Blue Eyes

A famously maddening logic puzzle — even people who "get" it argue about why it works.

On an island live 100 perfect logicians, all with blue eyes. But nobody knows their own eye color:

• There are no mirrors or reflections, and they may never discuss eye color.
• Every night at midnight, anyone who has deduced their own eye color must leave the island.
• Everyone can see everyone else's eyes, and all are flawless reasoners who know the others are too.

They've lived this way forever, in stable silence. Then one day a visitor, addressing the whole group, says out loud:

"At least one of you has blue eyes."

Everyone already knew that — each islander can see 99 pairs of blue eyes. The statement seems to add nothing.

On which night does everyone leave the island?

(Answer with the night number.)