Tuesday 28 January 2014

Fibonacci also plays football

An interesting story that mixes football, mathematics, art, communications, sports bettings and nature, all around the Fibonacci sequence.
(This post participes on the xxxth edition of the Carnival de Mathematics, hosted by the blog Cuentos cuánticos.)


Badge of the Athens soccer team Olympiacos F.C.new forward is coming. The board of Olympiacos F.C. (ΠΑΕ Ολυμπιακός) has decided to sign a new centre-forward, in order to strengthen the team for the next round of the Champions League.

The team owner, Evangelos, calls his friend, the editor of Protathlitis newspaper, to give him the scoop:

- Sotiris, we're signing a new striker. He’s a great Italian forward who plays in the Italian Serie B. Hardly anyone knows him, but he’s very good, plus we get him for little money. His name is 'Leo' (Leonardo).

New centre-forward of Olympiacos FC- Can I get ahead of the news?

- No, please. In 30 minutes we’ll make a press conference to announce the new signing, and I don't want many people know the news in advance.

- So I can’t tell anyone? Let me inform my best friends by SMS. I promise you I won't spread it massively.

- Ok. I let you spread it, but with the following conditions: you may not send it massively. You can only send one message at a time. And you must tell this rule to the people you send the message, for them to do the same.

- Right, Evangelos, I’ll do it that way. But you haven’t told me the player's surname yet...

- You'll find out his surname if you look at the rules I've given you for spreading the message.

After hanging up, Sotiris, the Protathlitis daily director, sends the first message to the editor in chief, to be ready for editing the news just when the conference takes place, in these terms:

The message that Sotiris, the newspaper director, has sent to his friends
'A player named Leonardo will be the new Olympiacos FC forward. I don’t know his last name, but apparently it has to do with how this message is send. You can resend this message to your contacts, but only one at a time, please.'

Cover of the newspaper Protathlitis where the new signing is announced
After sending the message to the editor, the director sends a new SMS, this time to a friend, a fan of Olympiacos. And so he continues sending messages for 30 minutes before the conference starts.

Likewise, the editor, after reading the message, resends it to the layout designer, in order to prepare the front page. And within the next minutes, he goes on resending the message to other people, always one at a time.

And that’s what happens to the rest of recipients of the message.

We know that SMS sending is instantaneous, that everyone takes aproximately 1 minute to read the message, and 1 minute more to decide the person to whom will forward it. And we know that nobody has received the message from two different issuers.

So the news spreads

With these data, and 30 minutes later, can you calculate how many people will know the news before the press conference starts? 100 people? 500? 1,000? And, most importantly, can you imagine what’s the second name of the Italian footballer?

Think for a moment the answer, you'll feel better even if you do not guess right


I don’t know how to start with this problem...

So we best start from the beginning :-)

We will call ‘minute 0’ to the moment the owner of Olympiacos calls the newspaper director (person 1 = P1). There's nobody who already knows the news, that is, 0 people know the scoop (excluding the members of the board of the club).

In the first minute, they talk about the new player. So, at the end of the first minute, there's 1 person who knows the news.

During the second minute, the newspaper director decides that the first person who should know the news is the editor in chief (person 2 = P2). He writes the SMS message and sends it to him. So when 2 minutes have passed, there's still only 1 one person who knows the name of the footballer.

Now let's see what happens in the next minute. The director decides to send a new message, this time to his friend, the fan of Olympiacos (P3). Meanwhile, the editor in chief has read the message the director sent him. Now, there are 2 people who know about the signing, and there's one more person that has received the message, but has not read it.

In the fourth minute, the director sends a new message, this time to other friend (P4). The editor, once he has read the message, has resent it to the layout designer (P5). And the fan of Olympiacos has received the message from the director and has already read it. Now there're 3 people aware of the deal, and 2 more to which the message has reached them, but haven’t read it yet.

What's happening during the first four minutes?

In the fifth minute, the director resends the message to another friend (P6). The editor does the same with a friend (P7). The fan of Olympiacos sends his first message, to P8. Meanwhile, P4 and P5 have read the message. Thus, we have 5 people who know the news.

In the sixth minute, that’s what happens: the director, the editor and the fan of Olympiacos send another new message (to P9, P10 and P11), P4 and P5 send their first messages (to P12 and P13), and P6, P7 and P8 read the message. Now there are 8 people in the know of the signingl.

In minute 7, the first five people send a new message. The next 3 make their first delivery. And the remaining five have just read the incoming message. Now there are 13 people who know the subject. And eight more will receive the message shortly.

what's happening during the first seven minutes?

We could go on until the 30th minute, but now we’ve got enough clues to figure out what's happening.

Yeah? Well, I just don’t see what happens... And with this rythm I don't think that many people will know the notice before the press conference starts

Wait a minute. Let's look at the number of people who know the name of the new player at the end of each minute:

0, 1, 1, 2, 3, 5, 8, 13...

I can't understand how these numbers are related to each other...

Try as follows: pick the first 2 numbers, add them up and tell me the result.

0 + 1 = 1

We’ve obtained the third number in the sequence. Now take the second and the third number, and add them up again.

1 + 1 = 2

Are we going to do the same with the third and fourth?

Yes, please.

1 + 2 = 3

I see how it works. Each number is obtained by adding the previous two, right?

That's right. In mathematical terms we’ll define it as follows:

F(n) = F(n-1) + F(n-2); F(0) = 0; F(1) = 1

This way, we’ll generate the following numbers until we reach number 30 in the series, which will be the one that indicates the number of people who knew the news before the press conference started.

Let's write down the numbers of the sequence, that mean the amount of people who know the news, and in brackets we'll write the corresponding minute:

0 (0), 1 (1), 1 (2), 2 (3), 3 (4), 5 (5), 8 (6), 13 (7), 21 (8), 34 (9), 55 (10), 89 (11), 144 (12), 233 (13), 377 (14), 610 (15), 987 (16), 1.597 (17), 2,584 (18), 4,181 (19), 6,765 (20), 10,946 (21), 17,711 (22), 28,657 (23), 46,368 (24), 75,025 (25), 121,393 (26), 196,418 (27), 317,811 (28), 514,229 (29), 832,040 (30)

Nearly a million people, in just 30 minutes, without performing mass messages!

Indeed, this sequence seems that progresses very slowly (in the first 10 minutes only 55 people know the news), but then it has an exponential behaviour.

People who know the story as time progresses

In fact, if we continue with the series, and assuming that all inhabitants of the planet have mobile, and network coverage, the news will be known worldwide in just 49 minutes.

But if there are several people who send the message to the same person, this no longer fulfills...

That’s why, at the beginning of the story, we said that we knew that nobody has received the message from two different issuers. So we avoid that multiple people send the message to the same recipient. This doesn’t happen in real life, so the figures would be somewhat different from those calculated in theory.

In addition, we must consider another detail. Not all the people are interconnected. There may be 'islands' of people who will never receive the message. Imagine, for example, that all the Greek people have solely contacts of Greek people in their phones, and none from another country. The message would not leave Greece! Or think that people living in Oslo are all interconnected, but don’t have contacts of other persons outside the city. The message would spread throughout the world, except by Oslo.

Leonardo Fibonacci, Leonardo Pisano or Leonardo of PisaDespite these drawbacks that can occur, it’s likely that, by the end of the press conference, almost everyone will already know the name of the new striker. However, the people who have guessed the surname of the player will be just a few.

And, what can be his surname?

You see, this sequence we've seen, in which each number is obtained by adding the two previous, is called Fibonacci sequence. Fibonacci was a mathematician of the 12th-13th centuries, significant because he introduced the use of Arabic numerals into Europe, and because he discovered, among other works, this succession when studying a problem about breeding of rabbits. However, this sequence was already noted by some Indian mathematicians some centuries before.

Then, the player is called Leonardo Fibonacci, right?

Actually his name was not Fibonacci. Fi-Bonacci means "son of Bonacci", which was how his father was known, an Italian tradesman. His real name was Leonardo Pisano (Leonardo of Pisa).

And by chance there’s an Italian striker whose name's: Leonardo Pisano. So, we have the full name of the forward!

The full name of the player: Leonardo Pisano

Well, I had never heard about Fibonacci nor his series of numbers.

It's astounding, because it’s a sequence that appears in the most unexpected places, which leads to many mathematicians and laymen to engage in researching its characteristics and applications.

Among its properties, which would fill an entire encyclopedia, there are some really amazing.

For example, the square of each number in the series is equal to the product of the two adjacent numbers, adding or subtracting 1 (the difference alternates positive-negative-positive-negative -...)

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...

1442 = 20,736             89 x 233 = 20,737 = 1442 + 1
2332 = 54,289             144 x 377 = 54,288 = 2332 - 1
3772 = 142,129           233 x 610 = 142,130 = 3772 + 1

Moreover, if we pick 4 successive Fibonacci numbers, we get that the difference of the squares of the two central numbers equals the product of the two extremes 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...

552-342 = 1,869 = 21 x 89
2,5842-1,5972 = 4,126,647 = 987 x 4,181

If instead of 4 consecutive numbers, we take 10 numbers and we add them together, we see that the sum is equal to the seventh number multiplied by 11.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...

89+144+233+377+610+987+1,597+2,584+4,181+6,765 = 17,567 = 1,597 x 11

If you look at the latest figures from the Fibonacci numbers, we can see that every 60 numbers we find the same figures. If you look at the last 2 figures, they repeat every 300 numbers. The last 3 figures are repeated in 1500 numbers. And so on.

If we forget about the first 0, the third number is 2, and we find a multiple of 2 every 3 numbers. The fourth number is 3, and every 4 numbers we get a multiple of 3. The fifth number is 5, and every 5 numbers we find a multiple of 5. The sixth number is 8, and we have a multiple of 8 every 6 numbers. And so on.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711,

The sum of all numbers up to a given one equals the number two positions farther, subtracting 1

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711..

0+1+1+2+3+5+8+13+21+34+55+89 = 232 = 233-1

And if we divide each number by the immediately preceding, we see that, as we move forward in succession, the result of this division is getting closer and closer to the value of the golden ratio, also known as golden section,  golden mean, golden number or Phi (φ).

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...

1/1=1 2/1=2 3/2=1.5 5/3=1.666... 8/5=1.6 13/8=1.625 21/13=1.615... 34/21=1.619 55/34=1.617...
φ = 1.61803398874989484820458683...

This ‘golden ratio’ sounds me a lot. It's widely used in art, right?
Vitruvian Man of Leonardo da Vinci
Yes. Many artists used this ratio in their art works. Thus we have the Vitruvian Man, in which another Leonardo, da Vinci this time, draws a man starting from rectangles formed with the golden ratio. We observe the same on the Mona Lisa, or on the picture of the Last Supper. And other painters such as Michelangelo and Dürer also applied it widely in their works.

So this proportion has been used from the Renaissance...

No. It was known long before. Thus, we find it in the Athenian Parthenon (s. V BC), when analyzing the relationship between the parts, the roof and columns. And the Babylonian and Assyrian civilizations also used it.

Proportions of the Parthenon temple, and the golden ratio

The golden ratio in the design of violins
So, it’s also applied to other various arts apart from of painting.

Yes, in addition to painting and sculpture, we can observe the golden ratio in the formal structure of many works of classical music of Mozart, Beethoven, Schubert, Béla Bartók and Debussy. And even in the arrangement of the shape of violins.

I see that this sequence has multiple mathematical and artistic applications.

And there’re more applications. We can find apps in other areas, truly amazing.

In nature, we can see how this series appears in the spiral arrangement of sunflower seeds. Thus there’re sunflowers containing 21 spirals in one direction and 34 in the other. And others have 55 and 89 spirals, or even 89 and 144, but always about consecutive Fibonacci numbers.

What a strange layout!

Spiral arrangement of sunflower seeds

Well, it turns out that this is the best way on a circular flower to fit as many seeds as possible. We also find this layout in the pinecones with their double set of intersecting spirals: the total of right-turning and left-turning spirals correspond to the Fibonacci sequence: 5 and 8, or 8 and 13. Daisies group their seeds in 21 spirals in one direction and 34 in the other, and usually have 13, 21, 34, 55 or 89 petals.

How the leaves are distributed in the branchesIf you look at the branches or leaves of plants, we see that they are always distributed so that they can receive the maximum sunlight and rainwater as possible. For this aim, its position around the stem or branch is determined by the Fibonacci sequence. And also occurs with roots, in order to cover the maximum possible terrain and therefore food, and so as to interfere as little as possible between them.

So the plant world is full of sequences of Fibonacci...

And also the animal world. For example, in human body, proportions between the distance from shoulder to fingers and the distance from elbow to the fingers, the height of the hip and knee, the joints of the hands and feet, or the height of being human and height of the belly button, among others, are determined by the golden ratio, so intimately linked to the Fibonacci numbers.

The golden ratio in human proportions

In the bees’ world, we find these numbers when we count the number of possible routes a bee can take by the hexagonal cells of a honeycomb.

We also find this pattern in how hares or rabbits multiply (that was the original problem in which Fibonacci stated this series) as well as other animals. And the DNA molecule measures 34 armstrongs long by 21 armstrongs wide!
Spiral shell of the nautilus
There’s only left to find these numbers in the marine world...

Indeed. And for that, nothing better than looking at the shell of the Nautilus.

In it, each full convolution is at a distance from the center 1.618 (φ) times of the previous round. This spiral, called golden spiral (or Dürer spiral), can be roughly drawn by using the numbers of the Fibonacci sequence. And can be found in many animals as in the shells of snails or in the horns of ruminants, and even in the form of some galaxies.

Spiral based on Fibonacci numbers

In stock market, when a certain value has been going up or down for long periods, and changes its trend, the limit of the estimated variation or correction tends to correspond to the inverse of the golden number 1/φ= 61,8%. And Fibonacci numbers are widely used to identify changes in market trends, by setting time periods of 5, 8, 13 and 21 years in the graphic indices or values.

And can we find any application of these numbers in the world of football?

The Fibonacci system on sports bettingNot so much in football itself, but in something closely related to it: sports betting. For them, the Fibonacci system has been determined as one of the safest methods, especially when you bet on favorite..

It works as follows: the player will be making bets whose amount is determined by the numbers of the Fibonacci sequence. If you lose a bet, you should keep betting using the following number and amount of the sequence. And if you win, you must go back 2 numbers in the sequence, and bet that amount.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269...

This is a very good system for betting on favorite, as it’s very difficult to miss more than 5 consecutive times. The main drawback is that the growth rate of potential gains is a little slow, and for having big results you have to invest more time, but instead losses are minimized.

And these Fibonacci numbers have something to do with fractals, right?

Yes, they do. Do you remember the tables we have been drawing to see the evolution of the people who knew the news as the minutes passed? Well, we could have studied the evolution of the spreading of the news in a more graphically way.

We will assign a small circle to each person involved in the communication. We'll leave them blank when they don’t yet known the news because they haven’t yet read them, and we’ll paint them green one minute after receiving the news, because we assume that they have already read them.

We draw a blue arrow between two circles to indicate that a person is sending a message to another. We’ll paint a yellow line to show that the person is reading the message, and a red line to indicate that the person is thinking on the next receiver of the message.

And now look at our new scheme:

Diagram of how the news spread

It’s clear that the layout of the scheme is a fractal: the main 'tree' (root in this case) repeats its form for any person you choose

Fractal formed with the spreading of the message through individual SMS

It's true. Just one more question. To go getting the terms of the Fibonacci sequence, we have been adding the previous two. But would be there a formula to calculate a specific term of the sequence, i.e. the 49th term, without having to make all the previous sums?

Yes, of course. And guess who we find in the formula!

I’ve got no idea.

Our beloved ‘golden number’ o Phi (φ):

This is the formula :

I like this storyƒn = [(φn-(1-φ)n]/√5

And if we apply this formula (attributed to Binet) to number 49, we get that, at minute 49 there will be 7,778,742,049 persons who would know the scoop.

It's clear that secrets are only secrets if you don’t tell them to anyone...

That 's true. In any case, this story is no secret, so you may spread it so widely as you want, even massively ;-) 

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