Blue Eyes and Brown Eyes

This is a puzzle I heard on a much smaller level while I was in my freshmen year of college. Georgia Tech has a high school mathematics competition every spring for potential incoming students. The competition comes in rounds – and those that don’t make it to the final rounds can attend fun mathematical talks. I was helping with the competition and happened to be at a talk on logic puzzles, and this came up.

I bring it up now because it has raised a lot of ruckus at Terry Tao’s blog. It doesn’t seem so peculiar to me, but the literally hundreds of comments at Terry’s blog made me want to spread it some more. There is something about this puzzle that makes people doubt the answer.

I have reposted the puzzle itself, as written by Terry. But for his included potential ‘solutions,’ I direct you back to his blog. Of course, the hundreds of comments there also merit attention.

Terry’s puzzle:

There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. 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 or her 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 devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

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?

Note 1:  For the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

Note 2: An essentially equivalent version of the logic puzzle is also given at the xkcd web site.  Many other versions of this puzzle can be found in many places.


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3 Responses to Blue Eyes and Brown Eyes

  1. Mathlete says:

    ALL Blue eyed know either there is 99 or 100 blue eyed..
    ALL Brown eyed know either there is 100 or 101 blue eyed…

    Now as they are perfect logician all blue eyed now either the suicide will occur on 99th day or 100th day..
    and same for vrown eyed they all know that either it will occur on 100th day or on 101st dat..

    After the comment of Visitor blue eyed will start reasoning and as on day 99 no blue eye wil turn up so on 100th days all will turn up because they will be confimed that only 100 can be the max and I am missing..

    • mixedmath says:

      While it is true that all each person knows the amount of each color, give or take 1, it is not so obvious that they all know that a suicide will occur even with the visitor’s additional information.

      A good question might be: what additional information did the visitor actually give since everyone already knew that there was a blue-eyed person among them?

  2. mixedmath says:

    At What’s New, the conversation ended up going to the “Unexpected Hanging Paradox,” and Terry posted a link to an interesting paper that examines this Paradox and proves Godel’s Second Incompleteness theorem with it.

    You can find the paper here.

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