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Ask all the guests in rooms >= 2 to move over one, then move into room 2.
There is always a vacancy.
Then housekeeping has to re-clean infinite rooms before everyone can resettle, which only works if you have infinite cleaners. Just walk down to the infinite + 1 room and no one has to move ;)
You will eventually reach a room with a worse situation than that one, and then your entire edifice falls apart again.
If they just have the guests stay in even numbered rooms, then they’ll always have an infinite number of rooms free.
Are they free if you can’t put people in them?
Asking the real questions here.
I wonder if the world would be any different if Hilbert had never lived.
