On a remote island live 100 people with blue eyes and 100 people with brown eyes. Every person on the island is highly logical and aware that there are 100 people with blue eyes and 100 with brown eyes.
However, nobody knows their own eye colour; they can only see the eye colours of other people, and there are no mirrors. There are no discussions of eye colour on the island, ever.
A visitor comes to the island and announces to everyone simultaneously: “At least one person here has blue eyes.” The visitor is telling the truth, and everybody hears it, but they only hear it once.
The islanders are aware of the following strict rule: if an islander ever discovers that they have blue eyes, they must leave the island at dawn the following day.
Everyone can observe who leaves the island each day at dawn (and they can keep count), but nobody knows why someone left unless they know that person’s eye colour for certain.
What happens after the visitor makes the announcement, if anything?
After the visitor makes the announcement, all of the people with blue eyes will leave the island on the 100th day after the announcement. Here’s why:
- Case of 1 Blue-Eyed Person: If there were only one person with blue eyes, they would look around, see no one else with blue eyes, and then realise that they must be the one. They would then leave the island at dawn the next day.
- Case of 2 Blue-Eyed People: If there were two people with blue eyes, each would see the other but would not leave on the first day (as each would be unsure if they themselves might have brown eyes). On the second day, however, they would each see that the other person had not left, realise that they must also have blue eyes, and both would then leave on the second day.
- General Case of N Blue-Eyed People: Following the same logic, if there are N blue-eyed people, they will all leave on the Nth day after the announcement.
In the original question, the visitor announces what everybody already knows (“At least one person here has blue eyes”). However, it acts as common knowledge, allowing all 100 blue-eyed people to leave on the 100th day.