Difficulties with hotel rooms (Posted on 2007-01-25)

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?

Take the numbers of rooms, hotels, and planets, and group their sums together. Then from there, divide each group into lines of people with the same room+hotel sum in the same line, placing the people with smaller room numbers at the head of the line. (The "first line" would be the line with smallest room+hotel sum, and the next line would be the next smallest, and so on)

Starting with room 2, and going room 4, 6, 8 and so on, give everyone from the first line from the first group a room, starting at the head of the line. When a line is done, go to the next line, and when done with all lines in one group, go to the next group.

Although there are an infinite number of groups, inside each group there are a finite number of lines, and a finite number of people in each line, so you will be able to give everyone a room. (Each group is separated by two lines, and each line by one line.)