Hard puzzle round, valued for the discussion rather than the answer.
The blue-eyed islanders
The question
On an island everyone sees others' eye colour but not their own, and anyone who deduces theirs is blue must leave that night. A visitor says aloud 'at least one of you has blue eyes'. With one hundred blue-eyed islanders, what happens?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
All one hundred leave on the hundredth night.
With one blue-eyed person, the statement is new information and they leave on night one.
With two, each sees one other, expects them to leave on night one, and when nobody does, both deduce their own colour and leave on night two.
Induction carries this all the way: with n blue-eyed islanders, all leave on night n.
The visitor's statement seems to add nothing, since everyone already saw blue eyes. What it actually adds is common knowledge: everyone now knows that everyone knows, to any depth, and that is what starts the count.
What it is testing
Whether you can distinguish knowledge from common knowledge.
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