The Adventures of Ion, Intergalactic Milk-Man.

At the mere mention of this name, Cari and Tom are probably recoiling in horror and have collapsed in drooling siezures. But as much as we all hated Dr. Mrs. Salter at the time, in retrospect, I realized I learned more in her classes than in any others. Additionally, the stories of Ion are very interesting studies in the concept of infinity and I thought that those readers of FU-News who are mathenatically inclined but haven't had the love/hate experience of a class taught by Dr. Mrs. Salter (Mom and Dad B, Mrs. Pickles, The Blue Fox) might find these stories interesting.

Ion was first introduced to me by a certain professor of mathematics. While I had only swerving loyalty towards the professor in question, after just a brief amount of time talking with Ion, I discovered he had a wealth of humorous anecdotes and tales of mathematical intrigue.

It was during one of our many conversations that followed that he related to me the tales of his friend Calvin Q. Lator, owner of a very interesting chain of hotels. Cal, as Ion referred to him, was the owner of the "Inn of the Infinite Happiness" chain of hotels (slogan: "We've got a room for you, guarenteed"). What made this chain so notable, was that not only were there an infinite number of hotels in the chain, but each hotel had an infinite number of rooms. The actual mechanism that accomplished this is something of a trade secret, but it is known to involve poison gas, a closed box, and a cat that may or may not be dead (perhaps both). Cal kept his secret from absolutely everyone, except his own mother, Seeia.

So after one particularly taxing milk delivery run, Ion stopped at Cal's hotel near the Horsehead Nebula to get a good night's sleep before heading home. It was with a feeling of great relief that he entered the hotel's lobby, looking forward to some relaxation, and headed to the front desk.

"Hello," Ion said to the receptionist. "I'd like a room for tonight."

"I'm sorry sir, but we're full," came the receptionist's reply.

Ion's feeling of relief gave way to an even greater sense of confusion. "How can you be full? There are an infinite number of rooms?" asked Ion.

Upon hearing the elevated volume of his good friend's voice, Cal emerged from the office behind the front desk. He was a short, rotund man, scarcely five feet tall, and having a weight-to-height ratio of at least 36 lbs./ft. His balding hair gave one the impression that a tiny man with a tiny lawn mower had started cropping Cal's hair, but had gotten up to get a lemonade half-way through, with the job still unfinished. He was also exceptionally well dressed, wearing a nicely pressed suit jacket and pants, with a striped neck tie sporting the two-tone blue colors of the hotel chain. A gold watch chain hung from his pocket; Ion knew that on the back was inscribed "Cal, thanks for keeping one of your infinite doors open for me. G." Who 'G.' was, Ion didn't know, but the rumor was it was a famous actress or model, of the gorgeous persuasion. Cal wouldn't answer any questions on the subject prefering to keep people guessing. The reality was much tamer than the rumors, and Cal liked it that way.

Cal was quite the rich man, as being the owner of the Universe's largest hotel chain is prone to make one. He also had far more wrinkles than a man his age should have, and much of his remaining hair was grey, as having to manage an infinite number of hotels and rooms is prone to make one look older than one's peers.

"Ion! I'm both happy and sad that you have come to my hotel today," said Cal, as he greeted Ion with a firm handshake and a slap on the back. "Happy to see my friend once again, but sad because you have come at the one time where it seems I can't accomodate you."

Ion's puzzlement had not subsided. "How can that be? I don't understand how a hotel with an infinite number of rooms could possibly be full."

"One would not think it possible, but yet it has happened. You see," Cal began explaining, "the Association of Infinite Members has chosen this hotel for their convention this year. Since there are an infinite number of them, they have completely booked up my hotel! Every room is currently occupied by one of the convention-goers. Thank goodness no one else has come to get a room tonight. I'm afraid this could ruin my reputation."

"But surely, you can squeeze me in somewhere," pleaded Ion. "I mean, with an infinite number of rooms there's got to be an empty one."

"Not a one," Cal responded, upset that he had to turn away his good friend.

"Can you move a guest to another room?" asked Ion.

"Quite easily, but as I mentioned, all rooms are full. To what room would I move the guest?" Cal said, not really understanding where Ion was going with his question.

"Well then, I'll wager that you do have room for me. I'll wager you a lifetime supply of milk for an unlimited number of free hotel stays."

"That sounds reasonable," said Cal. "Now what room shall I put you in."

"Simple," said Ion. "Room 1."

"But Room 1 is occupied," said Cal. "I would have to move the occupant to another room."

"Yes, you shall move him to Room 2."

"But Room 2 is also occupied. Where shall I put him?"

"Room 3," replied ion. "You see, all you have to do is move the guest staying in Room N to Room N+1. There is still room for everyone in your infinite hotel. Should anyone else come into the hotel asking for X number of rooms, you just simply need to put them in Rooms 1 through X, and move every other guest to Room N + X."

"Astounding, Ion! I hadn't thought of that. Eegads, man, you're a genius."

"It's all part of the job of an intergalactic milk-man," replied Ion.

And so Ion was checked in and given the key card to access his room. Ion remarked that for a hotel with so many rooms, it was ironic that they only had one door. The rooms themselves didn't exist within normal space, but were instead accessed by sliding your room's key card through the control panel of a teleporter. When you opened the one hotel room door, you would be transported to your own private room. As Cal explained it to Ion, the process of moving a guest from one room to another was actually quite simple, and in fact, the guest didn't even know he was being moved. All that was done was the guest's key card was changed so that when he activated the teleporter, he was ported to Room N + 1, instead of Room N.

* * * * *

Ion had just gotten comfortable in his hotel room and was flipping through the infinite number of channels on the Hotel TV, when his phone rang. He hadn't told anyone where he was staying for the night, so it was with a small amount of curiosity that he answered the phone, wondering who it was.

"Hello?" queried Ion of the unknown caller on the other end.

"Ion! You must help me!" the caller said in a frenzied panic. It took Ion a moment to recognize the voice as belonging to Cal.

"What's the matter, Cal?" asked Ion.

"Oh, you must help me, Ion. You are the only one I could think to turn to, after you solved my last problem. You see, we've had a problem with double-booking. The Infinite Membered Consortium has booked this hotel for their convention also! We have an infinite number of rooms, so we didn't think to check for room availability. But I don't know what rooms to assign the infinite number of people who have just come into my hotel. I thought about using your suggestion of assigning them rooms 1 through X, and moving everyone else to Room N + X, but I can't move someone to Room N + ∞! There is no such room!" Ion was taken aback not by the problem, but by Cal's ability to somehow speak all that without ever stopping for a breath.

"Well," said Ion, "that certainly is a pickle. On the one hand, you have an infinite number of rooms, so you should be able to fit them in, but on the other hand, how do you fit an infinite number of people into a hotel that is already full?"

"My dilemma exactly. I fear it cannot be done," came Cal's nervous reply.

"Oh, of course it can be done!" said Ion. "And quite simply, too."

"Then please, enlighten me. Where can I put the guests who are waiting in my lobby?"

"Well, the first person should be assigned to Room 1, the second person to Room 3, the third to Room 5, and so forth."

"Yes, yes, but what of the people in those rooms?"

"I was getting to that. The person in Room 1 should be moved to Room 2; the person in Room 2 should be moved to Room 4; the person in Room 3 should be moved to Room 6, and so on," said Ion, smuggly. "You see, a guest staying in Room N should be moved to Room 2N. That way, only the even-numbered rooms are occupied leaving the odd-numbered rooms, of which there are an infinite number, available for the newcomers."

"Ah! I see! Thank you once again, Ion! You have saved my reputation."

Ion was barely able to get out the "You're" of "You're Welcome" before Cal, in a frenzy to begin accomodating his infinite number of patrons waiting in the lobby, hung up.

* * * * *

Much time passed by after that night (I say "much time" and not "many months" or "years" because depending on where in the galaxy you were, the orbital frequency of a planet's moons or of the planet itself changed dramatically). Ion and Cal had been corresponding in the time since they had last seen each other, but as all things of that nature go, the frequency of their correspondence waned with the passage of time. Then one moment, (I say "moment" because "day" is a very relative term depending on which planet you are on and what direction the side you are on is facing) Ion received an IGE-mail (inter-galactic e-mail) from Cal pleading for his help. The letter stated:

'Ion,

'You must help me. The Intergalactic Business Regulatory Commission is complaining that I have a monopoly on infinite-roomed hotel chains and forcing me to close all of my hotels but one and consolidate my guests to that one hotel. I can easily move guests from one hotel to another, but at the moment, all my hotels are completely full with an infinite number of conventions of societies with infinite members. I don't know how I can move an infinite number of guests from an infinite number of hotels into only one hotel, even if there are an infinite number of rooms. I fear this cannot be done, but I cannot afford to reimburse an infinite number of guests. I would be ruined! Please help me.

'Your friend,
Cal L.'

Ion had to think for quite a bit, but eventually offered Cal the following solution.

'Cal,

'It is, in fact, possible, so you needn't worry about going bankrupt. The answer is not trivial, and may take some explanation.

'What you need to do is come up with a function that, given a guest's hotel number and room number, will give you a new number for their room in your one remaining hotel. To begin, let's start making a list of all the rooms and all your hotels. Such a list would start out looking something like this, where the rows (m) correspond to the hotel and the columns (n) correspond to the rooms:



'Now we need to come up with a numbering system so that each hotel/room combo is counted. We can accomplish this by numbering the rooms in diagonals, as such:



'Now to come up with a function to figure out the new room numbers based on the old hotel/room combination.

'Note that for any hotel/room (M/N) combination, the number of the hotel at which the diagonal it lies on ends is M + N - 1. For instance, Hotel 2, Room 3 lies on the diagonal ending at Hotel 4. 2+3-1=4. Another way of looking at this might be that each diagonal corresponds to all hotel/room combinations that add to a given number. We can use this to our advantage.

'Now, let's take a look at a hotel/room combination's value in the list of all rooms. Remember this list value is the room number in the one remaining hotel that the guest will be assigned to. This list number is equal to the list number at the end of the diagonal above it, plus the position of the combination on the diagonal it is on. For instance, the list number of Hotel 2, Room 3 is equal to the list number at the end of the diagonal above it, 6, plus its position along its diagonal, 2. 6+2=8, which is the list number associated with Hotel 2, Room 3.

'A combination's position on the diagonal is just equal to its row number (M), and the hotel number at the end of the diagonal above it is (M + N - 1) - 1, or M + N - 2. Now to come up with a function relating the list value at the end of a diagonal to the hotel number at the end of the diagonal. A list of these hotel numbers and corresponding list values looks something like this:

1: 1
2: 3
3: 6
4: 10
5: 15
6: 21

'We can re-write the list values (new room numbers) as such:
1 = 1+0
3 = 2+1
6 = 3+3
10 = 4+6
15 = 5+10
21 = 6+15.

'The first addend is just the hotel number. The second addend (and this is trickier to see) is equal to half the difference between the hotel number and its perfect square. 1^2=1, half the difference of 1 and 1^2 is 0. 2^2=4, half of 2^2 - 2=1. 3^2=9, half of 9-3=3. 4^2=16, (16-4)/2=6, etc. Generally speaking, the second addend can be expressed as:



'So the whole value is:



'Now, this just relates the hotel number to the value next to it. To calculate the new room number based on the current hotel and room number, we substitute the hotel number at the end of the diagonal above the one our hotel/room combination lies on (M+N-2) into the above equation and add the position on the diagonal that the hotel/room combination lies upon (M). So the function to calculate the new room number of a guest staying at Hotel M, Room N is as follows:



'Which simplifies to:



'Your friend,
Ion

'P.S. Perhaps a simpler way of expressing this is as a piece-wise defined function. If instead of listing the new rooms in a diagonal fashion, we use a square numbering method like such:



'Then we can define the new room function as:

* * * * *

Cal was quite thankful for Ion's reply. He was most thankful, however, that he was able to "stick it to the man" and avoid bankruptcy, despite a very stern commissioner for the Intergalactic Business Regulatory Commission who seemed intent on closing Cal's hotels down. Cal thought that perhaps he must have inadvertantly offended the commissioner's mother.

Eventually, the commissioner was fired in a scandal whose details were not widely know, but it involved a weasel, a chainsaw, and a watermelon. As this commissioner was not well-liked among the businessmen of the intergalaxy, a group of them decided to throw a party to celebtrate their good fortune. Cal was hosting the party and in a particularly good mood, as it was mere seconds after the commissioner's dismissal that he was allowed to re-open the rest of the hotels in his chain. Cal had asked Ion to provide the milk for the party, and so he did, saying "it's what I do best."

It was at this party that Ion was hastily pulled aside by Cal. Cal had been having a very intense, yet friendly, argument with his business manager, Warren, after they had both had a few too many glasses of milk. Cal had been extolling the praises of Ion and how he'd saved him from bankruptcy and humiliation countless times (which is not necessarily an exageration if you are the manager of an infinite number of hotels). Warren agreed that this Ion fellow must be pretty smart, but wagered that he couldn't solve the hotels' every problem. In truth, he was feeling a bit threatened that Cal so frequently had to go to Ion for help, completely skipping over his manager.

"Nonsense," said Cal. "If anyone can solve one of the hotels' problems, Ion can. I'd wager the secret to my infinite hotels on it!"

"I will have to take you up on that," said Warren. It was then that Cal pulled Ion into the conversation and explained to him the stakes.

"I've wagered the secret of my infinite-roomed hotels against an infinite supply of the best wine with Warren here that if anyone can solve a problem about my hotels, you can," explained Cal.

"You have quite a lot of confidence in me, Cal," Ion said, just a bit nervous about being put on the spot. Then turning to Warren he said, "expain the problem to me."

Warren said, "I have a problem that's been nagging at me for a bit. It has to do with maintenance and assigning cleaning staff to one hotel. I would like to have a list of all possible configurations of the rooms in the hotel. Each list of this entry should state whether each room is occupied or vacant and the list in its entirety would contain every possible combination of occupied and vacant rooms. This list would be very useful for assigning cleaning personnel and for booking purposes. Do you think you can make such a list, Ion?" Warren had a very smug expression on his face.

Ion stopped to think about it for a moment, rubbing his hand across his chin. After a minute he looked Warren straight in the eye and said without any emotion. "No, I can't."

Cal, who had been waiting for Ion with an increasing sense of nervousness, suddenly stared agape at Ion. "Ion! You can't let me down now. Are you telling me that I'll have to tell my secret to Warren?"

"Quite the opposite, actually," said Ion. "You've won the bet and he owes you an infinite supply of the best wine."

"But you yourself admitted you couldn't make such a list, Ion," interjected Warren. "How have I lost the bet?"

"The wager was that if anyone can do it, I can. But such a list cannot be made, not because I am incapable of it, but because it is impossible. As such, Cal has won the wager."

"A-ha! But I have you there," said Warren. "I have such a list! I made it myself. The way I accomplished this was as such:

"Each of my cleaning supervisors is assigned ten rooms to care for. Of course there are an infinite number of maintenance supervisors. I asked each supervisor to make a list of all possible combinations of his ten rooms, using 0 for vacant, and 1 for occupied. Each of these lists, therefore contains 1024 entries.

"I then combined each of these lists together, in every possible combination, into one master list, giving me a super-list of all possible configurations of vacant and occupied for the hotel."

"That's an very impressive attempt," said Ion. "But I can think of a configuration that isn't on your list."

"Oh yes? Please tell," said Warren, with an air of over-confidence.

"Well, it's very similar to your first entry, but it differs in the first digit. So if the first entry on your list begins with 1, my configuration begins with 0."

"Nonsense," said Warren. "I have that configuration somewhere else in my list. It's probably the second entry."

"No, it's not the second entry, either," said Ion. "My configuration differs from your second entry in the second digit."

"Then it's the third entry," said Warren.

"I don't think so either," said Ion. "My configuration differs from your third entry in the third digit. Actually, it differs from your nth entry in the nth digit."

"Well, even if your configuration is not on my list, I'll just add it to the list, and then I'll have a complete list."

"It's not that simple," said Ion. "You would have to add it as the nth entry, and then I would just create a new configuration that is not like it because it differs in the nth position. You see, no matter how many configurations you have, I can create a new configuration that differs from all your others in the nth digit. That is the first digit is different from the first entry, the second digit from the second entry, and so on. Such a list of all possible configurations cannot be made because all possible configurations are uncountably infinite which, by the very definition of such, cannot be listed, that is, put into one-to-one correspondence with a list that is countably infinite."

"Eegads, man, you're right!" proclaimed Cal. Warren just stood with a dumbfounded look on his face, trying to piece together what Ion had just said.

The rest of the party passed without any further incedent, except for the maid getting tipsy on the milk Ion had brought. In the end, Cal gave half his winnings from the bet to Ion, and still had an infinite amount of wine for himself. "That's the wonderful thing about infinity," Cal was known to say. "It doesn't matter how much you take out of it. You still always have an unending supply leftover."