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Math Help - Remote island with a happy tribe

  1. #1
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    Remote island with a happy tribe

    In a hypothetical scenario there exists an island with some number of inhabitants among which everyone knows everyone. If an inhabitant learns he/she has blue eyes, then he/she will flee at midnight; however none of the inhabitants will tell the other what color their eyes are, nor are there any mirrors for an inhabitant to learn the color of his/her eyes.

    One day everyone on the island is told by an outsider at the same time that there are only two colors of eyes among the island's population: blue and brown.

    43 days later many of the inhabitants flee at midnight.

    How many have fled and why?

    I don't see how any of the inhabitants can deduce they have blue eyes given only the information that there are two types of colors unless the outsider who tells this is lying and everyone has in fact brown eyes -- then everyone would believe they were the one and only person with blue eyes since there is supposedly at least one person with blue eyes.
    Although due to the way the problem text is stated it doesn't seem like it would allow for the possibility of the outsider to be lying. But even if the outsider were lying, why would 43 days be relevant?

    Is there some other way to consider the problem? Is there some other way for the inhabitants to deduce the color of their eyes? Why is 43 days important?

    [SOLVED]
    Last edited by posix_memalign; April 30th 2010 at 10:15 PM. Reason: Solved
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  2. #2
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    Quote Originally Posted by posix_memalign View Post
    In a hypothetical scenario there exists an island with some number of inhabitants among which everyone knows everyone. If an inhabitant learns he/she has blue eyes, then he/she will flee at midnight; however none of the inhabitants will tell the other what color their eyes are, nor are there any mirrors for an inhabitant to learn the color of his/her eyes.

    One day everyone on the island is told by an outsider at the same time that there are only two colors of eyes among the island's population: blue and brown.

    43 days later many of the inhabitants flee at midnight.

    How many have fled and why?

    I don't see how any of the inhabitants can deduce they have blue eyes given only the information that there are two types of colors unless the outsider who tells this is lying and everyone has in fact brown eyes -- then everyone would believe they were the one and only person with blue eyes since there is supposedly at least one person with blue eyes.
    Although due to the way the problem text is stated it doesn't seem like it would allow for the possibility of the outsider to be lying. But even if the outsider were lying, why would 43 days be relevant?

    Is there some other way to consider the problem? Is there some other way for the inhabitants to deduce the color of their eyes? Why is 43 days important?

    Hint - Assume there is only one person with 'Blue' eyes. What will happen? What if there are 2 people with 'Blue' eyes?

    Now can you generalize (think induction)?
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  3. #3
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    Quote Originally Posted by posix_memalign View Post
    In a hypothetical scenario there exists an island with some number of inhabitants among which everyone knows everyone. If an inhabitant learns he/she has blue eyes, then he/she will flee at midnight; however none of the inhabitants will tell the other what color their eyes are, nor are there any mirrors for an inhabitant to learn the color of his/her eyes.

    One day everyone on the island is told by an outsider at the same time that there are only two colors of eyes among the island's population: blue and brown.

    43 days later many of the inhabitants flee at midnight.

    How many have fled and why?

    I don't see how any of the inhabitants can deduce they have blue eyes given only the information that there are two types of colors unless the outsider who tells this is lying and everyone has in fact brown eyes -- then everyone would believe they were the one and only person with blue eyes since there is supposedly at least one person with blue eyes.
    Although due to the way the problem text is stated it doesn't seem like it would allow for the possibility of the outsider to be lying. But even if the outsider were lying, why would 43 days be relevant?

    Is there some other way to consider the problem? Is there some other way for the inhabitants to deduce the color of their eyes? Why is 43 days important?

    This island's inhabitants not only all know all but they also must be pretty intelligent and be capable of doing logic and also they must be sure the outsider told them the truth about their eyes.

    Let k be number of blue-eyed (BL) inhabitants and n-k the number of brown-eyed (BR) ones. Note that both k\,,\,n-k\geq 1 according to the outsider:

    ** if k = 1 this person will know at once that he's the BL one and he'll flee that very 1st night;

    ** if k = 2 each of the two BL's will wonder why the other BL didn't flee the first night (nobody did...) , so they both will realize each is a BL and they both will flee the 2nd night;

    ** if k = 3 the first night , as above , nothing will happen. Now, each BL will think that both BL's he/she sees will flee the 2nd night if he's a BR (we'd be back in the prior case!)...but the 2nd night, of course, nobody flees, so each BL realizes he's a BL and the three of them flee on the 3rd night .

    ........................... can you generalize now?

    Tonio
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  4. #4
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    Yes, I see it now, thanks. :-)
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