FIRST HALF

A

**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)**.- 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:

*'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.'*
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.
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?
SECOND HALF

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,

In the

During the

**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.
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.
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.
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.
Despite 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

And, what can be his surname?**have guessed the surname**of the player will be just a**few**.
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!

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...

144

^{2}= 20,736 89 x 233 = 20,737 = 144

^{2}

**+ 1**

233

^{2}= 54,289 144 x 377 = 54,288 = 233

^{2}

**- 1**

377

^{2}= 142,129 233 x 610 = 142,130 = 377

^{2}

**+ 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**
C

^{2}-B^{2}=AxD.0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...

55

^{2}-34

^{2}= 1,869 = 21 x 89

2,584

^{2}-1,597

^{2}= 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?

Yes. Many artists used this ratio in their art works. Thus we have the 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

**Vitruvian Man**, in which another**Leonardo**,**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.
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!

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.
If 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.
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!
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.
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?**

Not 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:

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
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 :

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**;-)
And if you're interested in learning more about Fibonacci, you can visit any of these wonderful links: Nature by numbers: the theory behind this movie, The Fibonacci sequence in nature, The Fibonacci numbers and golden section in nature, Nature, the golden ratio, and Fibonacci too..., Fibonacci in nature, Soccer betting system Fibonacci strategy, Fibonacci Numbers in Nature.