Hilbert's Dormitories
In Hilbert's Dormitories, there are an infinite number of rooms labelled numerically: 1, 2, 3, .... The current accommodation is rather strange:
- Every room labelled with a perfect square (like rooms 1, 4, 9, 16, ...) has 2 occupants: a boy and a girl.
- Every other room has just one occupant: a boy.
Hilbert, the superintendent, doesn't like this setting and wants every room to have exactly 2 occupants: a girl and a boy. Can he possibly do this, given an infinite amount of time?
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1 solution
Simply move every girl (initially occupying a perfect square labeled room) to the square root of her room. Of course leaving the girl in room 1 there.
Now every room (say N) has a girl (from room N x N) and a boy.
The rooms initially occupied by a girl would get a new girl (except room 1), but that's OK. Except maybe for the former roommates, but that is the downside to living in Hilbert's Dormitories!