Magic square for 4 out of 20. How the magic square works

This riddle quickly spread throughout the Internet. Thousands of people began to wonder how the magic square works. Today you will finally find the answer!

The mystery of the magic square

In fact, this riddle is quite simple and made with human inattention in mind. Let's see how the magic black square works using a real example:

Let's guess any number from 10 to 19. Now let's subtract its constituent digits from this number. For example, let’s take 11. Subtract one from 11 and then another one. The result is 9. It doesn't really matter which number from 10 to 19 you take. The result of the calculations will always be 9. The number 9 in the “Magic Square” corresponds to the first number with pictures. If you look closely, you can see that it is very a large number numbers are assigned to the same drawings.
What happens if you take a number in the range from 20 to 29? Maybe you already guessed it yourself? Right! The result of the calculation will always be 18. The number 18 corresponds to the second position on the diagonal with pictures.
If you take a number from 30 to 39, then, as you can already guess, the number 27 will come out. The number 27 also corresponds to the number on the diagonal of the so inexplicable “Magic Square”.
A similar algorithm remains true for any numbers from 40 to 49, from 50 to 59, and so on.

That is, it turns out that it doesn’t matter what number you guessed - “Magic Square” will guess the result, because in the cells numbered 9, 18, 27, 36, 45, 54, 63, 72 and 81 there is actually the same symbol .

In fact, this mystery can be easily explained using a simple equation:

Imagine any two-digit number. Regardless of the number, it can be represented as x*10+y. Tens act as “x”, and units act as “y”.
Subtract the numbers that make it up from the hidden number. Add the equation: (x*10+y)-(x+y)=9*x.
The number that comes out as a result of the calculations must point to a specific symbol in the table.

It doesn’t matter what number is in the role of “x”, one way or another you will get a symbol whose number will be a multiple of nine. In order to make sure that there is one symbol under different numbers, just look at the table and at the numbers 0,9,18,27,45,54,63,72,81 and subsequent ones.

MAGIC SQUARE

China is considered the birthplace of magic squares. In China, there is the teaching of Feng Shui, which states that the color, shape and physical placement of each element in space affects the flow of Qi, either slowing it down, redirecting it or speeding it up, which directly affects the energy levels of the inhabitants. To learn the secrets of the world, the gods sent Emperor Yu the most ancient symbol, the Lo Shu square (Lo - river).

MAGIC SQUARE LO SHU

Legend has it that about four thousand years ago, a large turtle, Shu, emerged from the stormy waters of the Luo River. People making sacrifices to the river saw the turtle and immediately recognized it as a deity. The considerations of the ancient sages seemed so reasonable to Emperor Yu that he ordered the image of a turtle to be immortalized on paper and sealed it with his imperial seal. Otherwise, how would we have known about this event?

This turtle was actually special because it had a strange pattern of dots on its shell. The dots were marked in an orderly manner, which led ancient philosophers to the idea that the square with numbers on the turtle’s shell serves as a model of space - a map of the world compiled by the mythical founder of Chinese civilization, Huang Di. In fact, the sum of the numbers in the columns, rows, and both diagonals of the square is the same M = 15 and is equal to the number of days in each of the 24 cycles of the Chinese solar year.

Even and odd numbers alternate: 4 even numbers (written from bottom to top in descending order) are in the four corners, and 5 odd numbers (written from bottom to top in ascending order) form a cross in the center of the square. The five elements of the cross reflect earth, fire, metal, water and forest. The sum of any two numbers separated by a center is equal to the number Ho Ti, i.e. ten.

Even numbers (Earth symbols) of Lo Shu were marked on the turtle's body in the form of black dots, or Yin symbols, and odd numbers (Heaven symbols) - in the form of white dots, or Yang symbols. Earth 1 (or water) is below, fire 9 (or sky) is above. It is possible that the modern image of the number 5, placed in the center of the composition, is due to the Chinese symbol of the duality of Yang and Yin.

MAGIC SQUARE FROM KHAJURAHO

East room

The magic of Joseph Rudyard Kipling, who created the images of Mowgli, Bagheera, Baloo, Shere Khan and, of course, Tabaka, began on the eve of the twentieth century. Half a century earlier, in February 1838, a young British officer of the Bengal Engineers, T.S. Bert, interested in the conversation of the servants carrying his palanquin, deviated from the route and stumbled upon ancient temples in the jungles of India.

On the steps of the Vishvanatha temple, the officer found an inscription testifying to the antiquity of the structures. Later a short time the energetic Major General A. Cunningham drew detailed plans for Khajuraho. Excavations began, culminating in the sensational discovery of 22 temples. Temples were erected by the Maharajas of their Chandel dynasty. After the collapse of their kingdom, the jungle swallowed up the buildings for a thousand years. The square of the fourth order, found among the images of naked gods and goddesses, was amazing.

Not only did this square’s sums along the rows, columns and diagonals coincide and equal 34. They also coincided along the broken diagonals formed when the square is folded into a torus, and in both directions. For such witchcraft of numbers, such squares are called “devilish” (or “pandiagonal”, or “nasik”).

Of course, this testified to the unusual mathematical abilities of their creators, who were superior to the colonialists. What the people in the white pith helmets inevitably felt.

DURER'S MAGIC SQUARE

The famous German artist of the early 16th century, Albrecht Durer, created the first 4x4 magic square in European art. The sum of the numbers in any row, column, diagonal, and also, surprisingly, in each quarter (even in the central square) and even the sum of the corner numbers is 34. The two middle numbers in the bottom row indicate the date of creation of the painting (1514). Corrections have been made in the middle squares of the first column - the numbers are deformed.

In the picture with the occult winged mouse Saturn, the magic square is composed of the winged intelligence Jupiter, which oppose each other. The square is symmetrical, since the sum of any two numbers included in it, located symmetrically relative to its center, is equal to 17. If you add up the four numbers obtained by the move of the chess knight, you will get 34. Truly, this square, with its impeccable orderliness, reflects the melancholy that has gripped the artist.

Morning dream.

Europeans were introduced to amazing number squares by the Byzantine writer and linguist Moschopoulos. His work was a special essay on this topic and contained examples of the author's magic squares.

SYSTEMATIZATION OF MAGIC SQUARES

In the middle of the 16th century. In Europe, works appeared in which magic squares appeared as objects of mathematical research. This was followed by many other works, in particular by such famous mathematicians, the founders modern science, like Stiefel, Baschet, Pascal, Fermat, Bessy, Euler, Gauss.

Magical, or a magic square, is a square table filled with n 2 numbers in such a way that the sum of the numbers in each row, each column and on both diagonals is the same. The definition is conditional, since the ancients also attached meaning, for example, to color.

Normal called a magic square filled with integers from 1 to n 2. Normal magic squares exist for all orders except n = 2, although the case n = 1 is trivial - the square consists of a single number.

The sum of the numbers in each row, column and diagonal is called magic constant M. The magic constant of a normal magic square depends only on n and is given by the formula

M = n (n 2 + 1) /2

The first values of the magic constants are given in the table

If the sum of numbers in a square is equal only in rows and columns, then it is called semi-magical. The magic square is called associative or symmetrical, if the sum of any two numbers located symmetrically about the center of the square is equal to n 2 + 1.

There is only one normal square of third order. Many people knew him. The arrangement of numbers in the Lo Shu square is similar to the symbolic designations of spirits in Kabbalah and the signs of Indian astrology.

Also known as Saturn square. Some secret societies in the Middle Ages saw it as the "Kabbalah of the Nine Chambers." Undoubtedly, the shade of forbidden magic meant a lot for the preservation of his images.

It was important in medieval numerology, often used as an amulet or divination aid. Each cell corresponds to a mystical letter or other symbol. Read together along a specific line, these signs conveyed occult messages. The numbers making up the date of birth were placed in the cells of the square and then deciphered depending on the meaning and location of the numbers.

Among pandiagonal, as they are also called, devilish magic squares, symmetrical ones are distinguished - ideal ones. The devilish square remains devilish if you rotate it, reflect it, rearrange the row from top to bottom and vice versa, cross out a column on the right or left and assign it to the opposite side. There are five transformations in total, the diagram of the latter is shown in the figure

There are 48 4x4 devilish squares with rotation and reflection precision. If we also take into account symmetry with respect to toric parallel translations, then only three essentially different 4x4 devilish squares remain:

Claude F. Bragdon, a famous American architect, discovered that by connecting one by one the cells with only even or only odd numbers of magic squares on a broken line, in most cases we get an elegant pattern. The design he invented for the ventilation grille in the ceiling of the Chamber of Commerce in Rochester, New York, where he lived, was built from the magical broken line of the Lo-Shu talisman. Bragdon used "magic lines" as designs for fabrics, book covers, architectural decorations and decorative headpieces.

If you lay out a mosaic of identical devilish squares (each square must be closely adjacent to its neighbors), you will get something like a parquet, in which the numbers in any group of 4x4 cells will form a devilish square. The numbers in four cells, following one after another, no matter how they are located - vertically, horizontally or diagonally - always add up to the constant of the square. Modern mathematicians call such squares “perfect”.

LATIN SQUARE

A Latin square is a type of irregular mathematical square filled with n different symbols in such a way that all n symbols appear in each row and each column (each once).

Latin squares exist for any n. Any Latin square is a multiplication table (Cayley table) of a quasigroup. The name "Latin square" comes from Leonhard Euler, who used Latin letters instead of numbers in a table.

Two Latin squares are called orthogonal, if all ordered pairs of symbols (a,b) are different, where a is a symbol in some cell of the first Latin square, and b is a symbol in the same cell of the second Latin square.

Orthogonal Latin squares exist for any order except 2 and 6. For n being a power of a prime number, there is a set of n–1 pairwise orthogonal Latin squares. If in each diagonal of a Latin square all elements are different, such a Latin square is called diagonal. Pairs of orthogonal diagonal Latin squares exist for all orders except 2, 3, and 6. The Latin square is often encountered in scheduling problems because numbers are not repeated in rows and columns.

A square made up of pairs of elements of two orthogonal Latin squares is called Greco-Latin square. Such squares are often used to construct magic squares and in complex scheduling problems.

While studying Greco-Latin squares, Euler proved that squares of the second order do not exist, but squares of 3, 4, and 5 orders were found. He did not find a single square of order 6. He hypothesized that there are no squares of even order that are not divisible by 4 (that is, 6, 10, 14, etc.). In 1901, Gaston Terry confirmed the hypothesis for the 6th order by brute force. But in 1959, the hypothesis was refuted by E. T. Parker, R. C. Bowes and S. S. Shrickherd, who discovered a Graeco-Latin square of order 10.

POLYMINO ARTHUR CLARKE

Polyominoes - in terms of complexity, they certainly belong to the category of the most difficult mathematical squares. This is how science fiction writer A. Clark writes about him - below is an excerpt from the book “Earthly Empire”. It is obvious that Clark, living on his island, he lived in Ceylon - and his philosophy of separation from society is interesting in itself, became interested in the entertainment that the boy’s grandmother teaches, and passed it on to us. Let us prefer this living description to the existing systematizations, which convey, perhaps, the essence, but not the spirit of the game.

“You’re a big enough boy now, Duncan, and you’ll be able to understand this game... however, it’s much more than a game.” Contrary to his grandmother's words, Duncan was not impressed by the game. Well, what can you make from five white plastic squares?

“First of all,” continued the grandmother, “you need to check how many different patterns you can put together from squares.”

– Should they lie on the table? – asked Duncan.

– Yes, they should lie touching. You cannot overlap one square with another.

Duncan began to lay out the squares.

“Well, I can put them all in a straight line,” he began. “Like this... And then I can rearrange two pieces and get the letter L... And if I grab the other edge, I get the letter U...”

The boy quickly made up half a dozen combinations, then more and suddenly discovered that they were repeating existing ones.

- Maybe I'm stupid, but that's all.

Duncan missed the simplest of figures - a cross, to create which it was enough to lay out four squares on the sides of the fifth, central one.

“Most people start with the cross,” the grandmother smiled. “In my opinion, you were too hasty in declaring yourself stupid.” Better think: could there be any other figures?

Concentratedly moving the squares, Duncan found three more figures, and then stopped searching.

“It’s definitely over now,” he said confidently.

– What can you say about such a figure?

Having slightly moved the squares, the grandmother folded them into the shape of a humpbacked letter F.

- And here's another one.

Duncan felt like a complete idiot, and his grandmother’s words were like a balm on his embarrassed soul:

– You’re just great. Just think, I missed only two pieces. And the total number of figures is twelve. No more and no less. Now you know them all. If you search for an eternity, you will never find another one.

Grandma swept five white squares into a corner and laid out a dozen bright, multi-colored plastic pieces on the table. These were the same twelve figures, but in finished form, and each consisted of five squares. Duncan was already ready to agree that no other figures really existed.

But since grandma laid out these multi-colored stripes, it means the game continues, and another surprise awaited Duncan.

– Now, Duncan, listen carefully. These figures are called "pentaminoes". The name comes from the Greek word "penta", which means "five". All figures are equal in area, since each consists of five identical squares. There are twelve figures, five squares, therefore, total area will be equal to sixty squares. Right?

- Hmm yeah.

- Listen further. Sixty is a wonderful round number that can be composed in several ways. The easiest one is to multiply ten by six. This box has such an area: it can hold ten squares horizontally, and six vertically. Therefore, all twelve figures should fit in it. Simple, like a composite picture-riddle.

Duncan expected a catch. Grandma loved verbal and mathematical paradoxes, and not all of them were understandable to her ten-year-old victim. But this time there were no paradoxes. The bottom of the box was lined with sixty squares, which means... Stop! The area is an area, but the figures have different shapes. Try to get them into a box!

“I’ll leave this task for you to solve on your own,” announced the grandmother, seeing how he sadly moved the pentomino along the bottom of the box. “Believe me, they can be assembled.”

Soon Duncan began to strongly doubt his grandmother’s words. He easily managed to fit ten figures into the box, and once he managed to squeeze in an eleventh. But the outlines of the unfilled space did not coincide with the outlines of the twelfth figure, which the boy was turning over in his hands. There was a cross, and the remaining figure resembled the letter Z...

After another half hour, Duncan was already on the verge of despair. Grandma was immersed in a dialogue with her computer, but from time to time she looked at it with interest, as if to say: “This is not as easy as you thought.”

At ten years old, Duncan was noticeably stubborn. Most of his peers would have given up trying a long time ago. (Only several years later did he realize that his grandmother had gracefully administered a psychological test to him.) Duncan lasted almost forty minutes without assistance...

Then the grandmother got up from the computer and bent over the puzzle. Her fingers moved the shapes U, X and L...

The bottom of the box was completely filled! All the pieces of the puzzle were in the right places.

– Of course, you knew the answer in advance! – Duncan drawled offendedly.

- Answer? – asked the grandmother. “How many ways do you think the pentomino can be placed in this box?”

Here it is, a trap. Duncan fiddled around for almost an hour without finding a solution, although during this time he tried at least a hundred options. He thought there was only one way. Could there be... twelve of them? Or more?

- So how many ways do you think there could be? – Grandma asked again.

“Twenty,” Duncan blurted out, thinking that now grandma wouldn’t mind.

- Try again.

Duncan sensed danger. The fun turned out to be much more cunning than he thought, and the boy wisely decided not to risk it.

“Actually, I don’t know,” he said, shaking his head.

“And you are a receptive boy,” the grandmother smiled again. “Intuition is a dangerous guide, but sometimes we have no other.” I can please you: it is impossible to guess the correct answer here. There are over two thousand different ways to fit pentominoes into this box. More precisely, two thousand three hundred and thirty-nine. And what do you say to this?

It is unlikely that his grandmother was deceiving him. But Duncan was so frustrated by his inability to find a solution that he couldn’t help but blurt out:

- I do not believe!

Helen rarely showed irritation. When Duncan offended her in some way, she simply became cold and distant. However, now the grandmother just grinned and tapped something on the computer keyboard.

“Look here,” she suggested.

A set of twelve multi-colored pentominoes appeared on the screen, filling a ten-by-six rectangle. A few seconds later it was replaced by another image, where the figures were most likely located differently (Duncan could not say for sure, since he did not remember the first combination). Soon the image changed again, then again and again... This continued until the grandmother stopped the program.

“Even at high speed, the computer will need five hours to go through all the methods,” explained the grandmother. “You can take my word for it: they are all different.” If it were not for computers, I doubt that people would have found all the ways through the usual enumeration of options.

Duncan stared at the twelve deceptively simple figures for a long time. He slowly digested his grandmother's words. This was the first mathematical revelation in his life. What he so rashly considered an ordinary child's game suddenly began to unfold before him endless paths and horizons, although even the most gifted ten-year-old child would hardly be able to sense the boundlessness of this universe.

But then Duncan's delight and awe were passive. The real explosion of intellectual pleasure happened later, when he independently found his first method of laying pentominoes. For several weeks, Duncan carried a plastic box with him everywhere. He spent all his free time only on pentominoes. The figures will turn into Duncan's personal friends. He called them by the letters that they resembled, although in some cases the similarity was more than distant. Five figures - F, I, L, P, N - were inconsistent, but the remaining seven repeated the sequence of the Latin alphabet: T, U, V, W, X, Y, Z.

One day, in a state of either geometric trance or geometric ecstasy, which was never repeated, Duncan found five styling options in less than an hour. Perhaps not even Newton, Einstein or Chen Tzu, in their moments of truth, felt more closely related to the gods of mathematics than Duncan Mackenzie.

He soon realized, on his own, without his grandmother’s prompting, that a pentomino could be placed in a rectangle with different side sizes. Quite easily, Duncan found several options for rectangles 5 by 12 and 4 by 15. Then he suffered for a whole week trying to fit twelve figures into a longer and narrower rectangle 3 by 20. Again and again he began to fill the treacherous space and ... get holes in the rectangle and “extra” figures.

Devastated, Duncan visited his grandmother, where a new surprise awaited him.

“I’m glad for your experiments,” said Helen. “You explored all the possibilities, trying to derive a general pattern.” This is what mathematicians always do. But you're wrong: solutions for a three-by-twenty rectangle do exist. There are only two of them, and if you find one, you will be able to find the second.

Inspired by his grandmother's praise, Duncan continued his “hunt for pentominoes” with renewed vigor. After another week, he began to understand what an unbearable burden he had placed on his shoulders. The number of ways in which twelve figures could be arranged was simply mind-boggling to Duncan. Moreover, each figure had four positions!

And again he came to his grandmother, telling her all his difficulties. If there were only two options for a 3 by 20 rectangle, how long would it take to find them?

“If you please, I’ll answer you,” said the grandmother. “If you acted like a brainless computer, doing a simple search of combinations and spending one second on each, you would need...” Here she deliberately paused. “You would need more than six million ... yes, more than six million years.

Earthly or titanic? This question instantly appeared in Duncan's mind. But what's the difference?

“But you are different from a brainless computer,” the grandmother continued. “You immediately see obviously unsuitable combinations, and therefore you don’t have to waste time checking them.” Try again.

Duncan obeyed, already without enthusiasm and faith in success. And then a brilliant idea came to his mind.

Karl was immediately interested in pentominoes and accepted the challenge. He took the box with the figures from Duncan and disappeared for several hours.

When Karl called him, his friend looked somewhat upset.

– Are you sure that this problem really has a solution? - he asked.

- Absolutely sure. There are two of them. Have you really not found at least one? I thought you were great at math.

“Imagine, I can figure it out, that’s why I know how much work your task requires.” We need to check... a million billion possible combinations.

– How did you know that there are so many of them? – Duncan asked, pleased that at least he managed to make his friend scratch his head in confusion.

Karl glanced sideways at a piece of paper filled with some diagrams and numbers.

– If you exclude unacceptable combinations and take into account symmetry and the possibility of rotation... you get a factorial... the total number of permutations... you still won’t understand. I'd better show you the number itself.

He brought another sheet of paper to the camera, on which an impressive string of numbers was large-scaled:

1 004 539 160 000 000.

Duncan knew nothing about factorials, but he had no doubt about the accuracy of Karl’s calculations. He really liked the long number.

“So are you going to give up this task?” – Duncan asked carefully.

- What more! I just wanted to show you how difficult it is.

Karl's face expressed grim determination. Having said these words, he passed out.

The next day, Duncan experienced one of the greatest shocks of his boyhood life. Karl’s haggard face, with bloodshot eyes, looked at him from the screen. It was felt that he had spent a sleepless night.

“Well, that’s all,” he announced in a tired but triumphant voice.

Duncan could hardly believe his eyes. It seemed to him that the chances of success were negligible. He even convinced himself of this. And suddenly... In front of him lay a three by twenty rectangle, filled with all twelve pentomino figures.

Then Karl swapped and turned the pieces at the ends, leaving the central part untouched. His fingers trembled slightly from fatigue.

“This is the second solution,” he explained. “And now I’m going to bed.” So good night or good morning - whatever you like.

The humiliated Duncan looked at the darkened screen for a long time. He did not know which way Karl moved, groping for a solution to the puzzle. But he knew that his friend had emerged victorious. Against all odds.

He did not envy his friend's victory. Duncan loved Karl too much and always rejoiced at his successes, although he himself often found himself on the losing side. But there was something different about my friend’s triumph today, something almost magical.

Duncan saw for the first time the power of intuition. He encountered the mysterious ability of the mind to break beyond the facts and throw aside interfering logic. In a matter of hours, Karl completed a colossal job, surpassing the fastest computer.

Subsequently, Duncan learned that all people have such abilities, but they use them extremely rarely - perhaps once in their lives. In Karl, this gift received exceptional development... From that moment on, Duncan began to take his friend’s reasoning seriously, even the most ridiculous and outrageous from the point of view of common sense.

This was twenty years ago. Duncan didn't remember where the plastic pentomino pieces had gone. Perhaps they remained with Karl.

Grandmother’s gift became their new incarnation, now in the form of pieces of multi-colored stone. The amazing, soft pink granite was from the Galileo hills, the obsidian was from the Huygens Plateau, and the pseudo-marble was from the Herschel ridge. And among them... at first Duncan thought he was mistaken. No, that’s how it is: it was the rarest and most mysterious mineral of Titan. My grandmother made the stone pentomino cross from titanite. This blue-black mineral with golden inclusions cannot be confused with anything. Duncan had never seen such large pieces before and could only guess what its cost was.

“I don’t know what to say,” he muttered. “What a beauty.” This is the first time I've seen this.

He hugged his grandmother’s thin shoulders and suddenly felt that they were trembling and she couldn’t stop the trembling. Duncan held her gently in his arms until her shoulders stopped shaking. At such moments, words are not needed. More clearly than before, Duncan understood: he was the last love in the devastated life of Helen Mackenzie. And now he flies away, leaving her alone with her memories.

LARGE MAGIC SQUARE

The 13th century Chinese mathematician Yang Hui was familiar with Pascal's triangle (arithmetic triangle). He left a description of methods for solving equations of the 4th and higher degrees; there are rules for solving a complete quadratic equation, summing progressions, and methods for constructing magic squares. He managed to construct a magic square of the sixth order, and the latter turned out to be almost associative (in it only two pairs of centrally opposite numbers do not give the sum of 37).

Benjamin Franklin constructed a 16x16 square, which, in addition to having a constant sum of 2056 in all rows, columns and diagonals, had one more additional property. If we cut a 4x4 square from a sheet of paper and place this sheet on a large square so that 16 cells of the larger square fall into this slot, then the sum of the numbers that appear in this slot, no matter where we put it, will be the same - 2056.

The most valuable thing about this square is that it is quite easy to transform it into a perfect magic square, while constructing perfect magic squares is not an easy task. Franklin called this square "the most charming magic of all the magic squares ever created by sorcerers."

MAGIC SQUARE, a square table of integers in which the sums of the numbers along any row, any column, and any of the two main diagonals equal the same number.

The magic square is of ancient Chinese origin. According to legend, during the reign of Emperor Yu (c. 2200 BC), a sacred turtle surfaced from the waters of the Yellow River (Yellow River), with mysterious hieroglyphs inscribed on its shell (Fig. 1, A), and these signs are known as lo-shu and are equivalent to the magic square shown in Fig. 1, b. In the 11th century They learned about magic squares in India, and then in Japan, where in the 16th century. Extensive literature has been devoted to magic squares. Europeans were introduced to magic squares in the 15th century. Byzantine writer E. Moschopoulos. The first square invented by a European is considered to be the square of A. Durer (Fig. 2), depicted in his famous engraving Melancholy 1. The date of creation of the engraving (1514) is indicated by the numbers in the two central cells of the bottom line. Various mystical properties were attributed to magic squares. In the 16th century Cornelius Heinrich Agrippa constructed squares of the 3rd, 4th, 5th, 6th, 7th, 8th and 9th orders, which were associated with the astrology of the 7 planets. It was believed that a magic square engraved on silver protected against the plague. Even today, among the attributes of European soothsayers you can see magic squares.

In the 19th and 20th centuries. interest in magic squares flared up with renewed vigor. They began to be studied using the methods of higher algebra and operational calculus.

Each element of a magic square is called a cell. A square whose side consists of n cells, contains n 2 cells and is called a square n-th order. Most magic squares use the first n consecutive natural numbers. Sum S numbers in each row, each column and on any diagonal is called the square constant and is equal to S = n(n 2 + 1)/2. It has been proven that nі 3. For a square of 3rd order S= 15, 4th order – S= 34, 5th order – S = 65.

The two diagonals passing through the center of the square are called the main diagonals. A broken line is a diagonal that, having reached the edge of the square, continues parallel to the first segment from the opposite edge (such a diagonal is formed by the shaded cells in Fig. 3). Cells that are symmetrical about the center of the square are called skew-symmetric. These are, for example, cells a And b in Fig. 3.

Magic squares of odd order can be constructed using the method of a 17th century French geometer. A. de la Lubera. Let's consider this method using the example of a 5th order square (Fig. 4). The number 1 is placed in the center cell of the top row. All natural numbers are arranged in a natural order cyclically from bottom to top in diagonal cells from right to left. Having reached the top edge of the square (as in the case of number 1), we continue to fill the diagonal starting from the bottom cell of the next column. Having reached the right edge of the square (number 3), we continue to fill the diagonal coming from the left cell in the line above. Having reached a filled cell (number 5) or a corner (number 15), the trajectory goes down one cell, after which the filling process continues.

The method of F. de la Hire (1640–1718) is based on two original squares. In Fig. Figure 5 shows how this method is used to construct a 5th order square. The numbers from 1 to 5 are entered into the cell of the first square so that the number 3 is repeated in the cells of the main diagonal going upward to the right, and not a single number appears twice in the same row or in the same column. We do the same with the numbers 0, 5, 10, 15, 20 with the only difference that the number 10 is now repeated in the cells of the main diagonal, going from top to bottom (Fig. 5, b). The cell-by-cell sum of these two squares (Fig. 5, V) forms a magic square. This method is also used to construct squares of even order.

If you know a way to construct squares of order m and order n, then we can construct a square of order mґ n. The essence of this method is shown in Fig. 6. Here m= 3 and n= 3. A larger square of the 3rd order (with numbers marked by primes) is constructed using the de la Loubert method. In the cell with the number 1ў (the central cell of the top row) fits a square of the 3rd order from the numbers from 1 to 9, also constructed by the de la Lubert method. In the cell with the number 2ў (right in the bottom line) fits a square of the 3rd order with numbers from 10 to 18; in the cell with the number 3ў - a square of numbers from 19 to 27, etc. As a result, we get a square of 9th order. Such squares are called composite.

MAGIC SQUARE
a square table of integers in which the sums of the numbers along any row, any column, and any of the two main diagonals equal the same number. The magic square is of ancient Chinese origin. According to legend, during the reign of Emperor Yu (c. 2200 BC), a sacred turtle surfaced from the waters of the Yellow River (Yellow River), on whose shell mysterious hieroglyphs were inscribed (Fig. 1a), and these signs are known as lo-shu and are equivalent to the magic square shown in Fig. 1, b. In the 11th century They learned about magic squares in India, and then in Japan, where in the 16th century. Extensive literature has been devoted to magic squares. Europeans were introduced to magic squares in the 15th century. Byzantine writer E. Moschopoulos. The first square invented by a European is considered to be the square of A. Durer (Fig. 2), depicted in his famous engraving Melancholy 1. The date of creation of the engraving (1514) is indicated by the numbers in the two central cells of the bottom line. Various mystical properties were attributed to magic squares. In the 16th century Cornelius Heinrich Agrippa constructed squares of the 3rd, 4th, 5th, 6th, 7th, 8th and 9th orders, which were associated with the astrology of the 7 planets. It was believed that a magic square engraved on silver protected against the plague. Even today, among the attributes of European soothsayers you can see magic squares.

In the 19th and 20th centuries. interest in magic squares flared up with renewed vigor. They began to be studied using the methods of higher algebra and operational calculus. Each element of a magic square is called a cell. A square whose side consists of n cells contains n2 cells and is called a square of nth order. Most magic squares use the first n consecutive natural numbers. The sum of S numbers in each row, each column and on any diagonal is called the square constant and is equal to S = n(n2 + 1)/2. It has been proven that n = 3. For a 3rd order square S = 15, 4th order - S = 34, 5th order - S = 65. The two diagonals passing through the center of the square are called the main diagonals. A broken line is a diagonal that, having reached the edge of the square, continues parallel to the first segment from the opposite edge (such a diagonal is formed by the shaded cells in Fig. 3). Cells that are symmetrical about the center of the square are called skew-symmetric. These are, for example, cells a and b in Fig. 3.

The rules for constructing magic squares are divided into three categories depending on whether the order of the square is odd, equal to twice an odd number, or equal to four times an odd number. A general method for constructing all squares is unknown, although various schemes are widely used, some of which we will consider below. Magic squares of odd order can be constructed using the method of a 17th century French geometer. A. de la Lubera. Let's consider this method using the example of a 5th order square (Fig. 4). The number 1 is placed in the center cell of the top row. All natural numbers are arranged in a natural order cyclically from bottom to top in diagonal cells from right to left. Having reached the top edge of the square (as in the case of number 1), we continue to fill the diagonal starting from the bottom cell of the next column. Having reached the right edge of the square (number 3), we continue to fill the diagonal coming from the left cell in the line above. Having reached a filled cell (number 5) or a corner (number 15), the trajectory goes down one cell, after which the filling process continues.

The method of F. de la Hire (1640-1718) is based on two original squares. In Fig. Figure 5 shows how this method is used to construct a 5th order square. The numbers from 1 to 5 are entered into the cell of the first square so that the number 3 is repeated in the cells of the main diagonal going up to the right, and not a single number appears twice in the same row or in the same column. We do the same with the numbers 0, 5, 10, 15, 20 with the only difference that the number 10 is now repeated in the cells of the main diagonal, going from top to bottom (Fig. 5, b). The cell-by-cell sum of these two squares (Fig. 5c) forms a magic square. This method is also used to construct squares of even order.

If you know how to construct squares of order m and order n, then you can construct a square of order mґn. The essence of this method is shown in Fig. 6. Here m = 3 and n = 3. A larger square of the 3rd order (with numbers marked by primes) is constructed using the de la Loubert method. In the cell with the number 1ў (the central cell of the top row) fits a square of the 3rd order from the numbers from 1 to 9, also constructed by the de la Lubert method. In the cell with the number 2ў (right in the bottom line) fits a square of the 3rd order with numbers from 10 to 18; in the cell with the number 3ў - a square of numbers from 19 to 27, etc. As a result, we get a square of 9th order. Such squares are called composite.

Collier's Encyclopedia. - Open Society. 2000 .

See what "MAGIC SQUARE" is in other dictionaries:

A square divided into an equal number n of columns and rows, with the first n2 natural numbers inscribed in the resulting cells, which add up to the same number for each column, each row and two large diagonals... Big Encyclopedic Dictionary

MAGIC SQUARE, a square MATRIX, divided into cells and filled with numbers or letters in a certain way, fixing a special magical situation. The most common letter square is SATOR, made up of the words SATOR, AREPO,... ... Scientific and technical encyclopedic dictionary

A square divided into an equal number n of columns and rows, with natural numbers from 1 to n2 inscribed in the resulting cells, which add up to the same number for each column, each row and two large diagonals. In Fig. example of M. k. s... ... Natural science. encyclopedic Dictionary

A magic or magic square is a square table filled with numbers in such a way that the sum of the numbers in each row, each column and on both diagonals is the same. If the sums of numbers in a square are equal only in rows and columns, then ... Wikipedia

A square divided into an equal number n of columns and rows, with the first n2 natural numbers inscribed in the resulting cells, which add up to each column, each row and two large diagonals the same number [equal as... ... Great Soviet Encyclopedia

A square table of integers from 1 to n2, satisfying the following conditions: where s=n(n2+1)/2. More general mathematical equations are also considered, in which it is not required that any number a be uniquely characterized by a pair of residues (a, b) modulo n(digits... Mathematical Encyclopedia

Book A square divided into parts, each of which contains a number that adds up to the same number along with others horizontally, vertically or diagonally. BTS, 512… Large dictionary of Russian sayings

- (Greek magikos, from magos magician). Magical, related to magic. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. MAGICAL magic. Dictionary of foreign words included in the Russian language. Pavlenkov F., 1907 ... Dictionary of foreign words of the Russian language

It is a three-dimensional version of the magic square. A traditional (classical) magic cube of order n is a cube of dimensions n×n×n, filled with various natural numbers from 1 to n3 so that the sums of numbers in any of the 3n2 rows, ... ... Wikipedia

Books

Magic Square, Irina Bjorno, “Magic Square” is a collection of stories and short stories written in the style of magical realism, where reality is closely intertwined with magic and fantasy, forming a new, magical style -... Category: Horror and Mystery Publisher: Publishing Solutions, eBook(fb2, fb3, epub, mobi, pdf, html, pdb, lit, doc, rtf, txt)

Introduction

The great scientists of antiquity considered quantitative relations to be the basis of the essence of the world. Therefore, numbers and their relationships occupied the greatest minds of mankind. “In the days of my youth, I amused myself in my spare time by making... magic squares,” wrote Benjamin Franklin. A magic square is a square whose sum of numbers in each horizontal row, in each vertical row and along each diagonal is the same.

Some outstanding mathematicians devoted their work to magic squares, and the results they obtained influenced the development of groups, structures, Latin squares, determinants, partitions, matrices, comparisons and other non-trivial areas of mathematics.

The purpose of this essay is to get acquainted with various magic squares, Latin squares and study the areas of their application.

Magic squares

A complete description of all possible magic squares has not been obtained to this day. There are no magic 2x2 squares. There is a single 3x3 magic square, since other 3x3 magic squares are obtained from it either by rotation around the center or by reflection about one of its axes of symmetry.

There are 8 different ways to arrange natural numbers from 1 to 9 in a 3x3 magic square:

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

In a 3x3 magic square, the magic constant 15 must be equal to the sum of three numbers in 8 directions: 3 rows, 3 columns and 2 diagonals. Since the number in the center belongs to 1 row, 1 column and 2 diagonals, it is included in 4 of the 8 triplets that add up to the magic constant. There is only one such number: it is 5. Therefore, the number in the center of the 3x3 magic square is already known: it is 5.

Consider the number 9. It is included in only 2 triplets of numbers. We cannot place it in a corner, since each corner cell belongs to 3 triplets: row, column and diagonal. Therefore, the number 9 must be in some cell adjacent to the side of the square in its middle. Because of the symmetry of the square, it does not matter which side we choose, so we write 9 above the number 5 in the central cell. On either side of the nine in the top line, we can only write the numbers 2 and 4. Which of these two numbers will be in the upper right corner and which in the left again does not matter, since one arrangement of numbers goes into another when mirrored . The remaining cells are filled in automatically. Our simple construction of a 3x3 magic square proves its uniqueness.

Such a magic square was a symbol of great significance among the ancient Chinese. The number 5 in the middle meant earth, and around it in strict balance were fire (2 and 7), water (1 and 6),

wood (3 and 8), metal (4 and 9).

As the size of the square (number of cells) increases, the number of possible magic squares of that size increases rapidly. There are 880 magic squares of order 4 and 275,305,224 magic squares of order 5. Moreover, 5x5 squares were known back in the Middle Ages. Muslims, for example, were very reverent about such a square with the number 1 in the middle, considering it a symbol of the unity of Allah.

Magic square of Pythagoras

The great scientist Pythagoras, who founded the religious and philosophical doctrine that proclaimed quantitative relations to be the basis of the essence of things, believed that the essence of man also lies in the number - the date of birth. Therefore, with the help of the magic square of Pythagoras, you can know the character of a person, the degree of health and his potential, reveal advantages and disadvantages and thereby identify what should be done to improve him.

In order to understand what the magic square of Pythagoras is and how its indicators are calculated, I will calculate it using my own example. And to make sure that the results of the calculation really correspond to the real character of a particular person, first I will check it on myself. To do this, I will do the calculation based on my date of birth. So, my date of birth is 08/20/1986. Let's add the numbers of the day, month and year of birth (excluding zeros): 2+8+1+9+8+6=34. Next, add up the numbers of the result: 3+4=7. Then from the first amount we subtract double the first digit of the birthday: 34-4=30. And again we add the digits of the last number:

3+0=3. It remains to make the last additions - 1st and 3rd and 2nd and 4th sums: 34+30=64, 7+3=10. We got the numbers 08/20/1986,34,7,30, 64,10.

and make a magic square so that all ones of these numbers go into cell 1, all twos into cell 2, etc. Zeros are not taken into account. As a result, my square will look like this:

The square cells mean the following:

Cell 1 - determination, will, perseverance, selfishness.

1 - complete egoists, strive to extract the maximum benefit from any situation.
11 - character close to egoistic.
111 - “golden mean”. The character is calm, flexible, and sociable.
1111 - people of strong character, strong-willed. Men with such character are suitable for the role of military professionals, and women keep their family in their fist.
11111 - dictator, tyrant.
111111 - a cruel person, capable of doing the impossible; often falls under the influence of some idea.

Cell 2 - bioenergy, emotionality, sincerity, sensuality. The number of twos determines the level of bioenergy.

There are no twos - the channel is open for an intensive collection of bioenergy. These people are well-mannered and noble by nature.

2 - people who are ordinary in bioenergetic terms. Such people are very sensitive to changes in the atmosphere.
22 - a relatively large reserve of bioenergy. Such people make good doctors, nurses, and orderlies. In the family of such people, there is rarely anyone who experiences nervous stress.
222 is the sign of a psychic.

Cell 3 - accuracy, specificity, organization, neatness, punctuality, cleanliness, stinginess, inclination to constant “restoration of justice.”

The increase of threes enhances all these qualities. With them, it makes sense for a person to look for himself in the sciences, especially the exact ones. The predominance of threes gives rise to pedants, people in a case.

Cell 4 - health. This is connected with the ecgregor, that is, the energy space developed by the ancestors and protecting a person. The absence of fours indicates that a person is sick.

4 - average health, it is necessary to harden the body. Swimming and running are recommended sports.
44 - good health.
444 and more - people with very good health.

Cell 5 - intuition, clairvoyance, which begins to manifest itself in such people already at the level of three fives.

There are no fives - the communication channel with space is closed. These people often

are wrong.

5 - communication channel is open. These people can correctly calculate the situation and make the most of it.
55 - highly developed intuition. When they see “prophetic dreams,” they can predict the course of events. Professions suitable for them are lawyer, investigator.
555 - almost clairvoyant.
5555 - clairvoyants.

Cell 6 - groundedness, materiality, calculation, a tendency towards quantitative exploration of the world and distrust of qualitative leaps and, especially, spiritual miracles.

There are no sixes - these people need physical labor, although, as a rule, they do not like it. They are endowed with extraordinary imagination, fantasy, and artistic taste. Subtle natures, they are nevertheless capable of action.

6 - can engage in creativity or exact sciences, but physical labor is a prerequisite for existence.
66 - people are very grounded, drawn to physical labor, although it is not obligatory for them; Mental activity or artistic pursuits are desirable.
666 is the sign of Satan, a special and ominous sign. These people have a high temperament, are charming, and invariably become the center of attention in society.
6666 - these people in their previous incarnations gained too much grounding, they worked very hard and cannot imagine their life without work. If their square contains

Nines, they definitely need to engage in mental activity, develop their intellect, and at least get a higher education.

Cell 7 - the number of sevens determines the measure of talent.

7 - the more they work, the more they get later.
77 - very gifted, musical people, have a subtle artistic taste, and may have a penchant for fine arts.
777 - these people, as a rule, come to Earth for a short time. They are kind, serene, and sensitive to any injustice. They are sensitive, like to dream, and do not always feel reality.
7777 - sign of an Angel. People with this sign die in infancy, and if they live, their lives are constantly in danger.

Cell 8 - karma, duty, obligation, responsibility. The number of eights determines the degree of sense of duty.

There are no Eights - these people have an almost complete lack of sense of duty.

8 - responsible, conscientious, accurate natures.
88 - these people have a developed sense of duty; they are always distinguished by the desire to help others, especially the weak, sick, and lonely.
888 is a sign of great duty, a sign of service to the people. A ruler with three eights achieves outstanding results.
8888 - these people have parapsychological abilities and exceptional sensitivity to the exact sciences. Supernatural paths are open to them.

Cell 9 - intelligence, wisdom. The absence of nines is evidence that mental abilities are extremely limited.

9 - these people must work hard all their lives to make up for their lack of intelligence.
99 - these people are smart from birth. They are always reluctant to learn, because knowledge is given to them easily. They are endowed with a sense of humor with an ironic tinge, independent.
999 - very smart. No effort is put into learning at all. Excellent conversationalists.
9999 - the truth is revealed to these people. If they also have developed intuition, then they are guaranteed against failure in any of their endeavors. With all this, they are usually quite pleasant, since their sharp mind makes them rude, unmerciful and cruel.

So, having drawn up the magic square of Pythagoras and knowing the meaning of all combinations of numbers included in its cells, you will be able to sufficiently assess the qualities of your nature that Mother Nature has endowed.

Latin squares

Despite the fact that mathematicians were mainly interested in magic squares, Latin squares found the greatest application in science and technology.

A Latin square is a square of nxn cells in which the numbers 1, 2,..., n are written, and in such a way that all these numbers appear once in each row and each column. Figure 3 shows two such 4x4 squares. They have interesting feature: if one square is superimposed on another, then all pairs of resulting numbers turn out to be different. Such pairs of Latin squares are called orthogonal.

The problem of finding orthogonal Latin squares was first posed by L. Euler, and in such an entertaining formulation: “Among the 36 officers there are an equal number of lancers, dragoons, hussars, cuirassiers, cavalry guards and grenadiers, and in addition an equal number of generals, colonels, majors, captains, lieutenants and second lieutenants, and Each branch of the military is represented by officers of all six ranks. Is it possible to line up all the officers in a 6 x 6 square so that in any column and any rank there are officers of all ranks?”

Euler was unable to find a solution to this problem. In 1901 it was proven that such a solution did not exist. At the same time, Euler proved that orthogonal pairs of Latin squares exist for all odd values of n and for those even values of n that are divisible by 4. Euler hypothesized that for the remaining values of n, that is, if the number n when divided by 4 gives in remainder 2, there are no orthogonal squares. In 1901 it was proven that there are no orthogonal squares 6 6, and this increased confidence in the validity of Euler's hypothesis. However, in 1959, with the help of a computer, orthogonal squares 10x10, then 14x14, 18x18, 22x22 were first found. And then it was shown that for any n except 6, there are nxn orthogonal squares.

Magic and Latin squares are close relatives. Let us have two orthogonal squares. Let's fill the cells of a new square of the same dimensions as follows. Let's put there the number n(a - 1)+b, where a is the number in such a cell of the first square, and b is the number in the same cell of the second square. It is easy to understand that in the resulting square, the sums of numbers in the rows and columns (but not necessarily on the diagonals) will be the same.

The theory of Latin squares has found numerous applications both in mathematics itself and in its applications. Let's give an example. Suppose we want to test 4 varieties of wheat for yield in a given area, and we want to take into account the influence of the degree of sparseness of crops and the influence of two types of fertilizers. To do this, we will divide a square plot of land into 16 plots (Fig. 4). We will plant the first wheat variety in plots corresponding to the lower horizontal stripe, the next variety in four plots corresponding to the next stripe, etc. (in the figure, the variety is indicated by color). In this case, let the maximum density of crops be in those plots that correspond to the left vertical column of the figure, and decrease when moving to the right (in the figure this corresponds to a decrease in color intensity). Let the numbers in the cells of the picture mean:

the first is the number of kilograms of fertilizer of the first type applied to this area, and the second is the amount of fertilizer of the second type applied. It is easy to understand that in this case all possible pairs of combinations of both variety and sowing density and other components are realized: variety and fertilizers of the first type, fertilizers of the first and second types, density and fertilizers of the second type.

The use of orthogonal Latin squares helps to take into account all possible options in experiments in agriculture, physics, chemistry, and technology.

square magic pythagoras latin

Conclusion

This essay examines issues related to the history of the development of one of the questions in mathematics that has occupied the minds of many great people - magic squares. Despite the fact that magic squares themselves have not found wide application in science and technology, they inspired many extraordinary people to study mathematics and contributed to the development of other branches of mathematics (the theory of groups, determinants, matrices, etc.).

The closest relatives of magic squares, Latin squares, have found numerous applications both in mathematics and in its applications in setting up and processing the results of experiments. The abstract provides an example of setting up such an experiment.

The abstract also discusses the issue of the Pythagorean square, which is of historical interest and possibly useful for drawing up a psychological portrait of a person.

Bibliography

1. Encyclopedic dictionary of a young mathematician. M., “Pedagogy”, 1989.
2. M. Gardner “Time Travel”, M., “Mir”, 1990.
3. Physical education and sports No. 10, 1998