Harry Potter and the Diagon(alization) Alley

Thursday, February 26, 2015 · 6 min read

It is a well-known fact among wizarding circles that the Gringotts Wizarding Bank has an infinite number of vaults, numbered from 1 onwards. Equally well-known is the fact that in the name of what goblins call efficiency and everyone else calls parsimony, each of the vaults is currently occupied.

As a result, there has formed a situation which Muggle economists describe as “scarcity”. Gringotts vaults, like heirloom-quality furniture, are prized in families. They only change hands at readings of wills of dead great-aunts.

With that in mind, it was, of course, reasonable for Mr. Hill Bertok to scoff at the old man who demanded of him a vault one chilly February morning.

Mr. Hill Bertok was a run-of-the-mill businessgoblin and teller at the Gringotts Wizarding Bank. The old man was Professor Albus Dumbledore, headmaster of the Hogwarts School of Witchcraft and Wizardry.

Mr. Bertok did not, of course, notice this at first. Had he realized who the old man was, he would perhaps have shown a little more respect before scoffing. The scoff, however, had been scoffed already, and a scoff once scoffed cannot be recalled.

Having conceded this, Mr. Bertok attempted mildly apologetic gesture with his eyebrows and presently returned to his paperwork. When he looked up a few minutes later, the old man was still there.

“Dumbledore,” he said, “as a goblin I hold you in much higher regard than most wizards. But I still think you’re crackers.”

“Very well, Mr. Bertok, but many millions of lives depend on the security of the object I have with me. Perhaps you can make an exception?”

“Exception! What object could possibly require so much security?”

“It’s a rock, Mr. Bertok. A very important rock.”

“…you are crazy, Dumbledore. I’ll tell you what: if you can find an empty vault, you can have it.”

“Mr. Bertok, I believe I have this problem sorted out already. If you’ll lead me down, perhaps I can demonstrate.”

“Crazy, completely and utterly bonkers,” thought Mr. Hill Bertok as he led the old man through the labyrinthine passages of Gringotts.

“713 sounds like a nice round number,” said Dumbledore, “Let’s stop here. Do I have your permission to use this vault? It’s a harmless procedure, really.”

“You are out of your mind.”

“Need I remind you that I am the Chief Warlock of the Wizengamot?”

“You need not,” seethed a very reluctant goblin, adding (under his breath) “Albus Dumbledore pulling rank—what is the world coming to?”

“Very well then. Presto-incremento!“ For a sliver of a second, Vault 713 of the Gringotts Bank glowed a pale electric blue. Then, with a gentle rumble, the door swung open, revealing absolutely nothing whatsoever.

“You asinine old man! That vault contained more galleons than your demented old brain can count!”

“Fear not, Mr. Bertok, you will find all of your galleons. They have simply been transported to Vault 714.”

“Impossible! Vault 714 contains statues of gold; you cannot fit a mountain of galleons in there!”

“Patience. The statues of Vault 714 have been moved to Vault 715. And—before you get started again—the contents of Vault 715 have been moved to Vault 716.”

“Excuse me?”

“Each vault’s contents have been moved into the next vault. Ad infinitum. That left Vault 713 open for me.”

“Wait, but… how can… our clients! How do you propose to notify them of the change?”

“I had the foresight to dispatch some Ministry owls ahead of my visit.”

“You cannot send owls to an infinite number of clients, Dumbledore.”

“It may interest you to know, Mr. Bertok, that the Ministry happens to have a fleet consisting of an infinite number of owls. They’ve been quite useful: just last week we used them to—ahem—persuade a Muggle family to send their nephew to Hogwarts.”

And with those words, Professor Albus Dumbledore dropped a small grubby bag inside the vault and closed it.

The news spread faster than the plague. The next morning, the tellers of Gringotts Bank were greeted at the front desk by Stan Stunpike. Unfortunately, they were also greeted by an infinite number of passengers on his infinitely long bus. These were restless passengers who wanted their magically-created Gringotts vaults and would not leave without them.

Owing to the lack of space and general stuffiness that was developing in the Gringotts lobby—and the inevitable threat being posed to his employment—Mr. Bertok decided to do something about it. After the first 50 new clients, though, he began coming to the realization that making all of these accounts would take an infinite amount of time.

He didn’t have an infinite amount of patience, but he knew someone who did. He sent an owl to Dumbledore.

“Hush, calm down, everybody.” The lobby was beginning to smell like the inside of a used coffin, so Dumbledore decided, for the first time in his life, to quit the whole pedagogical spiel and solve the problem.

“Presto-doublinato!“ he cried, and the lobby began to rumble as an infinite number of doors began to open.

“Now what have you done?”

“Vault 1’s contents have been moved to Vault 2. Vault 2’s contents have been moved to Vault 4. Vault 3’s contents have been moved to Vault 6. In general, each vault has been moved to the vault with double the number.”

“How does that help?”

“All the odd numbered vaults are empty. Since each of your angry bus passengers has a seat number, you assign them a vault based on those. The person in the first seat is assigned the first odd number (1), the person in the second seat is assigned the second odd number (3), and the person in the hundredth seat is assigned the hundredth odd number (199).”

Mr. Hill Bertok briefly considered hiring Dumbledore in the HR department.

Wizards, it has been found, have friends. As a result, after the miracles of the second day, news spread like proverbial wildfire and the parking lot behind Gringotts was quickly filled with an infinite number of infinitely long buses.

Mr. Bertok knew what to do this time, and before you could yell “cardinality”, Dumbledore had apparated to the lobby, holding a big bucket of black paint.

Without a word, he walked out to the parking lot and began painting large numbers on the buses’ windows.

“Hey, Mister, what do you think you’re doing?” cried an understandably distressed bus-driver. He was quickly and efficiently turned into a frog by Gringotts’ security team, and Dumbledore continued with his painting, undeterred.

After a few tense minutes, the parking lot resembled this (pardon my badly-illustrated row of buses):

+----+----+----+----+----
< 01 | 02 | 04 | 07 | ...
+----+----+----+----+----
< 03 | 05 | 08 | ...
+----+----+----+----
< 06 | 09 | ...
+----+----+----
< 10 | ...
+----+----
|... |

“As you can see,” he began (the outdoor parking lot was airy enough for his pedagogical side to shine), “I’ve numbered each bus window diagonally. If I keep this up, each window will get a number.

“Since I’ve already shown you how to deal with that with the use-all-odd-numbers trick, I think I’ll take my leave now. It’s almost lunchtime.”

He disappeared with a pop, leaving one frog and an infinite number of very confused wizards.

Like all good things, this one had to come to an end. The end came when an inhabitant of a Hogwarts painting overheard a conversation about Gringotts’ new policy. And so, the next morning, the portrait of Gringott the Goblin was inhabited by an infinite number of painting-people. In front of them was a dashing, athletic-looking young gentleman. A caption floated above his head. It read “C. Antor. Tennis player.”

“Greetings, Mr. Bertok. I’m Charles Antor, representing the paintings.”

“Welcome to Gringotts, Mr. Antor.”

“Well, we came to ask: can you work your magic and give us vaults, too?”

“What does a painting need a vault for?”

“If you’re going to denigrate us, we’ll take our business elsewhere.”

Mr. Bertok hesitated. “Alright, we will need you all to line up so that we can number you.”

“I’m afraid we can’t do that.”

“What?”

“We aren’t numbered with counting numbers like 1, 2, 3… We’re numbered with real numbers. Decimals and fractions. Each of us identifies himself or herself with a number between zero and one.”

“So? Why can’t you all stand in order?”

“Well, suppose we did order ourselves in a line, and suppose you assigned us all vaults.”

“Alright, then what?”

“What is the number of the painting at the front of the queue?”

“What? How can that possibly matter?”

“Please bear with me. Just invent a number between zero and one. It doesn’t matter which one.”

“If you say so. The painting at the front of the queue is painting 0.234567.”

“Well, the first digit of my number is a 1, not a 2 like his. So can we agree that I’m not at the front of the queue?”

“I suppose so.”

“Good. Now, who’s second?”

“Should I make up another number?”

“Sure.”

“0.1111111.”

“Well, my number’s second digit is a 9, not a 1. So I’m not second in line either.

“I don’t see where you’re going with this.”

“Mr. Bertok, no matter how you number us paintings, there will always be someone whose first digit is different from the first person’s first digit. And whose second digit is different from the second person’s second digit. And whose third digit is different form the third person’s third digit. And so on, and so forth.

In other words, he cannot have any position in your queue, and so he can’t be in the queue.”

Mr. Bertok scratched his head. “But if there’s no way to number you folks, there’s no way to assign you vaults.”

“Exactly. In a weird way, there are ‘more’ of us than there are of wizards in Stan’s bus. Even though both numbers are ‘infinity’.”

“Gosh, who know paintings could be so complicated? You aren’t even real.”

“On the contrary, Mr. Bertok, we are as real as can be.”

Epilogue. Mr. Hill Bertok, now inspired, went on to study infinities. After an interesting encounter with two old wizards named Banach and Tarski, he discovered a means of mining an infinite amount of gold. Surprisingly enough, he ended up living happily forever after.

Charles Antor went on to become a tennis star among the paintings. Unfortunately, he met his match when a certain B. Russell proved to the referee that none of his sets could exist.

The events of this story thoroughly confused Stan Stunpike, who decided to take an early retirement from bus-driving and instead perhaps go herd a finite number of goats in Mongolia. Fortunately, a pair of psychiatrists, Calkin and Wilf, convinced him that he can fit all the rational people in the world onto his bus. Though he tried hard, he never really got the hang of it, and one day his bus was found—destroyed—with an infinitely large tree growing out of the windshield.

This post was inspired by something my computer science teacher said. I forgot what exactly it was that he said. Most ideas were shamelessly stolen from a chapter in Ian Stewart’s delightful book, Professor Stewart’s Cabinet of Mathematical Curiosities.