Puzzles
A fully booked hotel with infinite rooms can always fit one more guest
The Grand Hotel has infinitely many rooms, numbered 1, 2, 3 and onward, and every single one is occupied tonight. A new guest arrives asking for a room. Can the manager accommodate them without turning anyone away?
Reveal the answer
Yes — ask every current guest to move from room n to room n+1, which empties room 1 for the newcomer, and the hotel is still completely full. The same trick, run further, can make room for infinitely many new guests all at once. David Hilbert used the paradox in a 1924–25 Göttingen lecture to illustrate the strange arithmetic of infinite sets; George Gamow popularised it in his 1947 book One Two Three... Infinity.