Emoticorpse is asking how you think the brown eyed people arrive at the conclusion that their eyes are brown as opposed to some other non-blue color.
Ah, I see. Each brown-eyed islander could speculate that the guru could have called out a third eye color, like pink, because that islander doesn’t know his/her own eye color. However, he/she knows for certain the guru could have called brown, and also knows for certain all the remaining islanders know this as well.
Right, I get all that. it’s just counterintuitive at first to think that if the lightning bolt happens right after the announcement, no new information was added with the guru’s statement. If the lightning bolt doesn’t happen until after the first night, then it stays common knowledge that at least one islander has blue eyes, and even if one is struck dead on the 2nd day, it doesn’t get ruined.
He knows for certain the guru could have named any human eye color.
A brown eyed person, once the guru speaks, will reason as follows:
There are either 100 blue-eyed-islanders, or 101. If there are 101, then (by the reasoning that’s been illustrated in this thread already, which you say you understand and accept,) then they will not leave til day 101.
Then, when they all leave on day 100, he will realize there were 100 of them, not 101. And from this he will conclude his own eyes are not blue. But that is all he will know–that his own eyes are not blue.
That only makes sense if you don’t understand why the blue eyed people are counting in the first place. It’s not because there was some arbitrary event they could coordinate their counts on. If that were the case, as I indicated before, there’s no logical reason for them to count from 0 to 200 instead of from 200 to 0. Or really to count at all. They’re counting from 0 because they learned something when the guru spoke. Each night they learn a bit more. But unless they know that there are only two eye colors, they learn only about blue eyes and not brown.
I don’t think they learn something each day. I think they’re waiting til day 99 to find out whether there are 99 or 100 b-islanders.
Why wait? Because if there are only 99 b-islanders, they’re waiting til the 98th day, knowing that if there are only 98 b-islanders, then they are waiting til the 97th day, knowing that if there only 97 b-islanders etc…
No, that’s quite clearly not true. The islander only knows for certain the guru could have called blue or brown (or possibly green). He can speculate he has a different eye color from anyone else on the island, but he doesn’t know this for certain. He also knows that if his eyes were pink, then his brown-eyed comrades would also know that he couldn’t know for certain that the guru could have called pink. The only logical course of action is for everyone to act as if the guru had called brown, because that’s the only other eye color they know for certain he could have called.
I am not sure what to say to this. It’s simply manifestly false.
Any eye color is possilbe. So the brown eyed islanders know, for certain (because they are perfect logicians), that the guru could have said any eye color at all.
I was initially confused with the A knows that B knows that C, etc. For me the easiest way to solve it is imagine that you are on day 98. You know that you see 99 blue eyes. You could have blue eyes which means there would be 100 blue eyes.
Day 99 arrives and nobody leaves.
So you (every islander) concludes that there must be a minimum of 100 blue eyes. Since you only see 99, then you realize that you MUST have blue eyes, so you leave the island that night because you are now sure of your eye color (along with all other blue-eyed islanders who reached the very same conclusion you did).
A brown-eyed islander sees 100 sets of blue eyes and is still unsure of his own eye color. So, on day 100, he doesn’t leave because he may have blue eyes himself and can’t be sure until day 101. But when 100 blue-eyed people leave on day 100, he realizes that he does not have blue eyes.
Frylock, remember that the guru is only allowed to speak the truth, so she can’t say that she sees an eye color that she doesn’t see.
That being said, Greg, the 100 brown eyed people do not know their own eye color, or that there are 100 brown eyed people on the island. Each individual could have brown, green, red, blue, black, hazel, amber, etc. Just because WE know there are 100 brown eyed individuals, and WE know that the guru could only call out either blue or brown, the brown eyed islanders DO NOT know this.
That’s not relevant, though. The guru has to speak the truth, but a brown eyed person does now know his own eye color. So as far as the brown person knows, the guru could have said “I see at least one person with amber eyes.” The brown eyed islander is certain that this is possible. (And he’s right. For all he knows, it definitely is possible.)
Basically, exactly what you say below:
Each night that passes without anyone leaving they all learn that everyone knows that there are more than that many blue eyed people.
I find this endlessly fascinating.
Suppose the blue-eyed and brown-eyed islanders speak different dialects of the same language. The dialects are very similar but where they differ is in the names of colors. Brown, for instance, in the dialect of the brown-eyed islanders is called blue.
Now the Guru speaks. “I see someone with blue eyes.” (We are unaware which dialect the Guru speaks.) The islanders (both sides completely oblivious to the ambiguity as they have never communicated) start their reasoning, the blue-eyes hearing ‘blue’, the brown-eyes hearing ‘brown’).
Do both blue-eyes and brown-eyes successfully leave the island after 100 or 101 days, just the blue-eyes still or is the chain of logic now somehow flawed and nobody can leave?
Yeah, that’s right now that you mention it–assuming you mean to “infinitize” that “everyone knows”.
Hmm, this is very frustrating. Let’s say we’re at the 100th day in my scenario. 120 brown-eyed islanders are waving goodbye to their blue-eyed friends. Every one of them knows there are brown-eyed islanders remaining. Do you agree with that? Good, we’re halfway there. Each one also knows that his own eyes may be any color (except blue), including brown. So the difference is, while they know the guru definitely could have called brown if he so chose, they know it to be only possible for the guru to have called another color.
I feel like that much should be easy to accept. That this knowledge allows them to escape the island may be harder to accept, but I haven’t spotted a flaw in my reasoning yet.
Hmmm. I think in this case they’d all leave on the hundredth day (each group eyeing the other suspiciously as they board the ferry), but that only works if there are the the exact same number of each eye colour.
Say we have only three people on the island. Alan has blue eyes, Bob and Brian have brown (which, of course, they call blue).
After the guru’s announcement:
Alan sees Bob and Brian with brown eyes, thinks “I must be the one with blue eyes!”, and leaves that night.
Bob sees Alan with brown (blue) eyes and Brian with blue (brown) eyes and thinks “Either Brian is the only one with blue eyes, and he’ll leave tonight, or and he and I both have blue eyes, and we’ll both leave tomorrow night.”
(Brian, of course, is following the same thought process about Bob)
Then Alan leaves. Confusion. Bob and Brian might think Alan made a mistake, but how could he if they are all perfect logicians? Perhaps they could infer that Alan speaks a different dialect, but how could Bob know that he and Brian are speaking the same one? What if Bob has red eyes, but Brian calls that colour blue?
I think the logic breaks down here if the islander’s cannot properly predict the thought processes of the others. Or at least adds another level of ambiguity. Does each islander know that there are two and only two languages? Does he know that the others know the same?
“It definitely could have happened” and “it was only possible for it to happen,” to me, mean exactly the same thing. If something is only possible, then it is definitely only possible, and so it definitely could be.
What do you take the difference to be?
What do you think of drew’s latest post and my response to him? They are relevant to this “definitely could have/only possible” issue.
The puzzle never stated they all wanted to leave? maybe they were having a good time on the island and just wanted to know their own eye colors. they figured it all out and then waited til the party was done to catch the ferry???
just a joke btw…
I think that intertwining the words “they” and “you” in this thread confuses everyone. the fact that they “all” left at the same time is incidental. you kind of have to think this one did this. forget everyone else. just pick one person and do what that person would do. by the time you figure out why you’d leave then look at the “they” part. don’t use the word they until you understand how “he” or “she” did it by themselves.
Chessic Sense, your “family Clue game” story really helped me. Along with Frylock’s “Bob and Carl” routine, it really clarified the following:
The phrase “everybody knows”, when put in a linked chain (e.g., “everybody knows that evrybody knows that everybody knows…”), is confusing and hard to grasp.
Actually, every time you use the phrase “everybody knows”, you are BUNDLING TOGETHER a bunch of SPECIFIC statements. Each specific statement has the form “Person X knows that person Y knows that person Z has blue eyes”, “Persons X knows that person Y knows that person X has blue eyes”, etc., etc.
It is much easier for me to understand the logic behind each specific statement, than the logic behind blanket term “everybody knows”. Yes, I can then make the rational leap to bundle this into a string of “everybody knows”, and I can be confident that it’s true, but as soon as we bundle it in this way, I’ve lost the direct grasp of what’s going on.