The Blue Eyes Puzzle

Problem

On an island, $100$ people have blue eyes and the rest have brown. Everyone can see everyone else’s eye colour but not their own. No one may communicate about eye colour. If a person discovers they have blue eyes, they must leave the island at midnight.

A visitor announces: “At least one of you has blue eyes.”

What happens?

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All $100$ blue-eyed people leave on the $100$ th night.

With $n$ blue-eyed people:

If $n = 1$ : that person sees no other blue eyes, knows they must be blue, leaves on night $1$ .
If $n = 2$ : each sees one blue eye. When that person doesn’t leave on night $1$ , they deduce they must also have blue eyes. Both leave on night $2$ .
By induction: all $n$ leave on night $n$ .

The visitor’s statement seems to add no information, but it establishes common knowledge — everyone knows that everyone knows that… everyone knows at least one person has blue eyes. This is the key to the chain of deductions.