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I've been discussing the logic puzzle where 100 logicians, all wearing blue hats, are trapped in a room and told: they can leave when a bell rings if they know they know they are wearing a blue hat, they are all perfect logicians and at least one of them is wearing a blue hat. The bell rings every five minutes.
I solved this by reasoning that if one person is in the room they can leave immediately as they know they are wearing a blue hat. If there are two people in the room then they will each reason that if they are wearing a red hat the other person would leave immediately and if the other person doesn't they can leave after the second bell ring. For three people they each reason that if they are wearing a red hat then the previous situation for two people would play out and if that doesn't happen they can leave after the third bell ring and so on until with 100 people they all leave after the 100th bell ring.
My solution to the problem relies on the fact that they are told one of them is wearing a blue hat but I've been told it should be enough that they all know that everyone can see a blue hat and therefore they don't need to be told at the start that at least one person is wearing a blue hat. Is it still solvable without that information?
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