A famous induction puzzle. Why does the guru's sentence start a countdown? From the free puzzle set on Think Fanny — next try Three switches, one bulb.
The puzzle
On an island, there are 100 blue-eyed people, 100 brown-eyed people, and a guru with green eyes. Everyone can see everyone else’s eyes, but no mirrors and no talk about eye color. If a person ever becomes sure their own eyes are blue, they must leave on the midnight ferry that night.
They are all perfect logicians. For years, nobody leaves.
One day the guru says, in public: “I can see someone who has blue eyes.”
What happens, and when?
Answer
All 100 blue-eyed people leave on the 100th night.
With 1 blue-eyed person, they leave night 1. With 2, each waits one night for the other to leave, then both leave night 2. By induction, n blue-eyed people leave on night n after the public announcement.
