Description of the presentation by individual slides:
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Secrets of magic squares. Author of the work: Yuneva Elizaveta Aleksandrovna Place of work: Soldato-Aleksandrovskoye village, Municipal Educational Institution "Secondary School No. 6 of Soldato-Aleksandrovskoye", grade 6 "a" Scientific supervisor: Denisova Natalya Valerievna, mathematics teacher of Municipal Educational Institution "Secondary School No. 6 of Soldato-Aleksandrovskoye"
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Introduction "Making magic squares is an excellent mental gymnastics for developing the ability to understand the ideas of placement, combination and symmetry." Leonard Euler Magic squares... This phrase immediately reeks of magic. The great scientists of antiquity considered quantitative relations to be the basis of the essence of the world. They saw that numbers have some kind of independent life, their own secrets. Later it turned out that by arranging the numbers in the correct rows, in the case of “magic” you can add them from left to right and from top to bottom, each time you get equal numbers. Thus, over the course of time, a magic square was formed, which we see to this day.
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Project goal: to study ways of filling magic squares and the history of their appearance; find out different ways to create magic squares; explore their areas of application. Project objectives: 1. Get acquainted with the history of the appearance and names of magic squares; 2.Study known methods of filling magic squares; 3. Find out the areas of application of the magic square. Research topic: filling in magic squares; Object of study: magic square; Hypothesis: to fill the magic square, there are special techniques that allow you to do this quickly
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During the work, the following methods were used: search method (use of reference and educational literature, as well as information resources of the global Internet); practical method (drawing magic squares based on acquired knowledge); research method (drawing up a psychological portrait of a personality using the Pythagorean square).
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The history of the appearance of the magic square 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, and these signs are known as lu-shu and are equivalent to a magic square . In the 11th century They learned about magic squares in India, and then in Japan, in the 15th century. Europeans learned about magic squares. The first square invented by a European is considered to be the Durer square, 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. 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.
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MAGIC SQUARE is a square table of integers in which the sums of the numbers along any row, any column, and any of the two main diagonals are equal to the same number. The name “magic” squares came from the Arabs, who saw something mystical in their properties and therefore took the squares as unique talismans that protected those who wore them from many misfortunes. Medieval Arab mathematicians also showed interest in amazing squares, citing examples of them in their writings. Various mystical properties were attributed to magic squares, as if they could even cure a person from terrible diseases. Making magic squares was a popular pastime among mathematicians, and huge squares were created. If in a square the sums of numbers only in rows and columns are equal, then it is called semi-magic
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Application of magic squares When I looked at the methods of composing magic squares, I became interested in the scope of their application. She seemed quite interesting to me. The Japanese puzzle Sudoku is very popular, the ancestor of which can be considered the Magic Square. It helps us develop logical thinking and computational skills. Nowadays, many newspapers publish these puzzles along with crosswords and other logic problems. Well, and, of course, in numerology. Even the great scientist Pythagoras believed that everything in the world is controlled by numbers. Therefore, the essence of a person also lies in the number - the date of his birth. He created a method for constructing a square, by which one can understand a person’s character, the state of his health and his potential, reveal his strengths and weaknesses, and thereby identify what should be done to improve him. In the time of Pythagoras, magic squares were created individually for each person. Now there is a special program where a person’s date of birth is entered, and a ready-made magic square is displayed on the screen. I'll make a magic square for myself.
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I was born on November 10, 2004. We add the numbers of the day of the month and year of birth, we get the first working number 9. Next we add the digits of the first working number and we get the second working number 9. From the first working number we subtract double the first digit of the birthday, so we get the third working number: 9-2=7. We get the fourth working number from the sum of the digits of the third working number: 7 Draw a 3 by 3 square. From our two lines, we count the number of ones in the numbers - we write them in the first square. The second cell contains twos, the third - threes, and so on. “111” – positive personality, stable character. “2” - I am a person sensitive to changes in the atmosphere, “4” - I have excellent health, “77” - I have everything - good and bad. I have taste, I draw well, I am very talented. In case of trouble, I can get away with it. “99” is smart from birth, knowledge comes easily. 111 4 77 2 - - - - 99
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Another traditional area of application for magic squares is talismans. For example, the Moon talisman has certain properties: it protects against shipwreck and illness, makes a person kind, helps prevent bad intentions, and also improves health. It is engraved on silver on the day and hour of the Moon when the Sun or Moon is in the first ten degrees of Cancer. A magic square of the 9th order fits into a hexagon (9 is the number of the Moon) and is surrounded by special symbols
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Types of magic squares There are no 2*2 magic squares. A square of size 2*2 would have to consist of the numbers 1,2,3,4, and its constant would be 5. Such a square would have two rows, columns and diagonals. For a square to become magic, you need to represent the number 5 as the sum of two given numbers in six different ways, but this is not possible! After all, there are only two such combinations: 1+ 4 and 2+3. There is only one 3*3 magic square, since the remaining 3*3 magic squares are obtained from it either by rearranging rows or columns or by rotating the original square by 90 or 180 degrees
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Algorithm for composing a 3x3 magic square 1) Write down the numbers in the order shown in the figure: 1 2 3 4 5 6 7 8 9 2) Swap the numbers at opposite ends of the diagonals: 1 and 9, 3 and 7: 9 2 7 4 5 6 3 8 1 3) Shift each of the numbers one step clockwise 4 9 2 3 5 7 8 1 6 Thus, we get a magic square, the magic sum of which (i.e. the sum of the numbers in any line, in any column and on each of the diagonals) is equal to 15. The direction does not matter, the main thing is to preserve the order of the numbers.
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Lo-shu square. A 3rd order magic square of the first 9 natural numbers (known in China as the Luo Shu talisman) is represented by a 3x3 matrix. The general method for constructing squares is unknown. The rules for constructing magic squares are divided into three categories depending on the order of the square. Squares can be: - odd, that is, consist of an odd number of cells, - even-even, that is, the order is equal to twice even; - even-odd, that is, the order is equal to twice the odd.
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Fourth order square. The 4x4 magic square depicted in Albrecht Durer's engraving "Melancholy I" is considered the earliest in European art. The two middle numbers in the bottom row indicate the date of creation of the painting (1514). The sum of the numbers on any horizontal, vertical and diagonal is 34. This sum also occurs in all 2x2 corner squares, in the central square (10+11+6+7), in the square of corner cells (16+13+4+1 ), in squares built by the “knight’s move” (2+8+9+15 and 3+5+12+14), in rectangles formed by pairs of middle cells on opposite sides (3+2+15+14 and 5+8 +9+12).
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Devil's magic square. The devil's magic square is a magic square in which the sums of numbers along broken diagonals in both directions also coincide with the magic constant. Such squares are also called pandiagonal. There are 48 4x4 devilish magic squares with rotation and reflection precision. Pandiagonal squares of the fourth order have a number of additional properties for which they are called perfect. There are no perfect squares of odd order.
From the depths of centuries Sacred, magical, mysterious, mysterious, perfect... As soon as they were called! “I don’t know anything more beautiful in arithmetic than these numbers, called planetary by some and magic by others,” wrote the famous French mathematician, one of the creators of number theory, Pierre de Fermat, about them.
An nth order magic square is a square table of size n×n, filled with natural numbers from 1 to n 2, the sums of which are the same across all rows, columns and both diagonals. There are magic squares of even and odd order (depending on the parity of n).
The “oldest” magic square that has come down to us is the Luo Shu table (about 2200 BC)
The magic square of the 4th order was known to the ancient Hindus. It is interesting because it retains the property of being magical after sequential rearrangement of rows (columns)
The Dürer square has a size of 4x4 and is made up of the first sixteen natural numbers, the sum of which in each row, column and diagonal is equal to
It turns out that 34 is also equal to the sum of other four numbers: those located in the center, in the corner cells, on the sides of the central square, and also forming four equal squares into which the original square can be divided
How to build a magic square? Many mathematicians are looking for ways to compose magic squares. The currently known rules for constructing such squares are divided into three groups depending on the order of the square. However, a general construction method still does not exist.
We write all natural numbers from 1 to 25 in cells diagonally (5 in a row) so that we get a diagonal square
Select a 5x5 square in the center. It will form the basis of the future magic square
We move each number located outside the central square inside - to its opposite side, moving by 5 cells
The magic square is ready
Let's fill the cells line by line with these numbers, moving from left to right and top to bottom, while skipping those that correspond to the filled cells
Let's fill the cells selected in the first step with the missing numbers in ascending order, moving from right to left and from bottom to top. The magic square is built
Let's consider ways to construct a magic square of any even order. In all cases, the n×n table is filled from left to right and top to bottom with natural numbers from 1 to n 2 in their natural order. Then, according to a certain rule, the numbers in some cells are rearranged, after which the square becomes magic.
Divide the square filled with numbers from 1 to 64 into squares of the 4th order
In each row and column of the upper left square, color two cells in a checkerboard pattern.
For each of the marked cells, highlight with the same color the one that is symmetrical to it relative to the vertical axis
We rearrange the number in each of the sixteen shaded cells with the number from the corresponding centrally symmetric cell
Construction of the square is completed
For example, let's take a 10x10 square. Divide the square filled with numbers from 1 to 100 into squares of the 5th order
In the upper left square we will paint three groups of cells with different colors, with each row and column containing two cells from the first group and one from the second and third. Use the same color to highlight the cells located along the diagonal of the square and the lines parallel to it
Cells that are symmetrical to the cells of the first group relative to the vertical axis will be painted with the same color.
The number in each of the marked cells is rearranged with the number from the corresponding centrally symmetric cell
The contents of each cell of the second group will be exchanged with the contents of the cell symmetrical to it relative to the horizontal axis of the square
The contents of each cell of the third group will be exchanged with the contents of the cell that is symmetrical to it relative to the vertical axis of the square
36 Questions While studying how to construct magic squares, I realized that it is important to know their constants, that is, the sum of the numbers in any row, column or diagonal. Of course, if the square is constructed and the value of n is small, then the sum can be calculated. A What to do if the square has not yet been built? And Or do you need to check whether a given square is magic? And how to construct the square itself without knowing its constant?
- Goals:
- 1. Get acquainted with magic squares.
- 2. Find out the history of the appearance of squares.
- 3. Learn to fill in magic squares correctly and quickly.
- Tasks:
- 1. Study the history of the emergence and development of magical
- squares;
- 2. Study the properties of magic squares;
- 3. Get acquainted with the basic construction methods
- magic squares.
- The order of the magic square.
- The word “order” in this case means the number of cells on one side of the square. The square 33 is of the third order, and the square 55 is the fifth, etc.
- The history of magic squares.
- The name “magic” squares came from the Arabs, who saw something mystical in their properties and therefore took the squares as unique talismans that protected those who wore them from many misfortunes.
- Magic squares originated in ancient times in China. Probably the “oldest” of the magic squares that have come down to us is the Lo Shu table (c. 2200 BC). It is 3x3 in size and filled with natural numbers from 1 to 9. In this square, the sum of the numbers in each row, column and diagonal is 15.
- According to one legend, the prototype was the pattern that adorned the shell of a huge turtle.
- Magic square 3rd order.
- The sum of the numbers in each row is 15
- Magic square 4th order.
- The sum of the numbers in each row is 34.
- Magic square 5th order.
- The sum of the numbers in each row is 65.
- 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 the nth order. For example, 3 cells are a 3rd order square, 4 cells are a 4th order square, etc. Most magic squares use the first 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. For a 3rd order square S = 15, 4th order – S = 34, 5th order – S = 65.
- At the beginning of the 16th century. the famous German artist Albrecht Durer immortalized the magic square in art, depicting it in the engraving “Melancholy”. The Dürer square has dimensions of 4 x 4 and is made up of the first sixteen natural numbers, the sum of which in each row, column and diagonal is 34.
- The traditional area of application of magic squares is talismans. For example, the Moon talisman has certain properties: it protects against shipwreck and illness, makes a person kind, helps prevent bad intentions, and also improves health. It is engraved on silver at the day and hour of the moon.
- Sudoku: Japanese puzzles. This game, also known as the magic square, was invented in 1783 by Swiss mathematician Leonhard Euler.
- Sudoku (Japanese “su” - number, “doku” - next to it, standing separately) - Japanese number puzzles, where in a square of 9x9 cells you need to arrange the numbers from 1 to 9 in a special way.
- Currently, Sudoku is widespread outside of Japan: both adults and children around the world love to solve them.
- Task 1. Write the missing numbers from 1 to 16 into the empty rectangles so that the total of all columns and rows and both diagonals results in the number 34.
- Answer:
- Nowadays, magic squares continue to attract the attention of lovers of mathematical games and entertainment. There has been an increase in the number of fun math books that contain puzzles and problems involving unusual squares. Their successful solution requires not so much special knowledge as ingenuity and the ability to notice numerical patterns. Solving such problems will serve as an excellent “mental gymnastics”.
- It was not the magic squares themselves that received practical use, but methods and entire sections of modern mathematics that arose and developed thanks to solving problems of compiling and analyzing the properties of magic squares.
- Like many centuries ago, magic squares are now used only by modern “magicians”, astrologers and numerologists.
- 1. Magic squares are something amazing, interesting and exciting.
- 2. Filling out magic squares is not difficult, but you need to know some rules.
- 3. The main features of magic squares are not only clarity, clarity and logic, but also aesthetics, harmony and beauty.
- From the presentation we received, we learned the types of magic squares, the history of their origin, as well as their use in the modern world.
- 1. Troshin V.V.. The magic of numbers and figures. M.: - Globus LLC, 2007.
- 2. Encyclopedia for children. – M.: Avanta Publishing Association, 2003.
- 3. Sarvina N.M. Unexpected mathematics // Mathematics for schoolchildren 2005, No. 4
- 4. Fainshtein V. A. Fill in the magic square // Mathematics at school, 2000, No. 3
- 5. Internet
...mathematical truths are immortal, not subject to decay and remain the same yesterday, today and forever
Eric Temple Bell (1883-1960)
Department of Education and Science of the Kemerovo Region
State budgetary educational institution
secondary vocational education
"Novokuznetsk Transport and Technology College"
Magic Squares (oral journal)
Naimushina Kristina Andreevna,
Melkov Maxim Sergeevich
"Historical"
1 page
Magic squares were highly respected and various mystical properties were attributed to them. .
"Cognitive"
2 page
- 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 in a square the sums of numbers only in rows and columns are equal, then it is called semi-magic . A normal square is a magic square filled with integers starting from 1.
From a filled magic square you can get a new magic square by increasing all the numbers of the square by the same number
M =15
M =21
From a filled magic square, a new magic square can be obtained by reflection relative to the symmetry axes
From a filled magic square, a new magic square can be obtained by reflection relative to the symmetry axes
From a filled magic square, a new magic square can be obtained by reflection relative to the symmetry axes
A filled magic square can be used to create a new magic square. turning around the center
"Practical"
3 page
Odd squares
- We build a square ABCD with 25 cells and temporarily expand it to a symmetrical stepped figure with steps of one cell.
- In the resulting figure, we place 25 integers from 1 to 25 in order in oblique rows from top to bottom - to the right.
- And now each number that is outside the square ABCD should be moved along the same row or column exactly so many cells from the cell it occupies, what is the order of the square, in our example - five. So, in accordance with this rule, we transfer these numbers...
Squares order, multiple of four
- Place the numbers in the cells of a given square in ascending order (in natural order).
- Select four squares with sides n/4 at the corners of a given square and one square with side n/2 in the center.
- In the five selected squares, swap the numbers located symmetrically relative to the center of the given square.
- Squares composed according to the specified pattern will always be magically symmetrical.
"Research"
4 page
Talismans Moon Talisman
Data protection Text encryption
O I R M E O S Y V T A L G O P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
ARRIVAL
Sudoku is a number puzzle game that has become very popular lately. Translated from Japanese, “su” means “number” and “doku” means “standing alone.”
Experiments in agriculture, physics, chemistry, technology.
Yield testing of 4 wheat varieties
"Entertaining"
5 page
Understanding a person's character:
Pythagorean square
MBOU "Vozhegodskaya SS"
Magic square
Math club lesson in 5th grade
Goal of the work:
Get acquainted with magic squares.
1. Find out the history of the appearance of squares.
2. Explore the properties of squares.
3. Learn the rules for filling in the squares.
3. Learn to correctly and quickly fill out a 3 by 3 magic square.
Formed UUD
Cognitive: prove, draw conclusions, build logically sound reasoning.
Regulatory: determine the goal, problem of activity; put forward versions; self-control and correction.
Communicative: express your opinion, organize work in pairs (ask questions, develop a solution).
Personal: respectful attitude towards classmates, awareness of the need to acquire new knowledge.
Progress of the lesson
1. Which of the concepts written on the board do we know:
- Mathematical sophistry(proof with error to be found)
- Mathematical paradox(a statement that can be considered both true and false)
- Möbius strip(topological figure having one infinite side)
- Magic square
The topic of our lesson is “Magic Square”
I’ll start with a legend according to which the Chinese Emperor Yiyu, who lived four thousand years ago, once saw on the bank of a river a sacred turtle with a pattern of black and white circles on its shell. The quick-witted emperor immediately understood the meaning of this drawing. Try to define it too.
Find the sum of the numbers represented by circles in each row, column and diagonal
The sum of the numbers in each row, column and diagonal is 15.
It is this square in mathematics that is called magic. The properties of magic squares were considered magical in both ancient China and medieval Europe. Magic squares served as talismans, protecting those who wore them from various troubles.
The engraving of the German artist Albrecht Dürer “Melancholy” (1514) also depicts a square. Prove that it is magical.
The sum of the digits in each row, column, and diagonal is 34.
There are other interesting properties in this square. Find the sum of numbers in 2 by 2 squares, in all corner cells.
And now that we have learned a little about what a magic square is, try to formulate the purpose of our lesson. (Learn to fill out). Tasks? (Learn the rule, practice).
How to make a magic square?
The number of cells along one side of the square is denoted by the letter n and is called the order of the square. There is a square of any order except 2nd. The simplest (trivial) is a square of the 1st order, consisting of one cell. The simplest magic squares fit natural numbers from 1 to n2 + 1
The sum of the numbers in each row, each column and on any diagonal of the magic square called the magic constant M.
The magic constant n is determined by the formula: 
Find the magic constant for a square of 3rd order (15), 4th order (34), 5th order (65).
We will start by constructing the simplest third-order magic square. We know that the sum of all numbers horizontally, vertically and diagonally is 15. Make up all possible sums of triplets of numbers from 1 to 9 that result in 15.
What number occurs most often? (5 - 4 times) This means that the number 5 should be at the intersection of 4 rows of the table. Where should it be? (In the center of the table). Distribute the remaining numbers yourself.
What squares did you get?
If you wrap a 4x4 “magic” square around a rectangular frame, you can discover a number of other properties.
the sum of the four numbers around the frame in any direction is 34
the sum of the four numbers that occur in each corner on the outside and in each corner on the inside is also 34
the sum of four numbers of the same color is 34
if you add the numbers in a spiral clockwise or counterclockwise around the frame, starting anywhere - 34.
Let's summarize. Have we achieved our goal?
Resource circle. What new things did you learn, your impressions of the lesson. We passed the tetrahedron to each other - this geometric body also has unusual properties. And we’ll find out what kind they are in one of the club’s classes.
Handout
| Magic square
n - square order Magic square, n = 3 | Magic square n - square order M - magic constant of the square Magic square, n = 3 9 = 1 + 5 + 9, 9 = ______________, 9 = ______________, 9 = 2 + 5 + 8, 9 = ______________, 9 = ______________, 9 = ______________, 9 = ______________. |
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