Hotel Infinity was a 4-star, 4-diamond resort sprawling over 40 acres of lush, landscaped grounds in the California desert. As its name implied it had an infinite number of rooms.
One night at Hotel Infinity, the Front Desk Agent was wrapping up for the evening. It was 10:30pm and his shift was nearly over. It had been a very busy day with many Check-outs and many Check-ins. In fact, the hotel was now fully occupied - there was a guest in every room.
At around 11pm a gentleman entered the lobby and approached the desk.
"May I have room for the night?" he asked.
"I am sorry sir, we are fully occupied tonight" said the agent.
[Guest] "But I've been traveling all day and I am very tired. I would certainly appreciate it if you could accommodate me."
[FD Agent] "Let me call the Manager on Duty and see what we can do."
[FD Agent, on radio] "Front Desk to MOD. Front Desk to MOD".
[MOD] "Go for MOD."
[FD Agent] "We need your assistance in the Lobby, please."
[MOD] "10-4, on my way."
When the MOD arrived in the Lobby, the Front Desk agent explained the situation.
[FD Agent] "We are fully occupied and we need to accommodate one more guest."
[MOD] "No problem at all. Remember, we work at Hotel Infinity. We can move the guest in Room #1 into Room #2, the guest in Room #2 into Room #3, the guest in Room #3 into Room #4, so on, and so on. Then you may give our new guest Room #1."
[FD Agent] "This is wonderful. Thank you."
[MOD] "You're welcome and let me know if you need any more assistance."
[Guest] "I am a little hungry. Is there a restaurant onsite?"
[MOD] "Yes sir, we have Restaurant Eternity in which you may dine."
[Guest] "Great, and at what time does it close?"
[MOD, smiling] "Restaurant Eternity is always open for the guests of Hotel Infinity." The guest and the FD agent nodded at the MOD in appreciation of his efforts and his quick dispatch of the problem.
All was well again at Hotel Infinity.
Shortly after 11:30, the Front Desk Agent heard a loud noise of a bus circling the porte-cochere. Going outside to investigate, the Agent met the Bus Driver who was on his way in to the Lobby.
[FD Agent] "May I help you sir; you are making quite a racket!"
[Bus Driver] "I am sorry, but you see I have a bus load of tourists and we need a place to stay for the night."
[FD Agent] "How many people do you have on your bus?"
[Bus Driver] "There is an infinite number of people on my bus."
[FD Agent] "Infinite?"
[Bus Driver] "Yes, infinite."
[FD Agent] "Front Desk to MOD, come in MOD."
[MOD] "This is the MOD."
[FD Agent] "We have another situation and need your assistance."
[MOD] "10-4, I'm on my way."
Arriving, the MOD was filled in by the FD Agent. An infinite number of guests to accommodate but not one room was vacant!
[MOD] "Not to worry. Remember, we work at Hotel Infinity. Move the guest in Room #1 into Room #2, the guest in Room #2 into Room #4, the guest in Room #3 into Room #6, ..."
[FD Agent] "Ah, I see where you're going with this, you're moving all the guests into the even-numbered rooms, thereby leaving all the odd-numbered rooms available for our infinite number of guests waiting on the bus. Brilliant, fantastic, thank you."
[MOD] "You're welcome. I'll be out on my rounds if you need me."
All was well again at Hotel Infinity.
At around midnight, a tremendously loud sound of many, many buses were heard coming round the porte-cochere. The FD agent went out to the first bus and spoke to the driver.
[FD Agent] "Good evening, sir. How many buses are in your group?"
[Bus driver] "Hi there! We have an infinite number of buses, each containing an infinite number of people."
[FD Agent] "Oh no, this will be a challenge to accommodate all these guests. You see we are fully occupied tonight. I must summon the MOD."
[FD Agent] "Front Desk to MOD. Front Desk to MOD."
[MOD] "Go for MOD."
[FD Agent] "We have yet another situation that requires your attention please."
[MOD] "On my way."
Arriving at the Main Entrance the MOD quickly surmised the situation and realized the FD Agent was in a quandary.
[MOD] "Relax, we can handle this. Remember we work at Hotel Infinity."
[FD Agent] "But an infinite number of buses with an infinite number of guests each?"
[MOD] "Do not worry. We will make use of the prime numbers. All our in-house guests will be associated with the first prime number, 2. The guest in Room #1 will be moved into Room #2. The guest in Room #2 will be moved into Room #4. The guest in Room #3 will be moved into Room #8."
[FD Agent] "Yes, ok, I see. 2, 4, 8, 16, 32, the powers of two, correct?"
[MOD] "Yes. Then for the guests on the first bus we will use the next prime number, 3."
[FD Agent] "You mean we will put them in rooms 3, 9, 27, 81, etc.?"
[MOD] "Precisely, now you've got it. Then just move on to successive prime numbers for the successive buses."
[FD Agent] "That will take a while."
[MOD] "It will take an eternity, but I know you can handle it."
[FD Agent] "But wait, we still have an infinite number of buses."
[MOD] "Yes we do."
[FD Agent] "So how do we know there is an infinite number of prime numbers to handle this method?"
[MOD] "We had an employee named Euclid who proved it 23 centuries ago."
[FD Agent] "Did he work at the Front Desk?"
[MOD] "No, he worked in the IT department, but the hotel was under different management back then."
[FD Agent] "I see. Thank you for your help tonight."
[MOD] "You're welcome."
All was well again at Hotel Infinity.
After the FD Agent printed out his final reports, he notified the MOD that he would be leaving soon.
[FD Agent] "Front Desk to MOD. Sir, I'll be leaving property shortly and wanted to thank you for helping me to accommodate all our guests tonight."
[MOD] "You are quite welcome."
[FD Agent] "The Night Auditor has just arrived, and I will update him that we have a full house so he is prepared..."
[MOD] "Hold on there. Have you checked your In-House Guest Register lately?"
[FD Agent] "Let's see. This is interesting, I did not realize it until now. Room #1 is vacant. Room #6 is vacant. So are Rooms #10, #12, #14, #15, and many, many more. It appears we now have an infinite number of vacant rooms ?!?"
[MOD] "Correct. Since we used just the prime numbers for our last check-in procedure, all rooms that are not a prime number, or a power of a prime number, are now vacant. The Night Auditor should not have any issues checking in additional guests. Remember, we work at Hotel Infinity. Have a good night."
[FD Agent] "Good night, sir."
All was well again at Hotel Infinity.
All was always well at Hotel Infinity...