I didn't mean they'd track him down to force him to perform his duty, I meant that they would have to track him down as part of a maximal good-faith attempt to commit suicide themselves. Maybe I'm belaboring an worn-out point.

Actually, you are on the track I had in mind. Let me change the story a little. After the gasp has gone through the crowd, and each person, in spite of themself, looks around at the appalled faces of the rest of the tribe, they all begin to realize the full consequences of what is going on. At this point, a tribe member with no dependents, and one of the quicker thinkers, stands up and announces "well I'll be leaving now. I'm going to make a boat and sail to the mainland, and never return. Having missed tomorrow's meeting, I can no longer consider myself part of this tribe, I am banished. Perhaps Microsoft is hiring. Please remember me. For the rest of your sakes, I hope I have blue eyes." Explain why he took this selfless act, and what did that last bit mean? (Easier than part 2 of the puzzle by a long shot. I claim this is probably what would happen in reality, but maybe even more reasonable solutions are possible.)

Well, I didn't realize people coud just up and leave. But his/her departure means that the rest of the tribe will not "benefit" from the information provided by the lack of suicide of that person...

The puzzle only states that there are 5 blue-eyed people, but it doesn’t state how many brown-eyed or other-colored-eyed people there are.

If there are 5 blue-eyed people that we know of, and if we assume there are 15 brown-eyed people in the tribe, I don’t see how the professor’s statement would cause anyone to know what color their own eyes are, because both the blue-eyed people and the brown-eyed people would already know there are blue-eyed people and brown-eyed people in the tribe, but that wouldn’t tell anyone what color their own eyes are. So under the given fact of 5 blue-eyed and the assumption of 15 brown-eyed, what has the professor revealed that they didn’t already know?

Remember that the professor was only talking about the group’s similarity to his own blue-eyed/brown-eyed family, and not about their similarity to all the various eye-colored people in the world.

Now, suppose there are 5 blue-eyed people in the tribe, 14 green-eyed people in the tribe, and only one brown-eyed person in the tribe. The professor would have then revealed information that the one brown-eyed person did not already know, and since that person could see that no other person in the tribe is brown-eyed, that must mean that he is brown-eyed.

For the purposes of simplicity, let's assert that the only possibilities are brown and blue. The tribe is allowed to know this, they're just not allowed to know which is their own color.

That's the puzzle, exactly. Quite subtle, no? Kudos for montebianco, see his earlier post.

Yes, I figured this was a bit of a swindle, but the idea is, it's better that someone renounces their own place in the tribe and in the religion rather than condemning them all to death. An interesting moral quandary in its own right, but let's go with it. Follow it through-- what will have to happen to save this tribe, given that they have a love of ethics as strong as their love of logic and religion? There can be benefit, I'm not sure I understand what you meant by that.

That’s changing the rules of the puzzle. The puzzle actually said, “Imagine a tribe of people who are legendary logicians, but who have a curious religious commitment that if they are ever able to determine the color of their own eyes, they must commit ritual suicide in front of the whole tribe at the tribe's daily meeting.”

The puzzle doesn’t say anything about an option of leaving the island. If we are going to introduce possibilities that aren’t given in the premise to the puzzle, we can make up anything we want, and we might as well say that when the professor said what he did, all 20 people laughed and said, “Well, heck, there goes our religion, cause I ain’t gonna kill myself!”

If you want to change the rules of the puzzle, then we might as well say the professor just talked them out of killing themselves.

If you really want the “leaving the island” solution, you need to state that option in the premise of the puzzle, or not rule it out, because you said "they must kill themselves". You didn't say "they must either kill themselves or leave the island." You’ve already said they must kill themselves. Now you are saying they wouldn’t kill themselves. So the puzzle and the solution are both invalid.

Well, no. the OP doesn't preclude leaving the island. It does lay out some very specific things that are required for suicide. Leaving just invalidates one of these required things and precludes suicide.

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

You misunderstand, Sam5. They must still kill themselves if they determine their own eye color. What we are looking for now is a way to prevent them from figuring out their eye color. That will require drastic measures, like leaving the island (which seems less drastic than suicide or murder). So the current question is, who must leave the island?

I don't think it changes things that much - the blues and browns still end up dead, people of other colors do not (provided they acknowledge at least the possibility that there are four colors, even if there really are only three).

Well, we all ought to pursue that, but before doing so, I think I must dissent with much of the thought that is going on here.

Much discussion about how to save the tribe revolves around a blue effectively being removed (or removing him/herself) from the game, to prevent the learning process that would otherwise take place. The discussion is then about what methods of removal are allowed, e.g., is murder acceptable, is indirect murder (my solution, telling a blue s/he's blue) acceptable, is exile allowed by the rules of the game, etc. But quite apart from that, I don't think removal of a blue from the game stops the catastrophe; it only postpones it.

Right after the initial revelation, a learning process begins which, unless somehow stopped, leads to the mass suicide of the blues. Now, the same process by which the blues learn they are blue is also taking place among the browns. Furthermore, the learning process of the blues does not depend on the actions of the browns, and the learning process of the browns does not depend on the actions of the blues. But both take place simultaneously. However, because there are more browns than blues, the fuse on the brown bomb is longer than the fuse on the blue bomb, and the blue bomb therefore detonates first. The detonation of the blue bomb also causes the brown bomb to go off the next day, so the learning process taking place among the browns is wholly irrelevant. However, if we find a way to remove a blue from the game, thereby defusing the blue bomb, the brown bomb is still ticking, and will go off in fifteen days. So now everyone lives longer, and the blues live longer than the browns this time, but they still all end up dead.

So, unless I have fallen into error, it is not enough to take a blue out of the game somehow; both a blue and a brown must be taken out...

Nick

Let's work it up. Suppose the leaver was a bluey.
- If he was the only bluey, then everyone would only see brown. However, noone would know their own own colour (each would reason that they and the leaver may have been the only 2 blues), and would live happily ever after.

- If there were 2 blues, the remaining bluey would see all browns (but still no idea that he was blue, as above). The others would see 1 blue (and not know whether they were blue as well). Both live happily ever after (even after 1 day, bluey would still have no idea).

... And so on.

Neat, so it does restore the uncertainty.

If the leaver was (as everyone knows) a brown, then the blues know that nothing's changed. However, once the leaver has gone, another tribe member may make a similar gesture (and then another, and another ...) until the tribe sees that a blue-eyed hero has departed, whereupon they all breathe a big sigh of relief

(I still think it might have been easier for the visitor to say "You know, my colour-blindness stops me from distinguishing blue and green!")

[ Edited to remove incorrect preamble]

There is not enough information in the original puzzle for any of them to determine their own eye color, if we know that 5 are blue-eyed and we assume, as Ken says, that the other 15 are brown-eyed.

Because 15 brown-eyed ones already know that 5 are blue-eyed and that 14 are brown eyed.

And 5 blue-eyed ones already know that 15 are brown-eyed and 4 are blue-eyed.

They don’t have to wait 4 or 5 days to learn this. They already know this before the professor arrives on the island.

But no one knows if they themselves are blue-eyed or brown-eyed, because the professor didn’t tell them and they didn't tell each other and the professor didn't tell them the number of how many had which color. This is something that WE know, if Ken’s assumption about the 15 is correct, but this is not something that THEY know if the professor didn’t tell them.

Based on the original puzzle and Ken’s assumption that 15 are brown-eyed, the visitor to the island gives them no new information that would allow any of them to determine their own eye color.

So, I think something is missing from the premise of the puzzle or that Ken’s assumption is not correct.

Well, that's correct. But the new information is knowledge about the knowledge of others about the knowledge of others (etc., to more levels for the browns than the blues).

This is how I see it, yes, I view that as the solution to this last bit, though there is one further complexity (read the next post). I admit other answers are also good. For example..
Or a similar suggestion by 01101001 that there is too much uncertainty in any anecdotal testimony. But I think that beyond a reasonable doubt, anyway, such efforts would be viewed as flimsy efforts to introduce uncertainty where there really is none, such as is done by CTs all the time.

Well, certainly for this mass suicide scenario to work, you need to assume that everyone heard and understood the anthropologist, everyone knows that everyone heard and understood, everyone knows that everyone knows that, everyone knows that everyone knows that, etc., to a certain number of levels. If people are sure everyone was paying attention, or if they aren't sure that everyone else knows that everyone was paying attention, etc., the whole thing breaks down...

How can any of them determine their own personal eye color, based on what the professor said?

Based on that alone, they can't. They need that plus several days of non-suicides to infer what everyone else knows, what everyone else knows, what everyone else knows everyone else knows, etc.

Begin with some counter-factual scenarios, but these need to be considered to figure out what happens.

Case 1 - only one blue. If this is the case, then the next day the only blue commits suicide, because s/he can't see anyone else with blue eyes, but knows a blue exists. Knowledge required - everyone knows that there is at least one blue (actually, that's more than good enough, but let's go with it).

Case 2 - only two blues. The two blues are not aware that there are two blues, they think there might be only one. But if there were only one blue, that person would have committed suicide on the first day. It didn't happen. The two blues now know there must be two blues, but they can only see one. Guess who the other one is? Time to commit suicide on the second day. Knowedge required - everyone knows there is at least one blue (guarantees suicide will take place the first day if there is one blue), and everyone knows that everyone knows there is at least one blue (so they will correctly interpret the non-suicide of the first day).

Case 3 - three blues. The three blues don't know there are three blues, they think there might be only two. If there are only two, the two blues will commit suicide on the second day. They didn't. There must be three blues, and since they can only see two other blues, each blue has now figured out that s/he is a blue. Knowledge required - same knowledge as in case 2, and also everyone knows that everyone knows the same knowledge as in case 2 (so they will correctly interpret the two days of non-suicides).

Keep going this way until we get to five, and presto, all the blues off themselves on the fifth day...

Nick

But Ken has already said in the puzzle itself that there are 5 blues and 15 browns. There are no other "cases" but that. The professor gives them no information to tell any of them which they are, blue or brown, since he mentions no numbers to them. Ken mentioned the numbers to US but the professor didn't mention any numbers to THEM. And they already knew there were blues and browns on the island.

Your point that the two populations are doing the same thing is well taken, even though each individual doesn't know which population to belong to. If a person sees {n,m,o,p} etc. different colors, all of which were mentioned, then they must wonder if they are the n+1, or m+1, etc., and the same process occurs for each. Truncating one just reverts to the next numerous type. So you will need to have people leave, as their sense of duty compels them, until one of each of the mentioned types has left. But the statement "I hope I'm a blue" by the first leaver is another charged statement, and may be disallowed by their religion, but it targets blues as the minority and therefore does truncate the process for all the colors. In other words, once the visitor has spoken, it is better to admit to all that blues are the minority.

Do they all know that? Do they all know that they all know that? Do they all know that they all know that they all know that? These other scenarios arise as each member tries to assess what the others know, what the others know the others know, what the others know the others know the others know, etc.

The anthropologist gives them knowledge about the knowledge of others about the knowledge of others and so on. That's the trick.

Do you agree with my earlier post? If there were only one blue, s/he would commit suicide on the first day? If there were only two, those two would commit suicide on the second day? And so on?

Well, if we are allowed to send people into exile, the tribe could ask the anthropologist to select one blue and one brown and tell them to ship out. But, the anthropologist must make sure that each exile never learns the identity of the other, if the exiles are to be saved...

I only have a minute. I've been away for quite a few hours, and I have to leave again in a minute. I haven't read most of the recent posts. This point caught my eye, however. If the defector was the only bluey, then he would have already realized his eye color. I thought anyone who had realized his eye color must commit suicide. If we cancel this rule, then everyone can relax for a few days and the five blueys can just leave after day four. What occurs to me is that stating that you will leave is not the same as actually leaving. Sorry, gotta go. Hope this comment isn't obsolete by now.

Based on what Ken said about 5 blues and 15 browns, this is what they know:

15 brown-eyed ones already know that 5 are blue-eyed and that 14 are brown eyed.

And 5 blue-eyed ones already know that 15 are brown-eyed and 4 are blue-eyed.

So they ALL know that there are brown-eyed and blue-eyed ones on the island. It’s Ok for them to see each-others eyes, but not their own.

All the professor says is that there are blue-eyed and brown-eyed ones. But they already knew that. He said nothing that would tell them which color their own eyes are, and they don’t say anything to each other to tell each other what each one sees.

I think there might be a defect in this version of the puzzle.




I already gave that solution. I said that would be the case if there were only 1 brown, 5 blues, and 14 greens, but Ken said I couldn't say that because there are 15 browns and 5 blues. So you can't say there is only 1 blue because Ken says there are 5 blues and 15 browns.

Ken said, "For the purposes of simplicity, let's assert that the only possibilities are brown and blue. The tribe is allowed to know this, they're just not allowed to know which is their own color."

Yes, that is a good solution, economy of banished people. And I need to correct my previous post-- knowing that blues are the minority has its own problems. Consider: what if the visitor had simply said "Like in my world, in yours I can see that it is more likely for people to have brown eyes". Knowledge that browns outnumber the other colors dooms all the browns to suicide, so again people would have to be exiled! There's no way around it, people have to leave at random until one member of all specifically mentioned colors must leave (or the anthropologist could finger them, and tell them that one of every color is going, when in fact it is only the mentioned colors). Once again, montebianco is right! The bottom line is, every color that is established to exist in the tribe in front of any group of that color dooms the members of that group until one of them leaves to break the chain.

The key case to consider is, what if there were 2 blues? Then everything you've said would hold, yet they would still be able to figure out their color. Let's say the two blues are you and one other. You think "poor devil, he'll probably have to die tomorrow", but then he's thinking that about you, so when tomorrow comes and neither of you commit suicide.... see? It's meta-knowledge, or knowledge about knowledge, that is imparted by the visitor.

Then we've got a completely different puzzle.

First you stated 5 blues and 20 people (the professor indicates at least one brown).

I solved that one.

Then you said 5 blues and 15 browns.

I solved that one too.

But I don't have time to solve all the different versions of your puzzle.

With 5 blues and 15 browns, none will commit suicide. Case closed. Puzzle solved. Next thread please.

Hi Ken G,

Have a look at the post above this one. I don't think you have a willing pupil...

Nick

The interpretation of the puzzle I've been following is that the announcement was "you're all only blue- and brown-eyed". Previously, each individual could assume they were green, orange, pink with flashing dots, whatever. The announcement limited their options, and so started the clock ticking to their doom.

And doom is right, becuase DOH! Sam5 - you're right !!! My solution doesn't work, because I was wrong with the initial case. Since a sole bluey will promptly kill himself (and so doesn't have the option of leaving), there is no uncertainty left, and the whole pack of cards collapses.

Suppose there are 2 blue-eyed tribesmens on the island and you're one of them. Before the visitor arrives you always assume that the blue-eyed tribesman you observe is the only tribesman with blue eyes -- ie, you're safe. (That tribesman is of course thinking the same thing about you.) When the visitor announces that there are blue eyes or the island you expect the *one* (so you think) blue-eyed tribesman to commit suicide. When they don't, you realize you're in trouble. They must have been watching someone else and waiting for that person to commit suicide. The only person that they could have been watching is you -- you can see that everyone else on the island has brown eyes so that rules them out. So because the other blue-eyed tribesman didn't kill herself on day 1, you deduce on day 2 that your eyes are blue and you kill yourself. The other tribesman went through identical reasoning watching you and so will also commit suicide on day 2.

Supposing there are 3 blue-eyed tribesmen, each is watching the other 2 and expecting, by the reasoning above, that they will both die on day 2. When they don't, she concludes that they must not be watching just each other. Since it's symmetric, the other 2 tribesmen are each watching you and the other one and thinking the same thing. Hence, you all 3 know to kill yourselves by the 3rd day. etc...