A hotel has infinite rooms numbered 1, 2, 3,…, and they are all occupied. A new guest arrives. Is it possible to host it without kicking anyone out?
Home > Riddles > Hilbert I's Hotel
Hilbert I's Hotel
There are few problems so brief and so revealing. Here infinity is not presented as something enormous, but as something strangely flexible: moving everyone still leaves room for one more.
Hints
Show hints
- It is not necessary to empty many rooms; It is enough to release just one.
- The obstacle in a finite hotel would be the last room. There isn't one here.
- Moves a each guest of the room n a the n+1.
Solution
Show full solution
Answer: Yes. Explanation: Simply move each guest one room forward: - the one from 1 goes to 2,
- the one from 2 goes to 3,
- the one from 3 goes to 4,
- and so on. That way no one is left without a room and room 1 is free for the new guest. The apparent paradox arises because in a finite hotel this maneuver would not work, but in an infinite one it would: there is no “last room” that blocks the shift.
Related riddles
Keep practicing
If you enjoyed this one, try more pure-logic riddles, explore this theme, browse the full archive, or read the riddle-solving guide.