The visitor who says she sees at least two blue-eyed people

Question

On an island live perfect logicians. Five have blue eyes and the rest have brown. There are no mirrors and eye colour is never discussed. Anyone who deduces their own eye colour must leave the island on the ferry at midnight that night. Everyone can see everyone else's eyes. One day a visitor addresses the whole island and says: "I can see at least two people with blue eyes." On which night do the blue-eyed islanders leave? What changes compared with the classic version where the visitor says "at least one"?

Accepted Answer

All five blue-eyed islanders leave on the fourth night. Two blue-eyed people. Each sees exactly one other blue-eyed person. The visitor says there are at least two. So each concludes on the spot: "the second one is me." Both leave on night 1. Three blue-eyed people. Each sees two others and thinks: "If I have brown eyes, there are exactly two blue-eyed people, and they will leave on night 1." When night 1 passes and nobody leaves, each knows there are at least three, so they are blue-eyed. All three leave on night 2. In general. With k blue-eyed people and the announcement "at least two", the…

The visitor who says she sees at least two blue-eyed people

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