A galactic prize

The mathematical language is so amazing that, with a few figures, and a pair of operations, we can create formidable numbers.

(This post participates on the 121th edition of the Carnival de Mathematics, hosted by the blog Life through a mathematician's eyes.)

FIRST HALF

The players of Los Angeles Galaxy team are very happy. They've just won the Major League Soccer Cup. 

Once the celebrations for the victory have finished, president Chris Klein has promised them an economical reward for their victory:

- I've come to the changing room to tell you our decision of giving you a gift for your success.

We're going to give you as many cents of dollar as the greatest number you may be able to form with three of your shirts (you can't use those with double figures, nor use the number zero or the infinity symbol, right?).

- For example, if you put togheter the shirts with numbers 5, 2 and 7, we'll give you 527 cents of dollar.

- For everyone?

- Oh, no! Our budget is not so big. It will be 5.27 dollars for all the team. You can share the amount the way you decide.

- Can we repeat numbers?

- No, you can't use three shirts of the same player.

- And can use mathematical symbols?

- Ok, you can, as long as you don't repeat them. And if you use one symbol, you must remove one number.

- That is, if you use, for example, the plus sign, then you must remove one figure, so you can use only two digits: 5 + 9.

- So for every sign we use, we must use one digit less, right?

- Right.

- And when should we tell you the answer?

- Tomorrow morning.  The wishes of the club are that you can enjoy your well deserved reward as soon as possible...

- All right. We see you tomorrow.

The players will meet in the evening, after the training session, to see what amount they will ask the club.

In addition, they've called Joe Vitruvius, who is on holiday by the West coast, to help them with this problem.

SECOND HALF

The sun starts to fall down, and the football players met on the top floor of a skycraper in the city of Los Angeles.

- Well, let's see what we can do.

- I see that, if we're going to use only numbers, it's clear that the highest number is 987. So we'll get 987 cents, that is, 9.87 dollars. A high fortune, to be shared between all of us!

- Yes. And if we use some mathematical symbols, it seems that we'll only lose money.

- We can leave apart the subtraction sign. If we use it, we'll also lose money. The best option, without using the zero, would be 9 - 1 = 8 cents. 

- The sign of addition doesn't seem to be very helpful. From our chosen number, if we replace any of the digits with this sign, we certainly lose a lot of money again. At best, we'll get 9 + 8 = 17 cents, far from the initial 987.

- And with the multiplication symbol (here we'll use a dot) we can't either improve our prize. In this case, among all the possible products, the most beneficial one would be 9·8 = 72 cents. Still far from the 987 cents.

- Obviously, if we introduce the division symbol, we won't achieve anything positive, not to mention the square root...

- Don't think more about it, We'll ask 987 cents, and problem solved!

- Do you agree with us, Joe? You've been very quiet, during our discussions...

- Oh, I haven't said anything because all your arguments were right. All, except the final decision you've taken.

- Why?

- I think you've forgotten a very useful mathematical operator:  the exponentiation. It's a particular case of product in which one of the numbers (the exponent) shows how many times we have to multiply the other number (the base) by itself.

Thus, we have for example 9³, which means 9·9·9 = 729.

All we have to do is to place some numbers at a higher level from the others, and that's all.

Althought it has not always been like this. Babylonians, Greeks (especially Diophantus), Arabs, or medieval mathematicians used different symbols and words to describe the squares, the cubes, and the rest powers of a number.

Michael Stifel, in his ‘Integra Arithmetica’ of 1544 introduced the concept of exponent. James Hume, in 1636, decided to set the exponent on a higher level in relation to the base, although he used Roman numbers to designate them. And finally Descartes, in his work ‘Géométrie’ replaced the uncomfortable Roman numerals for the Hindu-Arabic numbers.

So, thanks to Descartes, this operation will be free for us, because we don't have to use any extra character so we don't have to eliminate one of the figures. Now, our problem is how to place the digits in the most favourable order.

As you can see, I've eliminated some combinations. For example, we know that 89⁷ has to be smaller than 98⁷, so we don't calculate it.

- Joe, I've got a scientific calculator in my backpack.

- Yes, and I downloaded a calculator app for my mobile. We could use it.

- I'm afraid that none of them will be useful for our purpose. Both apps will get blocked with the operations.

In fact, there are lots of internet websites that offer services of calculation for huge numbers, and they wouldn't be able to make these calculations. But there's a powerful one. This is the HyperCalc JavaScript website. We'll use it for getting the results of the various options.

Now we're going for the last row of my table of possible combinations: 9⁸⁷, 9⁷⁸, 8⁹⁷, 8⁷⁹, 7⁹⁸and 7⁸⁹. In this case, each combination of numbers can be calculated by 2 different ways. If we take number 9⁸⁷, we can calculate first 9⁸ and raise the result to 7, or we can calculate first 8⁷ and then take number 9 and raise it to the result we got.

Let's see the results:

- According to this, we get the highest number with the combination 7⁽⁸⁹⁾. So we'll ask the club 7⁸⁹ dollars.

- Great! We can now go home!

- Don't hurry, strangers! There's still something to be done.  

- What do you mean?

- Well, I think that you're fogetting another mathematical symbol.

- Ah, yes? I can't think of anything.

- Well. There's a symbol that will cause admiration to some of you... It's the sign: 

- Is it a mathematical sign, too? And what is it for?

- This symbol represents the factorial function of a number. The factorials consist of the product of all positive integers from 1 to the related number.

That way, the factorial of 5, written 5!, will be the product of the first five integer numbers: 

5! = 1·2·3·4·5 = 120

And the factorial of 9 will be:

9! = 1·2·3·4·5·6·7·8·9 = 362,880

- We know when to use the signs of addition or multiplication. But, when can we use these factorials?

- They've been used for a long time, especially in Combinatorics and in Mathematical Analysis. Hindu Mathematicians used them yet in the twelfth century. And they're used, for instance, to calculate the number of different ways of arranging some distinct objects.

For example, if we want to know how many differents forms you can enter the pitch (the 11 players going one by one, not by pairs, and assuming that the starting lineup is always the same), we get a total of:

11! = 1·2·3·4·5·6·7·8·9·10·11 = 39,916,800 different ways of entering the pitch.

I mean that, if you play a match every 2 hours, 24 hours a day without resting, you could be over 4,556 years playing football without repeating the entry order to the field.

- Blow it!

- And why are factorials represented by the exclamation mark?

- Well, the first person who used this symbol was Christian Kramp in 1808. He was a Math teacher at Strasburg, who made a in-depth study on factorials. Initially he used a symbol in the form of a right angle, placed on the lower right corner of the number.

But he had problems with the printer's, so for the edition of his book Elements d'arithmétique universelle, he decided to use the exclamation mark just after the number.

What I ignore is why he chose this sign, used to express happiness. The symbol comes from the Latin word iocundum, which means happy, pleasant, nice. First it was shortened to the expression (io), that became an expression of happiness, and then the capital 'I 'was placed over the small 'o', and that's how this symbol was born, as a sign of exclamation.

Maybe the relationship between the symbol and the factorial function is the wonder caused in people when calculating the result of the operation. Just look at what happens when we change one of our digits, i.e. number 7, by an exclamation mark:

98! = 9.4268 · 10¹⁵³

That is, a 154-digits number!

- Yes, it's a huge number. But I remember that with potences we got a higher amount:

7⁸⁹ = 7,3785 · 10^113,427,138    >    98! = 9,4268 · 10¹⁵³

- Then it seems quite clear: we get 7⁸⁹ cents. Or do you have one more surprise for us, Joe?

- Yes, we can still mix both operations to try to get the highest number.

If we want to observe the imposed conditions, it seems that there are six possible combinations. But it's very difficult to choose one of them.

I don't know what's better: to put the highest digit as the base or as the coeficient:

2³ (8) < 3² (9)     but     3⁴ (81) > 4³ (64)

And I don't know where to place the factorial symbol, after the base or after the exponent:

2!³ (8) < 2^3! (64)     but     3!² (36) > 3^2! (9)

We'll have to use again the HyperCalc JavaScript website and get the different results:

- I think that we finally will choose the option of 8⁹! better than 7⁸⁹ 

8⁹! = 1,7828 · 10^{1.032.606.169} > 7⁸⁹ = 7,3785 · 10^113.427.138

- It seems a great amount of cents, but I'm sure it won't be a problem for Los Angeles Galaxy, right?

- I'm afraid this is going to be a problem. Are you aware of what this number means?

- Hmmm…

- It's a number of over 1,032 billion digits.

- It's not very big, isn't it?

- Do you know how many atoms are there in the universe?  About 10⁸⁷, that is, a number 1 followed by 88 zeros.

- Well, I can't see much difference between both numbers. Just a few more digits in the exponent.

- With scientific notations it's quite difficult to be conscious of the magnitude and the proportion of numbers. To get started on this subject, there's a great website that explains it very well: The scale of the universe.

Look: the volume of an electron is estimated to be 9.36·10⁻⁴⁴ m³, and the volume of the universe is about 7·10⁸¹ m³. Well, if we fill the whole universe with electrons, the amount of these would be of 7.4786 · 10¹²⁴. This will be as long as we leave no space between them, and we know this is not possible because of the laws of sphere packing. 

Now let's imagine that all the universe, full of electrons, was compressed to the size of an electron, and we fill up again another universe like ours with those little universes full of electrons.

And imagine we repeat this process 8 million times. The final universe, full of small universes, full of small universes, full of small universes….(8 million times)… full of small universes full of electrons would contain as many electrons as the cents the club should give us.

- That way, it does seem a very high amount. Perhaps we'd better choose another combination of numbers to get an amount that the club can afford…

- All the money in circulation worldwide is estimated to be about 80 trillion dollars (80·10¹² $). So there you have the upper limit of your request.

And you also know which is the lower limit: when I was entering in the skyscraper, I saw lots of young fans at the main entrance because they knew you've met here, and have begged me to ask if you can sign some autographs for them.

- Yes, perhaps we'd better forget those big numbers, because we already have got enough reward with the smile of these supporters. I think that our satisfaction and the children's can be also calculated with a potence, don't you think so, Joe?

- Let's go down immediately!

If you're interested in learning more about this topic, you can visit any of this wonderful articles: The History of Mathematical Symbols, History of Mathematical Notation, The origins and development of mathematical notation, The History of Exponents.

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

Don't forget to take a walk by the 121th Carnival of Mathematics. There you'll find lots of excellent math posts that you'll surely like too.

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.

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

FIRST HALF

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 !

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.

- 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.

- 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.

- 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.

- 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? 

SECOND HALF

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.

- 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.

- 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.

- 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. 

- 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.

- 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.

- 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. 

- 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. 

- 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.

- 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.

- 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..." 

- 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.

- 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.

- 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). 

- 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.

- 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’. 

- 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...

- 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?

- 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.

- 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?

- 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?

- 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.

- 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':

- 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.

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

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

- 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 !

If you're interested in learning more about the topics of this story, you can visit any of these wonderful articles: A symmetric number in Dante, Elgar's 515 cyphers, Shakespeare's secrets: a hidden cipher in literature's greatest works?, Bacon Cipher - Decoder, encoder, decrypt, encrypt, Binary system, Bacon's cipher encoder/decoder, How to make anything signify anything.

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.