Math Investigation eTwinning. Mission: Infinity...Affairs! Volume: Infinity | Page 7

Now the first room is empty, and we can move in. Of course this wouldn’t have worked in any
finite hotel. We could still ask all guests to move up into the next room, but the person in the last
room would be left without a room. The Hilbert doesn’t have a last room, and infinity plus one is
still infinity.
This idea can be extended: If 10 new guests arrive, the reception just asks all guests to move up 10
rooms. If 100 new guests arrive, all current guests have to move up 100 rooms – and similarly for
any other number. Thus even if the hotel is full it can still offer rooms to any number of new guests!
Let us suppose that the hotel is full again, and that infinitely many new guests arrive. Now the
previous method won’t work anymore: you can’t ask the guests to move up infinitely many rooms –
they would never arrive in their new room.
Instead the hotel does something very clever: they ask their guests to move into the room
with twice the number of their current room. Thus the guest in room 1 will move into room 2, the
guest in room 2 will move into room 4, and so on. The guest in room n will move into room 2n.

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LYKEIO SCHIMATARIOY, GREECE

- VALLAURI FOSSANO, ITALY

Σελίδα 7