On Pigeon Island, in Lake Agua, a tribe of several hundred people live happily, completely cut off from the rest of the world. Some (at least one) of these people have a purple spot on the back of their heads. It is impossible to see whether you have such a spot, and nobody will ever tell you, because it is a tribal rule that if you learn that you have a purple spot on the back of your head, you are required on that very day to hurl yourself off of the highest cliff into the lake, thus drowning.
Life progresses in a stable and happy manner, until one day Dondre, who is out for a sail from the mainland, is caught in a storm and runs his ship aground on Pigeon Island. The islanders help him push off again. Just as he is leaving, he says, "My goodness! I just saw a purple spot on the back of someone's head!"
He doesn't say any more than that, and no islander ever mentions the incident either. But a certain number of days later all of the islanders who have purple spots hurl themselves off of the cliff into the sea.
Explain this. You don't have to know how many islanders had purple spots. Your explanation should work for any number.
Solution
Take a deep breath, and consider this fact. If an islander knows about 15 purple spots, then he knows that there are either 15 or 16 of them, the 16th being on his own head.
Now, we know that there is at least one purple spot because Dondre saw it. If there is only one, then there is an islander, we'll call him islander A, who has never seen any at all, so he learns that the one spot must be his, and he hurls himself into the lake.
Now here's the tricky part. Suppose that there are two purple spots total. Suppose that islander B has one of those spots. This means he knows there is only one spot, on islander A, or maybe two, if that second one is on his own head. Now, if the only spot is on islander A, then as we saw islander A will learn that he has a spot and will hurl himself into the lake. But suppose the whole day goes by and he doesn't? That means he has seen a spot too, and it must be on islander B. He waits for islander B to hurl himself into the lake, but B doesn't do it because he's waiting for A to do it. So on day 2, both A and B hurl themselves into the lake.
We continue with this reasoning and see that if there are n islanders with purple spots, on the nth day they will all hurl themselves into the lake.