Matifutbol. Maths, logic and football. Matifutbol. Maths, logic and football.


A black (and red) hole.


Silvio Berlusconi, president of the AC MilanSilvio Berlusconi worries about the low number of spectators to San Siro in recent games.

After the elimination in the Champions League, and the inability to win the Scudetto (Italian League), increasingly fewer people go to the stadium to watch the matches of AC Milan.

San Siro Stadium half inningTherefore, Il Cavaliere has thought of a way to convince people to return to the stadium.

So, in the next matches, he will give away half of the gate money to one of the fans attending the match.

The lottery system is as follows:

Each person, when entering the stadium, will choose a number between 0 and 9 (inclusive), with which we'll form a number.

Thus, if the first spectator entering the stadium chooses number 1, next one picks number 4, next chooses number 3..., we will be forming the number:

1 4 3 ...

This number will have as many digits as spectators in the stadium.

AC Milan badgeOnce all the fans have entered, we'll have our number formed, and we will go ahead as follows:

We'll obtain a new number, whose first digits will be the amount of even digits our number has, the following figures will be the quantity of odd digits we can find in it, and finally we'll add the total number of digits (odd + even).

So, if we have number

1 4 3 5 4 8 6 9 2 3 1 2 4 3 0 7 9

The number that we will get is:

8 9 17 (8 even, 9 odd, 17 figures in total)

With this number we'll do again the same method, and thus three more times.

In total, we will proceed 5 times.

San Siro stadium completely crowdedThe prize of half the proceeds will be given to the spectator whose seat number matches with the number resulting from this process. (All San Siro seats are numbered from 1 to 81,277).

With this draw, Berlusconi achieved an attendance success for some matches. Fans were delighted with the opportunity to take home a great prize. But gradually the attendance started to decline again.

What do you think is the reason?

Go to the SECOND HALF to discover the solution

homo mathematicus


It seems that fans will return only when the AC Milan regain its fitness level of a few years ago. Because it's clear that the financial incentive was not enough to recover the attendance to San Siro.

Silvio Berlusconi wins the prizeSpecially when the list of the winners has been made public. In fact, all the awards have been delivered so far to the same spectator: Silvio Berlusconi.

How is this possible?

Let's see what happens once the stadium is filled.

We have a number of 81,277 figures:


























..........................................................and so 40 more times.

There are various possibilities, as you can see:

all the spectators have chosen an odd number: 08127781277

there are less even than odd numbers, such as: 1258115281277

even and odd figures are similar: 406394063881277

there are more even numbers than odd ones, for instance: 71763951481277

everyone has chosen an even figure: 81277081277

Thus we can check that the number we get after the first process has 11 - 15 figures. In case that this number has the highest number of digits possible, that is, 15 digits, and following the same reasoning, after the second process we will get a 4 or 5 digits number:

01515, 11415, 21315, ..., 7815, ..., 15015

We choose the number with the maximum digits as possible, which is 5 digits. We apply again our method and we obtain a 3-digits number:

055, 145, 235, 325, 415, 505

After applying the same procedure a fourth time, we get a 3-digit number, which will be among the following:

033, 123, 213, 303,

In the first case: 033, if we apply the process a fourth time, we get the number 123, (1 even digit, 2 odd digits, 3 digits in total).

In the case of 123, we have the number 123 again (1 even digit, 2 odd digits, 3 digits in total).

In the case of 213, we get again 123 (1 even digit, 2 odd digits, 3 digits in total).

And in the case of 303, we will obtain also 123 (1 even digit, 2 odd digits, 3 digits in total).

So after five applications of the method, and no matter the number from which we have started, we obtain in all cases the number 123, which matches, curiously, with the seat number of Berlusconi.

Silvio Berlusconi sitting in seat number 123

This is a case of 'mathematical black hole'. As in Physics we find black holes from which nothing can escape, not even light, also in Mathematics there is a similar scientific curiosity.

Thus, there are mathematical expressions and sequences of operations that always result in a numeric 'black hole' that attracts the other numbers, no matter the number you start from.

In fact, for each number we can generate a sequence of steps to convert it in a black hole.

In addition, we find black holes in arithmetic and geometric processes, and even in alphanumeric sequences. Here we have dealt with the black hole of 123 (or 213, if you write first the odd figures than the even ones), but there are many other equally curious cases.

For example, we start with any number. We write all its divisors, including 1 and itself, and we add all the digits of these divisors. With the result we go back through the same process, and so till we reach the 'black hole'. We will see that we always get number 15.

We can also choose any number, we write its numeral in words, and count the characters in its spelling. With the number of characters obtained, we will do the same. At the end of this process, we will see that we get to the 'black hole': 4-FOUR-4-FOUR-....

We also find the Collatz problem, which is as follows: we will build a sequence such that an+1 = an/2 if an is even, and an+1 = 3 an+1 if is odd. At the end, we always arrive at a time when the term with value 1 will be repeated forever.

Or the famous Kaprekar constant. We select a 4-digits number, we order its figures from highest to lowest, and then we order them from lowest to highest, we subtract these 2 numbers, and then we repeat this process with the result obtained, the times necessary, until the number repeats. Here, and after 7 iterations at most, we get number 6174, called Kaprekar constant, in honor of its discoverer, the Indian mathematician C.R. Kaprekar.

homo mathematicus

If you are interested in this subject, you can see also:

In English:

Even, odd and total number of digits Cut the Knot

Kaprekar constant Wikipedia

Kaprekar Wikipedia

Mysterious number 6174 - Yutaka Nishiyama)

Kaprekar routine Mathworld - Weisstein, Eric W.

Collatz conjecture Wikipedia

Another black hole number

Black hole number 15 - Sunil Kumar

Mathematical black holes Recreational and educational computing - Dr. Mike Ecker

In Spanish:

Magia y agujeros negros Pedro Alegría

Constante de Kaprekar Wikipedia

Kaprekar Wikipedia

Conjetura de Collatz Wikipedia

El 123. Un agujero numérico - Malena Martín

Una curiosa propiedad del 123

In Catalan:

Constant de Kaprekar

Conjectura de Collatzr

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