On an island, there are k people who have blue eyes, and the rest of the people have green eyes. Initially, no-one on the island knows their own eye colour. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island at dawn the next day. On the island, each person knows every other person's eye colour, there are no reflective surfaces, and there is no discussion of eye colour.
At some point, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement: "At least one of you has blue eyes". The outsider, furthermore, is known by all to be truthful, and all know that all know this, and so on. It is common knowledge that he is truthful, and thus it becomes common knowledge that there is at least one islander who has blue eyes. The problem: assuming all persons on the island are completely logical and that this too is common knowledge, what is the eventual outcome?
Solution
The answer is that, on the kth dawn after the announcement, all the blue-eyed people will leave the island.
The solution can be seen with an inductive argument. If k = 1 (that is, there is exactly one blue-eyed person), the person will recognize that they alone have blue eyes (by seeing only green eyes in the others) and leave at the first dawn. If k = 2, no one will leave at the first dawn. The two blue-eyed people, seeing only one person with blue eyes, and that no one left on the 1st dawn (and thus that k > 1), will leave on the second dawn. Inductively, it can be reasoned that no one will leave at the first k-1 dawns if and only if there are at least k blue-eyed people. Those with blue eyes, seeing k-1 blue-eyed people among the others and knowing there must be at least k, will reason that they must have blue eyes and leave.
What's most interesting about this scenario is that, for k = 2, the outsider is only telling the island citizens what they all already know: that there are blue-eyed people among them. However, before this fact is announced, the fact is not common knowledge. Until the announcement, each blue-eyed person knows that there is someone with blue eyes, but they do not know that the other blue-eyed person has this same knowledge.