Blue-eyed Islanders Puzzle

A group of people have what is called "common knowledge" of something when they all know it, they all know that they know it, they all know that they all know that they know it, and so on ad infinitum.

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 color. 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 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?


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