Answer to blue-eyed suicide - a math problem:
If there are n blue-eyed persons on the island, they will all commit suicide on the n-th night.
The proof is by induction:
Clearly, if n=1, the one blue-eyed person, would commit suicide the night of the oracle's proclamation. This blue-eyed person knows that everybody else on the island has brown eyes; but since the oracle has said there must be at least one blue-eyed person, our blue-eyed person knows it must be him/her.
Now, let some n be given, and assume that the following statement is true: "If there are n blue-eyed persons on the island, they will all commit suicide on the n-th night (i.e. each of them can prove to his/herself on the n-th day that s/he has blue eyes)". I will show that this implies that the statement is also true for n+1.
Suppose there are n+1 blue-eyed persons on the island. On the n+1-th morning, each blue eyed person would look around and see that none of the n blue-eyed people that s/he can see had committed suicide. But by our inductive hypothesis, that means there must be at least one more, and they can see only n blue-eyed persons, each has proof that s/he is also blue-eyed. They all commit suicide that night.
This is a pretty tricky problem. To get a better grasp on it, consider the case where there are only three islanders, two of whom have blue eyes. On the morning after the oracle's fateful pronouncement, each blue-eyed person sees that the other blue-eyed person did not commit suicide. This tells blue-eyes #1, for example, that blue-eyes #2 must know that someone else has blue eyes. This can only be #1 himself, and thus he has the fatal proof.
Food for thought:
What is the importance of the oracle in this story? (I'm not telling here... /msg me and I might ;)