The hotel with infinitely many rooms

Question

A hotel has infinitely many rooms, numbered 1, 2, 3, and so on without end, and every room is occupied. Guests can be asked to move, all at the same time, to a new room number that you announce. 1. A new guest arrives. How do you fit her in? 2. An infinite coach arrives carrying passengers numbered 1, 2, 3, ... How do you fit them all in? 3. Infinitely many such coaches arrive at once, each carrying infinitely many passengers. How do you fit everyone in?

Accepted Answer

All three are possible, because an infinite set can be put in one-to-one correspondence with a proper part of itself. One new guest. Ask the guest in room n to move to room n + 1. Every guest still has a room (room 1's guest goes to 2, room 2's to 3, and so on), and room 1 is now empty for the newcomer. Nobody is pushed out at the far end, because there is no far end. An infinite coach. Ask the guest in room n to move to room 2n. The current guests now occupy exactly the even rooms, and the odd rooms 1, 3, 5, ... are all free. Put coach passenger k in room 2k minus 1. Infinitely many coaches.…

The hotel with infinitely many rooms

Your answer

Discussion

Learn the concepts

Related questions