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Home > Student activities > National Maths Day > National Maths Day 2014 > Hilbert’s paradox of the Grand Hotel

Hilbert’s paradox of the Grand Hotel

The Grand Hotel has an infinite number of rooms, all of which are occupied. Can the hotel accommodate any more guests?

Move the person occupying room 1 to room 2, the person occupying room 2 to room 4, and, in general, the person occupying room n to room 2n. Then all the odd-numbered rooms will be free for the new guests.