Characteristics of numbers Number 1 is red. The point of reality, the basis, the core of the entire digital superstructure, which determines the Type of this or that energy flow. The purpose of number 1 is to determine the meaning, importance and weight of the emerging reality. So in the world of business

“Magical Proofs” or “Proofs of Magic” “You are a bad person!” Or: “He is a bad person” Or: “He is a good person!” Or: “You are a good person!” Choose! What do you prefer? Isn’t it funny to watch the “ritual Zulu dances on

5.2. Magic squares in Chaturanga. Chaturanga as fortune telling 5.2.1 About the magic of numbers. What are magic squares? You can talk a lot about the magic of numbers. As an example, at the beginning of this study we already mentioned the number 4. Much can be said in a similar way about any

5.2.2. Magic squares in Chaturanga 5.2.2.1 The magic of a non-magical square It is curious that the simplest (non-magical) square 5?5, where the numbers simply go one after another - from 1 to 25, can also have unusual properties. So, in this simple square the sum of the “Elephant Cross”

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 only one 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 we 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 - a 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. Suitable professions for them are lawyer, investigator.
• 555 - almost clairvoyant.
• 5555 - clairvoyants.

Cell 6 - groundedness, materiality, calculation, a penchant for quantitative exploration of the world and distrust of qualitative leaps, and even more so of 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 is the 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 comes easily to them. They are endowed with a sense of humor with an ironic tinge and are 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 an 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

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.

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 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.

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:

1. 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 a very large number of numbers are assigned the same pictures.
2. 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.
3. 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”.
4. 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:

1. 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”.
2. Subtract the numbers that make it up from the hidden number. Add the equation: (x*10+y)-(x+y)=9*x.
3. 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.