On an island, there are k people who have blue eyes, and the rest of the people have green eyes. There is at least one blue-eyed person on the island (k ≥ 1). If a person ever knows they have blue eyes, they must leave the island at dawn the next day. Each person can see every other person's eye color, there are no reflective surfaces, and there is no discussion of eye color. 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?