Saturday, 6 December 2014

A very special greetings card

A greetings card for the new year takes us into the history of the binary numbering system, the steganographic encryption methods, the numerology in Renaissance works and the origins of the sign of subtraction.
Galileo Galilei. Mathematics is the alphabet in which God has written the universe.

(This post participates on the 117th edition of the Carnival de Mathematics, hosted by the blog Plus Magazine)


Luis Suarez is back! The player confirms his return to the Liverpool F.C. at the end of the season.

It's been a frantic evening at the editorial office of the Liverpool Post, since Luis Suarez launched this publication on twitter: 

FeLiZ Año NueVO – haPPy NEW YeAr – BON aNY NOU !

Luis Suarez wishes a Happy New Year 2015.

10 minutes after he has posted the tweet for wishing a happy new year, the newspaper staff met urgently.

- As you can see, on the card there are only 10 players, because Luis has crossed out his photo. This can only mean that he wants to leave his current team.

Detail of FC Barcelona players.

- Also, in the image we can see lots of red devils, and the name given to the Liverpool FC team is: 'the reds'. You must remember that he played for this club during the last 2 seasons, and he felt very happy there.

One player less on May 15.
Number one with a happy face on it.
- On the other hand, we've got the encrypted notation he has written: 1 – 515. If the Spanish League ends this year on 15/5, the message is clear: the F.C. Barcelona will have 1 player less on May 15th.

- That's true! And he has titled his image: 'Happy New Year', that is, he's very happy because this coming year he will return back to his former club. We can also see a happy face in the number 1, which represents the player less at the end of the season, none other than himself.

- In addition, he has chosen a representation of the hell for the background image. I guess that this year, with the disciplinary punishment and the adaptation to the new team, it has been something like a hell for him, and that's why he wants to return to Liverpool, where he was so well.

Probability and sunrise.
- However, I think we should talk to him before publishing the news, in order to confirm everything we're saying.

- We've tried it, but he's gone on holiday to a remote Pacific island  with his friend Laudrup, and we can't get in touch with him, the same way as in the story Probability and sunrise.

- Anyway, and considering that the message is made in a mathematical language, we should ask Joe Vitruvius if he thinks similarly as we do.

The wonderful Joe Vitruvius
- O.K. Have you got his telephone number?

- Yes, now I'm going to ask him for coming here with us.

While Joe arrives to the editorial office, do you think that the interpretation of the message that the journalists of the Liverpool Post have thought is correct? 


Finally Jose Vitruvius arrives to the central office of the newspaper.

- Hello everybody. What happens?

- We've got a doubt with Luis Suarez. Let's have a look at the picture he has published on twitter.

Luis Suarez's greetings card for the new year 2015..

- He tells that on 15/5 the FC Barcelona team will have one less player, who will go to the Liverpool F.C. to play with 'the reds', and he's very happy for this reason, because this season has been a hell for him.

- Hmm... Have you spoken with the player?

- We can't contact with him. That's why we want to ask you if you can find out something else in these numbers, or if you confirm our analysis, before we publish the news

- Let's see, the message just says Happy New Year in three languages: Spanish, English and Catalan. I can't see what's the problem.

Wishing a Happy New Year.

- Yes, but look at the picture: 1-515. One less on 5/15, the day that the Spanish League ends.

- Well, it may mean something different.

Red devils.
- And what can you tell me about the red demons? It's obvious that this is a clear reference to his former club, the Liverpool.

- Yes, they're all red...

- And the photos of his teammates, among which he has crossed out his face, this has no other possible explanation...

- I don't know. I think there's something hidden on this message. Let me think for a moment... As you told me, the background of the image is the hell. That's why devils are all around. But it's not just a simple hell. It's a representation of the hell that Dante describes on his book The Divine Comedy

Botticelli's hell.

- Dante was an Italian poet (1265-1321) whose masterpiece is the Divine Comedy, considered as one of the fundamental works of the universal literature, in which we can study the transition from the Medieval to the Renaissance world.
Dante Alighieri.
- As for number one, I can get no clue from it for the moment, or perhaps I should say I get too many clues: it represents the unity, is a happy, triangular and pentagonal number, appears 2 times in the Fibonacci sequence, is a Bell number, a Catalan number, a factorial and regular number, It's also the neutral or identity element of the product, and is the most common digit on a set of data according to the Benford's Law... Too much data. There's also a face inside it, but I don't know how to interpret it for now. 

- And as regards the sign that separates number 1 from 515, I doubt if it's a minus sign or if it's a simple dash between both numbers.

Medieval sign for representing the subtraction.
- In fact, at the time when Dante wrote the Divine Comedy, nobody used that symbol for representing the subtraction, but the letter M (for the Latin word minus), to which later was added a line written over it to avoid confusions. It was in 1518 when Henricus Grammateus used it for the first time in solitary, eliminating the m, as the symbol of subtraction.

- On the other hand, we've got number 515. You say that it refers to the expression 5/15, that is, the 15th day of the 5th month, the day when the Spanish League ends. But I think there should be a separator between the 5 and the 15. 

On player less on May 15.

- I think we should focus on this number, the 515, because it can give us more clues about what he wants to tell with his message

- As we said, the picture that is on the background of the tweet is a Botticelli's painting, inspired by ‘The Divine Comedy’ of Dante. As a good man from Renaissance, Dante, was a writer, but also a good fan of mathematics. Thus, in his work grants a great symbolic value to certain pairs of numbers, such as 3 and 9, 7 and 22, or 515 and 666. 

3 versus 9

- We find number 3 in many aspects of the book. It's a number that refers to the Holy Trinity, and that means balance and stability. This way, the poem is composed of tercets, articulated in 3 chapters: Hell, Purgatory and Paradise.

- Meanwhile, number 9 is the number Dante assigns to Beatrice. 9 is the square of 3, and that way he enhances the beauty and importance of his beloved. In the work, this figure appears as the number of divisions or circles of the Hell, or the amount of different heavens of Paradise.

7 versus 22

- The number 7 is a sacred number (seven liberal arts, seven colors of the rainbow, seven days of the week), which he uses to name the divisions of the Purgatory, while the number 22 is a number closely related to the 7, since the result of the ratio 22/7 is very close to number pi

- So with these 2 numbers that Dante refers to the circle, the most perfect figure for the Pythagoreans. That way, he defines the heaven and the hell as 2 worlds formed by concentric circles.

515 versus 666

- And finally we arrive to the most interesting numbers. Everybody knows number 666 is considered the number of the Beast or Devil in the Apocalypse. His antagonist or enemy must be the 'Veltro', or the one sent by God, who is marked by number 515 according to the prediction Beatrice makes in the book: "Un cinquecento diece e cinque, messo di Dio..." 
The Duke fighting the Beast.
- If we write 515 en Roman numerals we've got DXV, and if we change the order of the letters we get DVX, the Duke in command of the Holy Roman Empire, who will be chosen to fight the evil represented by the 666. 

- Therefore, number 515 must be taken as if it was a positive force, opposed to the hell of the picture's background. This may mean that Luis is overcoming the bad times of the past.

- So everything seems to be connected, but we still haven't found what Luis Suarez wants to tell with his message.

- The one link I see between this number and the new year 2015, is that we can get this number by adding 15 hundreds to 515, but there's no longer relationship between them beyond the concentration of ones and fives in the operation.

1500 + 515 = 2015
- And if we study the properties of the number 515, we get that it's a composite number, with 4 divisors (1, 5, 103 and 515), so it's not a prime number. It's neither a Fibonacci number, nor a Bell , Catalan or Lucas number. It's not perfect, nor abundant, nor polygonal. But it's deficient, undulated, odious, palindromic, square-free... and unhappy

- As a curiosity, we've got that 515 is the addition of 9 consecutive prime numbers (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73),  that the Moon's orbital plane is inclined at 5.15 degrees to the plane of Earth's orbit round the Sun,.and that if we multiply number pi by 0.515 we get approximately the golden number 'phi' (1,618), that you can see on my t-shirt, and that is so related to Fibonacci sequence.

pi multiplied by 0.515 approximately equals phiThe magnificent Joe Vitruvius.
- And if we analyze the number 2015, the new year, we've got that it's a composite number, with 8 divisors: 1, 5, 13, 31, 65, 155, 403 and 2015. It's a deficient, evil, unhappy and square-free number, but it's not a Fibonacci number, nor Bell, nor Catalan, nor regular, nor perfect, nor polygonal number...

- It seems that we won't get any clue from these two numbers, either.

- On the other hand, if we focus on the face of one of the devils, we can see that it's Francis Bacon, an amazing English philosopher, politician, lawyer and scientist (1561-1626). 

Red devil with Francis Bacon`s face.The philosopher Francis Bacon.

- His work Novum Organum was a starting point for modern science, with the development of the empirical method based on the observation and experimentation.

- Yes, we've already seen it. We find no relationship between the philosopher and Luis Suarez, but there's some connection with other Francis Bacon (1909-1992), the modern painter of the 20th century, whose paintings are exhibited at the museum of Liverpool, and is so admired by Luis.

Francis Bacon's paintings.

- Certainly, we think that it's a clear sign that he wants to return to this city to visit more often Bacon's collections. Also, there's a movie about the life of this author, titled: ‘Love is the devil’. So it's obvious that he still loves his last club, the Liverpool FC.

- What have in common the Liverpool team and the devil?

- Well, the authentic team called 'red devils' is Manchester United. But Liverpool FC is also related to them. Just before playing a match against the Anderlecht team, the coach Bill Shankly had the idea of dressing his players completely in red, to make them look like devils to frighten more their opponents. Thus they changed forever their uniform, and since then the club is known by the name of ‘the reds’. 

Uniform change of Liverpool FC

- I understand. But the thing I can't comprehend is the meaning of the circles on the numbers: 2 filled circles on the fives, and an empty circle on the one...

One less player on May 15.

- Or maybe yes. I think that this is the key to understand what's all this about: Francis Bacon, evil numbers, devils...

- Have you noticed the layout of the players on the top of the card? There are five visible faces, one crossed out face, and another five visible faces, right?

- That's true.

- Just the same layout that he repeats with the figures of number 515 and with the circles set over the numbers 5, 1 and 5. What if we draw 5 full circles, an empty circle, and 5 other full circles, and we compare them with the players' photos?

Correspondence of circles and players.

- What a strange coincidence!

- Well, I said before that Francis Bacon was an outstanding philosopher. But he was also the inventor of the binary encoding system, not in its present notation, with ones and zeros, but with the letters A and B, known by the name of Bacon's Cipher, described in his book The Advancement of Learning.
Francis Bacon's Cypher.
- You surely know that the binary system is a system of numeration in which numbers are written only using two types of figures: ones and zeros. We can say that all our today world, with the boom of the digital technology, consists nowadays of an infinite sequence of ones and zeros.

- And that's what Suarez has made in this message: a binary greetings card. Let's transcribe the number 2015, which corresponds to the new year, to its expression in the binary system

2015 10 = 11111011111 2

- What do you think? Haven't you seen before this distribution of ones and zeros: a row of 5 ones, a zero, and another row of 5 ones again?

- That's right. With the photos of his teammates, and with number 515, Luis is trying to represent the new year 2015.

- In addition, it's a very nice binary number, because it's a palindromic binary number (you can read it the same forwards as backwards), the largest you can write with only 11 digits. The next one will have 12, and will be the 100000000001 (number 2049 in decimal system).

- All right. And does it mean the number one before the 515?

Football and happiness.

- For number one, as everybody who have read my story about the happy numbers knows, is a happy number. In fact, it's the happiest number, because all happy numbers converge at number 1.

- Therefore, 1 – 515 means in mathematical language Happy 2015!

- So, this is only a mathematical greetings card for the New Year, isn't it?

- Exactly. 

- Ok, but how can we be sure that he doesn't wants to go back to Liverpool? Why else he would put so many red demons in the card?
Diablos rojos.
- Well, he might have a good memory of his former team, and he has made it a little homage. But we must look at the text he has written: some letters are written in uppercase, and other in lowercase, without any apparent order, right?

- Yes, it's true, but we have given it no importance.

- Yes it has, a lot. Before I mentioned Bacon's Cipher. With it, the philosopher replaced the characters with sequences of 5 letters, specifically As and Bs, to encrypt confidential documents. Later, Leibniz devised a detailed study on binary systems, using ones and zeros, thus establishing the foundations of the current binary numbering system.

Bacon's cypher.
- For example, if we want to encode the word 'hello' with the Bacon's Cipher, first we replace each letter by its code of 5 letters with As and Bs. Then we think on a phrase, with the same length as the formed series of As and Bs (it will have 5 times more characters than the original text). In our case, I've chosen the phrase of 25 characters (excluding the spaces) 'I like very much country music'.

- Now we form groups of 5 letters, which will match each one with the code A or B of the series. If the first letter of the series is an A, we will write the corresponding first letter of our invented text in capital letters, and if it's a B, then we write it in lowercase.

- We do the same with the remaining letters, in order, and finally we separate again the words of the resulting text, to mislead our 'enemy':

Text cyphered with Bacon's Cypher.

- This way, we've converted the word 'Hello' into the sentence 'I Like VErY MUcH cOUnTrY MusiC'.

- We've set that the uppercase letters correspond to an A, and lowercase letters to a B, but we could also use other methods to differentiate the characters, such as using different fonts of letters, or different colors...

- Now let's apply this steganographic method of Bacon's Cipher to the sentence of Luis Suarez. For that, we separate the letters of the text in groups of five letter, and write an A at the position where there's a capital letter, and a B for lowercase. The exclamation sign will be considered as lowercase, because it's very tiny.

Encoded text with Bacon's Cypher.

- What have we got? Love FCB! I think you have lost your cover page...

- Don't matter. We just have to change the headlines:

New cover of Liverpool Post.
- Many thanks for all, Joe!

- Thank you for thinking of me to solve the problem. And I wish you all a HAPPY 13*5*31 !

I wish you a Happy New Year 2015.

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 like too.

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.)


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?


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.