In the universe Roomeron, there are infinitely many planets. Each planet has an infinite number of hotels, and each hotel has an infinite number of rooms. Since the business is so great, you decide to build a hotel of your own, also with an infinite number of rooms. To keep track of the rooms, each is numbered starting at 1. The hotels and planets are similarly numbered.

During the current tourist season, every room of every hotel, (including yours) on every planet is full. A freak catastrophe occurs in every other hotel besides yours and their rooms become trashed. The guests from those hotels ask to stay in your unwrecked hotel.

How can you put the infinitely many guests from infinitely many hotels from infinitely many planets in your already full hotel?

What is required is a way to assign each incoming guest to a unique unoccupied room.

Move all your current guests into the room numbered one less than double their original room number.  This frees up all your even numbered rooms.

Number each planet, hotel, and room number for every incoming guest starting at 1.  Each incoming guest is then assigned an ordered triple (planet,hotel,room) as follows:

(1,1,1) -> 2
(1,1,2) -> 4
(1,2,1) -> 6
(1,2,2) -> 8
(2,1,1) -> 10
(2,1,2) -> 12
(2,2,1) -> 14
(2,2,2) -> 16
(1,1,3) -> 18
(1,2,3) -> 20
...
(3,3,3) -> 54
(1,1,4) -> 56
...

Exhaust all the triples with each integer before advancing the maximum integer.

A formula assigning (p,h,r) to a unique room # would not be very hard to find.

Posted by Jer on 2007-01-25 12:58:46