Monday, 27 October 2014

Mathematical Halloween

Mathematics often inspire fear to many people. And sometimes even rightly. So, in a game that takes place in Transylvania we discover some very special and mysterious figures: the vampire numbers.
Mathematical Halloween: vampire numbers

(This post participes on the 111th edition of the Carnival de Mathematics, hosted by the blog Boole's Rings.)

FIRST HALF

Terrifying numbers Never a few numbers had caused so much terror.

Today's Halloween, and an interesting match of the Champions League is about to be played between the teams of CFR Cluj and Manchester United FC.

CFR Cluj badgevs.Manchester United FC badge


Dr. Constantin Rădulescu stadium of Cluj CFR
The bus of the English team has arrived to the stadium a couple of hours before the kick off.

The players have put their tracksuits on and have come to the pitch to do the warm-up exercises.

It was in that moment when Juan Mata has fled in panic into the changing room, and has taken refuge in it. He says he doesn't want to go back to the field again.

The Manchester United coach, Louis van Gaal, is very worried because Mata is an essential piece of the team. He doesn't know what to do, nor the reason of Mata's behaviour.

It seems that the player returned to the dressing room when he saw some advertisements with bloodied numbers.

Commercials with bloodied numbersJuan Mata, Manchester United player

Knowing that Juan Mata is fond of Mathematics, Louis decides to call his friend Joe Vitruvius with his last generation phone, to see if he can find any solution.

Louis van Gaal, the Manchester United coachLast generation mobileJoe Vitruvius 

What do you think is happening?




SECOND HALF


Van Gaal phones Joe Vitruvius, and tells him all the details of what happened.

- So, where did you tell me you play?

- We're in Cluj. It's the capital of Transylvania, in Romania, a very beautiful Carpathian town.

Historical center of ClujHistorical center of Cluj

- I see. And, what time is it there now?

- It's half past eight. Just fifteen minutes to start the game, and though it's cold, the sky is clear and the moon shines perfectly.

- Is there a full moon?

Full moon over Cluj's stadium

- Yes, today it's a full moon.

- And what happened when you came to the pitch to train?

- Well, I had noticed that Juan was a little nervous. But when he saw those billboards, he began to run to the changing room.

- Which billboards?

- Some billboards with red numbers, as if they were written with blood.

Billboards with bloody numbers

- Were there only numbers?

- Yes. I though that they were some telephone numbers of a local on-line car insurance company, or something like this

- OK. Do you remember what numbers were they?

- No. But wait a moment, I took several photos of the warm-up session with my mobile. Look, here I've got them. I can see, among others, the following numbers:

Vampire numbers
- Hmm... I think I have an idea of what happens. Let me have a look at the wonderful website of the OEIS... Yes, that's what I though.

- What happens, then?

- It's clearly obvious. Everybody knows that Juan Mata is a big fan of Mathematics. And that he's also a little superstitious. The Cluj's managers also know this fact, and they know how important is his participation in the game of Manchester United. That's why they've put those ads on the panels of the stadium.
Map of Transylvania
- I don't understand.

- All right. You are in Transylvania, it's night, full moon, and Halloween. A really sinister combination, and very disturbing for someone so superstitious as Juan Mata. But they needed to frighten him a bit more.

- And they've succeeded with a few simple numbers?

- Yes. They could have put into the pitch someone dressed like a vampire, but it would have been too obvious. They have devised something more subtle.

Transylvanian vampire
- Why are these numbers so special? I don't see them very sinister...

- Yes, they are so dark. These numbers are known as vampire numbers.

- Huh? Why?

- Well, this name was suggested by the great mathematician Clifford A. Pickover. Vampire numbers are numbers that satisfy some properties:

1.- They must have an even number of digits.

2.- They're the result of the product of 2 numbers, which we call fangs.

3.- They have the same digits as their fangs.

4.- Not all their fangs end in zero.

- I haven't understood anything.

- I'll explain it to you with an example: the number 1395. This number has 4 digits, and can be obtained multiplying numbers 15 and 93, that will be its fangs.

15 · 93 = 1395

- You can see that the vampire number has the same digits as its fangs (1, 3, 5 and 9), and that all of its fangs don't end in zero.

- That's true.


- Now look at number 125460. This vampire number can be decomposed into these products:

204 · 615 = 246 · 510 = 125460

- Wow, it has 2 pairs of fangs!

- Yes, and there're still some numbers with more fangs. Thus, let's look at this one, with 3 pairs of fangs:

1620 · 8073 = 1863 · 7020 = 2070 · 6318 = 13078260

- There's even a 70-digit number, that has 100.025 pairs of fangs...

- There're also pseudovampire numbers, whose fangs don't need to have an even number of digits:

8 · 86 = 688

- Or prime vampire numbers, those that can be decomposed into a product of prime fangs, like this one:

167 · 701 = 117067

- So all those numbers on the billboards were vampire numbers, weren't they?

- That's right.

- Now I understand why Juan was so scared.

- Yes. I think you should talk to the Cluj's manager and tell him that we have discovered their ploy for intimidating Juan Mata, and then try to reassure the player.

- Perfect. That's what I'll do. But I don't know if I'll be able to calm Juan. He was very nervous.

- Good. I know that FIFA forbids playing with rings, bracelets, watches and other objects that could injure the opponents. But maybe there's no rule about playing with a garlic collar on the neck...

- I'm going to go over the rules. Thank you very much, Joe.

- It has been a pleasure. And good luck in the match!



If you're interested in learning more about this topic of the Will Rogers phenomenon and about the game theory, you can visit any of this magnific articles: Vampire numbersVampire numbers visualizedVampire numberVampire numbers NumberphileVampire numbers for Dummies,

Below this lines  you will find other links, for if you liked this story and you want to share it with your friends.

And don't forget to take a walk by the Carnival of Mathematics. There you'll find lots of excellent math posts that you'll surely also like.


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