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Math Help - eyes

  1. #1
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    eyes

    There is an island upon which a tribe resides. The tribe consists of 1000 people, 100 of which are blue-eyed and 900 of which are brown-eyed. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and highly devout, and they all know that each other is also highly logical and highly devout.

    One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.

    One evening, he addresses the entire tribe to thank them for their hospitality.

    However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

    What effect, if anything, does this faux pas have on the tribe?
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    It shouldn't have any if they are so devout. A blue-eyed person can see that 99 other people have blue eyes, and 900 people have brown eyes. A brown-eyed person can see 100 ppl with blue and 899 with brown. Why would someone take what the foreigner said as they are talking about him/her?
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  3. #3
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    Or 100 days after the traveler speaks, they all commit suicide.
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    Quote Originally Posted by shilz222 View Post
    Or 100 days after the traveler speaks, they all commit suicide.
    Why?
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    he is looking at a blue-eyed person when he is saying the statement. And so for  n = 1 the blue-eyed person commits suicide the next day. No its not suicide. Its ritual suicide. Then proceed by induction. A person will reason that if he is not blue-eyed, then there are  n-1 blue eyed people in the island, and they will commit suicide  n-1 days after the traveler's note. But none of them do commit suicide on the  n-1^{th} day because they do not know they have blue eyes. So there is a collective suicide on the  n^{th} day (they figure that they must have blue eyes).
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    Quote Originally Posted by shilz222 View Post
    he is looking at a blue-eyed person when he is saying the statement. And so for  n = 1 the blue-eyed person commits suicide the next day. No its not suicide. Its ritual suicide. Then proceed by induction. A person will reason that if he is not blue-eyed, then there are  n-1 blue eyed people in the island, and they will commit suicide  n-1 days after the traveler's note. But none of them do commit suicide on the  n-1^{th} day because they do not know they have blue eyes. So there is a collective suicide on the  n^{th} day (they figure that they must have blue eyes).
    I didn't know one person was being pointed out as blue eyed. In that case, that person would commit ritual suicide, but why would anyone else follow? I do not see how logic or induction plays a role here. Call me dense, but I'm not seeing it?

    If the foreigner stays and says this everyday, then of course there will be a ritual suicide each of the following mornings. Any of the blue-eyed people will see 98 people with blue eyes the next day. Hmmm, lost?
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    Quote Originally Posted by colby2152 View Post
    Call me dense, but I'm not seeing it?
    Make it easy. Two people. Say you have unknown color eyes. You see your fellow have blue colored eyes. If he does not do kill himself then it means you must have blue eyes because if you had brown eyes he would have reasoned that he must have blue eyes because at least one of you have blue eyes and killed himself. But the reason why he does not kill himself is because you have blue eyes and he is in doubt, meaning he can have blue or brown. Thus, you conclude that you must have blue eyes because you brought doubt upon the person.
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    It becomes clear when you simplify the problem. In the case in which there are 901 tribe members, 900 of which have brown eyes and 1 has blue eyes, the one with blue eyes sees that the other 900 do not have blue eyes, so he figures, he must have blue eyes, and he commits suicide the very next day.
    In the case in which there are 902 tribe members, 900 brown-eyed, and 2 blue-eyed, the ones with blue eyes see 900 others with brown eyes and 1 with blue eyes, so they don't know that they were addressed by the foreigner, but they both don't know. And when, the following day neither of the two commits suicide, they both figure they must have blue eyes as well, otherwise the other one should and would have committed suicide, so they commit suicide themselves, at day 2.
    With 3 blue-eyed tribe members, they all commit suicide on day 3, because they figure that if neither of the other two committed suicide on the first two days, there have to be at least three tribe members with blue eyes, and they can only see two of them, so they know they have to be the third and they have to commit suicide.

    And by induction, with 100 blue-eyed tribe members, they all commit suicide on the 100th day, because they found out then that there are 100 blue-eyed tribe members, whereas they can only see 99 of them.

    Koko
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    Quote Originally Posted by Koko View Post
    It becomes clear when you simplify the problem. In the case in which there are 901 tribe members, 900 of which have brown eyes and 1 has blue eyes, the one with blue eyes sees that the other 900 do not have blue eyes, so he figures, he must have blue eyes, and he commits suicide the very next day.
    In the case in which there are 902 tribe members, 900 brown-eyed, and 2 blue-eyed, the ones with blue eyes see 900 others with brown eyes and 1 with blue eyes, so they don't know that they were addressed by the foreigner, but they both don't know. And when, the following day neither of the two commits suicide, they both figure they must have blue eyes as well, otherwise the other one should and would have committed suicide, so they commit suicide themselves, at day 2.
    With 3 blue-eyed tribe members, they all commit suicide on day 3, because they figure that if neither of the other two committed suicide on the first two days, there have to be at least three tribe members with blue eyes, and they can only see two of them, so they know they have to be the third and they have to commit suicide.

    And by induction, with 100 blue-eyed tribe members, they all commit suicide on the 100th day, because they found out then that there are 100 blue-eyed tribe members, whereas they can only see 99 of them.

    Koko
    and the rest kill themselves the next day?

    RonL
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    Quote Originally Posted by CaptainBlack View Post
    and the rest kill themselves the next day?

    RonL
    Why would they?
    They reason that since 100 members killed themselves on the 100th day, there must have been no more than 100 tribe members with blue eyes.

    Koko
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    Quote Originally Posted by Koko View Post
    Why would they?
    They reason that since 100 members killed themselves on the 100th day, there must have been no more than 100 tribe members with blue eyes.

    Koko
    Well if eyes come in only two colours all the remaining islanders now know that they have brown eyes.

    Of course there could be the posibility of more than two colours.

    RonL
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    Thank you Koko and TPH, I understand this induction problem now!
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    If you want to see more discussion, terry tao has the problem in his blog: Here
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  14. #14
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    Of course, if you think of this problem inductively (or if you suppose that these tribesmen would do so), then it's clear that all of the blue-eyed people will commit suicide one hundred days after the foreigner speaks. However, I don't believe that the islanders would necessarily think inductively - by assuming that they would, we are assuming that induction in the "right" way to think through this problem. The contraint on the situation is that these people are logical, but induction is not the only logical way to analyze or prove something.

    In the case of n=1, there is no doubt that the person would commit suicide the next day, as there is no doubt he will realize he must have blue eyes.

    In the case of n=2, there is little doubt that the people will commit suicide after the second day, as they will reason that they must have blue eyes.

    In the case of n=3, there is good likelihood that the people will commit suicide after the third day, as they may reason the n=2 case (from the n=1 case) and will discover they must have blue eyes.

    However, as n becomes larger, it becomes less likely, as a consequence of it being less logical, to reason through this situation with induction. If all of the tribesmen think exactly the same way, and know with absolute certainty that every other tribesmen will think in that same way, and if they all consider the n=1 case, then it is necessarily true that all blue-eyed people will commit suicide. However, if either they do not all think exactly the same way, they do not know they all think the same way, or they do not consider the n=1 case, they would not commit suicide. And realize, thinking about a thing by induction is not obvious or necessary if there happens to be a more direct or deductive way of reasoning. In the case of n=100, the more obvious and direct way to reason through what the foreigner has said is simply, "Of course there are people with blue eyes. I see them every day," in which case, no consequece or mass-ritualistic suicide is necessary.

    Note: after those with blue-eyes commit suicide, it is not necessarily the case that those with brown eyes will commit suicide the next day. Unless the tribespeople know that the only possible eye colors are blue and brown, they would not necessarily know their eyes were brown because their eyes can conceivably be some other color, like green.
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