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Do you like mathematical brain teasers? This one should keep you busy for a while.
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 worldâ€.
What effect, if anything, does this faux pas have on the tribe?
I never really understood what was wrong with argument 1, though I understood and believed argument 2, until my friend Alon Amit explained it to me.
In the case of 1 blue-eyed person, it’s not the case that everyone knows that someone has blue eyes.
In the case of 2 blue-eyed people, everyone knows that someone has blue eyes, but not everyone knows that everyone knows that someone has blue eyes: the 2 blue-eyed people don’t know that everyone knows that someone has blue eyes.
And so on. With 3 people, not everyone knows that everyone knows that everyone knows that someone has blue eyes. WIth 100 people … you get the idea. And that’s the information that the foreigner’s comment changes.
You don’t have common knowledge without the foreigner. Therefore argument 1 is flawed.