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?

Not that I disagree with any of the previously posted solutions, but here's yet another way to do it:

Make all your guests move to double their room number. Hence all the odd numbered rooms are free.

Then each incoming guest is assigned a triple of (x,y,z) like in the other solutions: x = planet, y = hotel, z = room number. Assign each of these a room number equal to (3^x)*(5^y)*(7^z). This has the advantage that each guest can easily compute their own room by that simple formula, and each of them has a unique room, plus you have infinitely many rooms left over.

Posted by Avin on 2007-01-31 12:01:29