as you don't have the triggering scenario of 1 person noticing that he has to be the odd one out (B1 changes completely, and so does the rest).

Ah, interesting, the difference between B and C. We all can see that there are two colours, and yet the mere possibility that there was only one colour saves us. This is highly counterintuitive to me, but as I can't find the flaw, I'll leave it for your scrutiny.

Ken G, what was the situation exactly before the anthropologist came along? A, B, or C, or something I didn't think of?

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Knowledge is a curse, but ignorance is worse

I already suggested coloured lenses, but I guess that isn't it. Breeding isn't fast enough. Meeting only at night is probably against the religion. So no idea!

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Knowledge is a curse, but ignorance is worse

The original statement of the problem does not linguistically preclude the possibility of there being more than two eye colors among the members of the tribe. Ken later agreed to that interpretetion. If the members of the tribe don't know that blue and brown are the only two colors, the five blue-eyed people woild still commit suicide on the fifth day, but the others would then live on.