A fully booked hotel with infinite rooms can still always find room for one more
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. The manager insists there's no problem at all, full as the hotel is. How can that possibly be true?
Reveal the answer
Ask every current guest to move from room n to room n+1. Room 1 is now empty for the new arrival, and because the hotel has infinitely many rooms, nobody runs out of a room to move into. David Hilbert devised this thought experiment in a 1924 lecture to show how counting rules that work perfectly for finite sets break down, in genuinely useful ways, once infinity enters the picture; the same trick can even make room for infinitely many new guests at once.