The island of the ojos azules

1. Start with the smallest case: what would happen if there were only one blue-eyed person?
2. Then try two blue-eyed people and compare what each one sees.
3. Generalize from \(n-1\) to \(n\): if no one leaves before, what does that force on day \(n\)?
4. From there, the argument chains by induction.

Answer: The 100 blue-eyed people leave the island at dawn on the 100th day. Explanation: The key is induction. - If there were 1 person with blue eyes, upon hearing “I see at least one” she would immediately know that person was herself and would leave the next day.

• If there were 2, each one would see one. I would think: “If I weren't blue, the other one would be the only blue one and she would be gone tomorrow.” Since no one leaves on day 1, they both deduce on day 2 that they are also blue.
• The same reasoning continues by induction. With 100 blue-eyed people, each one sees 99 and waits for 99 sunrises without anyone leaving. Since no one leaves in those 99 days, everyone simultaneously concludes that they also have blue eyes. That is why the 100 leave the island at dawn on the 100th day.