AC. No Vacancy

A hotel with an infinite number of rooms, all filled, but room for more?

We answer last week’s puzzle on coins, and begin to discuss the amazing Hilbert Hotel!

The Hilbert Hotel has an infinite number of hotel rooms (numbered 1, 2, 3, … and so on forever) Every room is filled. But if one more guest shows up, you can easily fit her in!

Not by sending her to the “last room”! You have to tell her specifically what her room number is!

But you can shuffle the other guest around: the guest in room number 1 moves to room number 2; the guest in room number 2 moves to room number 3, and so on. Each guest knows exactly what room to go to.

But what if an infinite number of new guests shows up? (The guests have nametags labeled 1, 2, … Each old guest and each new guest needs to know what their room number will be)

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