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PuzzlesBrain Teasers

Suppose you have a hotel which has one floor with an infinite number of rooms in a row (numbered $1, 2, 3, \ldots$ ), and all of them are occupied.

A new customer wants to check in. Can you accommodate her?
What if an infinite number of people want to check in?
Suppose an infinite number of buses arrive at the hotel, each carrying an infinite number of people. Can you accommodate all of them?
In which of these cases can you accommodate all the new guests (every current guest must still have a room)?

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