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.
Hilbert I's Hotel
Riddle statement
A hotel has infinite rooms numbered 1, 2, 3, …, and they are all occupied.
A new guest arrives.
Is it possible to accommodate him without kicking anyone out?
Show solution
Solution
Answer: Yes.
Explanation:
It is enough to move each guest one room forward:
- the one from room 1 goes to room 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 move.