Hilbert's Hotel: A puzzle developed by the mathematician David Hilbert to illustrate infinity.
Imagine you arrive at a hotel late at night that prides itself on its infinitely many rooms. To your disappointment, however, you are told that there are no vacancies.
Astonished, you ask the receptionist: "But don't you have infinitely many rooms?"
"That's right," he replies, "and we would love to offer you one. Unfortunately, we also have infinitely many guests at the moment. Thus, every available room is currently occupied; even the broom closet is taken. There are absolutely no rooms available at this hotel."
You consider his reply thoroughly. How in the world could you still manage to get a room at Hilbert's hotel without having to share it with some other guest?*
And if you are a travel guide for an infinitely large group of tourists (having arrived in an infinitely long tourist bus), how could you possibly accommodate all members of your group in this fully booked-up hotel?**
* Here is the solution: You talk (or bribe) the receptionist into moving the guest from room # 1 to room # 2, the one from room # 2 to room # 3, and so on. Since there are infinitely many rooms, all guests can be moved and room # 1 will be available for you.
** Here is the solution: You propose to the receptionist that he assign all guests currently at the hotel to rooms with even room numbers, thus freeing up the infinitely many odd-numbered rooms for your group. You can read up on why this will work in the article on ►countability.