A nice logic puzzle Terry Tao reposted on his blog.
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?
There are two plausible-sounding solutions. The logic part of the puzzle is figuring out which is correct and why the other is wrong. As a first solution, the foreigner gives no one any additional information about their own eye color. The other solution follows by induction on the number of people with blue eyes. (If one person has blue eyes, he deduces the foreigner is talking about himself, and so kills himself. If two people have blue eyes, they assume the foreigner is talking about the other person. When the other person doesn’t kill themselves the next day, they deduce they must both have blue eyes, and so they both kill themselves on the second day. If three people have blue eyes, they expect the other two people to kill themselves on the second day. When this doesn’t happen they deduce all three had blue eyes, and all three people kill themselves on the third day. So on and so forth, the other solution implies that the 100 blue eyed people will all kill themselves 100 days after the foreigner’s address.)
