A fully booked hotel with infinite rooms finds room for one more guest. How?
Imagine a hotel with infinitely many rooms, numbered 1, 2, 3 and so on, and every single room is occupied. A new guest arrives asking for a room, and the manager agrees without hesitation. How?
Reveal the answer
Ask every current guest to move from room n to room n+1: the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on forever, which frees up room 1 for the newcomer without evicting anyone. Mathematician David Hilbert devised this thought experiment in a 1924 lecture to illustrate the strange, non-intuitive properties of actual infinity.
— David Hilbert, Hilbert's paradox of the Grand Hotel — Popularized from a 1924 lecture