A stranger tells 100 blue-eyed islanders something they all already knew
On an island, 100 people have blue eyes and know the local rule: anyone who works out their own eye color must leave that midnight. No one ever leaves, because everyone can see everyone else's eyes but not their own, and no one ever mentions eye color. One day a visitor announces, truthfully: 'At least one of you has blue eyes.' What happens next?
Reveal the answer
On the 100th midnight after the announcement, all 100 blue-eyed islanders leave. The visitor's statement adds no new private information — everyone already saw at least one blue-eyed person — but it creates common knowledge: everyone now knows that everyone knows that everyone knows... Working through the logic by induction, starting from what would happen if there were only 1 or 2 blue-eyed islanders, the whole group can only deduce their own eye color on day 100. Popularized by mathematician Terence Tao, who traces versions of the puzzle back decades.
— Terence Tao, The blue-eyed islanders puzzle — terrytao.wordpress.com, 2008