MAGIC SQUARE

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 both diagonals is the same.

The sum of the numbers in each row, column, and diagonal is called the magic constant, M.

The smallest magic constant of a 3x3 magic square is 15, a 4x4 square is 34, a 5x5 square is 65,

If the sums of numbers in the square are equal only in rows and columns, then it is called semi-magic.

Building a 3 x 3 magic square with the smallest

magic constant

Find the smallest magic constant of the 3x3 magic square

1 way

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = (1+9) + (2+8) + (3+7) + (4+6) + 5 = 45

4
5 : 3 = 15 1 2 3 4 5 6 7 8 9

M = 15.

The number written in the middle is 15 : 3 = 5

Determined that in the middle, the number 5 is written.

where n is the number of rows

If you can build one magic square, then it is easy to construct any number of them. Therefore, remember the construction techniques

3x3 magic square with constant 15.

1 way construction. Put the even numbers in the corners first

2,4,8,6 and 5 in the middle. The rest of the process is simple arithmetic.

15 – 6 = 9; 15 – 14 = 1 15 – 8 = 7; 15 – 12 = 3

2 way solutions

Using the found magic square with a constant of 15, you can set a lot of diverse tasks:

Example. Build new different magic squares 3 x 3

Solution.

Adding each number of the magic square, or multiplying it by the same number, we get a new magic square.

Example 1 Construct a 3 x 3 magic square whose number in the middle is 13.

Solution.

Let's build a familiar magic

square with constant 15.

Find the number that is in

the middle of the desired square

13 – 5 = 8.

To every magic number

add 8 squares.

Example 2 Fill the cages of magic

squares, knowing the magic constant.

Solution. Let's find the number

written in the middle 42: 3 = 14

42 – 34 = 8, 42 – 30 =12 42 – 20=22, 42 – 36=6 42–24=18, 42–32= 10

tasks for independent decision

Examples. 1. Fill the cells of magic squares with magic

constant M =15.

1) 2) 3)

2. Find the magic constant of magic squares.

1) 2) 3)

3. Fill in the cells of the magic squares, knowing the magic constant

1) 2) 3)

M=24 M=30 M=27

4 . Construct a 3x3 magic square knowing that the magic constant is

equals 21.

Solution. Recall how a magic 3x3 square is built according to the smallest

constant 15. Even numbers are written in the extreme fields

2, 4, 6, 8, and in the middle the number 5 (15 : 3).

According to the condition, it is necessary to construct a square according to the magic constant

21. In the center of the desired square should be the number 7 (21 : 3).

Let's find how much more each member of the desired square

each term with the smallest magic constant 7 - 5 = 2.

We build the required magic square:

21 – (4 + 6) = 11

21 – (6 + 10) = 5

21 – (8 + 10) = 3

21 – (4 + 8) = 9

4. Construct 3x3 magic squares knowing their magic constants

M = 42 M = 36 M = 33

M=45 M=40 M=35

Building a 4 x 4 magic square with the smallest

magic constant

Find the smallest magic constant of a 4x4 magic square

and the number located in the middle of this square.

1 way

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 +13 +14 + 15 + 16 =

(1+16)+(2+15)+(3+14)+(4+13)+(5+12)+ (6+11)+ (7+10)+(8+9) = 17 x 8= 136

136: 4= 34.

where n is the number of rows n = 4.

The sum of numbers on any horizontal,

vertical and diagonal is 34.

This amount also occurs in all

corner squares 2×2, in the central

squared (10+11+6+7), squared from

corner cells (16+13+4+1).

To build any 4x4 magic squares, you need to: build one

with the constant 34.

Example. Build new different 4 x 4 magic squares.

Solution.

Adding up each number found

magic square 4 x 4 or

multiplying it by the same number,

get a new magic square.

Example. Build a magical

a 4 x 4 square that has a magic

the constant is 46.

Solution. Built a familiar magical

square with constant 34.

46 – 34 = 12. 12: 4 = 3

To each number of the magic square

let's add 3.

Before proceeding to solve more difficult examples on magic squares 4 x 4 check again the properties that it has if M = 34.

Examples. 1. Fill the cells of the magic square with magic

constant M =38.

H = 38-(10+7+13)=8 d = 38-(17+4+11)=6 c = 38-(17+4+14)=3

e \u003d 38- (12 + 7 + 8) \u003d 11 p \u003d 38- (17 + 6 + 10) \u003d 5 c \u003d 38- (3 + 12 + 8) \u003d 15

b = 38-(11+7+16)=4 d = 38-(5+7+12)=14 c = 38-(6+11+12)=9

property 1,3,1 properties 2,1,1 t =38-(14+9+13)=2

properties 1,1,1,1

Answer.

Tasks for independent solution

Fill in the cells of the magic square with if the magic is known

constant

K = 46 K = 58 K = 62

Meet the magic squares 5x5 and 6x6

Tasks:

1. Teach how to fill in magic squares.

2. Develop observation, the ability to generalize.

3. Instill a desire for new knowledge, interest in mathematics.

Equipment: computer, multimedia projector with screen, PowerPoint presentation (Appendix 1).

In ancient times, having learned to count and perform arithmetic operations, people were surprised to find that numbers have independent life, amazing and mysterious. By adding different numbers, placing them one after the other or one under the other, they sometimes received the same amount. Finally, dividing the numbers with lines so that each was in a separate cell, they saw a square, any of the numbers of which took part in two sums, and those located along the diagonals even in three, and all the sums are equal to each other! No wonder the ancient Chinese, Indians, and after them the Arabs attributed mysterious and magical properties. (slide 1)

Magic squares appeared in the Ancient East even before our era. One of the surviving legends tells that when Emperor Yu of the Shang Dynasty (2000 BC) was standing on the banks of Luo, a tributary of the Yellow River, a large fish (in other versions, a huge turtle) suddenly appeared, on which there was a drawing of two mystical symbols - black and white circles (slide 2), which was then realized as an image of a magic square of order 3. (slide 3)

The first special mention of such a square was found around the 1st century BC. Until the 10th century AD. magic squares were embodied in amulets, spells. They have been used as talismans throughout India. They were painted on jugs of good luck, medical mugs. Until now, they are used by some eastern peoples as a talisman. They can be found on the decks of large passenger ships as a playground.

So, by magic we mean squares in which the sums of numbers in any column or in any row, as well as along the diagonals, are the same.

Until now, you have used magic squares most often for mental counting. At the same time, several numbers, including the central one, are already placed in the cells of the square. It is required to arrange the remaining numbers so that in any direction a certain amount is obtained.

Task 1. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are given. Some of them are arranged in cells. It is required to arrange the remaining numbers so that the total is 15. (slide 4)

It turns out that all other magic squares made up of the same numbers can be obtained from the given one by symmetry with respect to a row, column, or diagonal, so the numbers in all squares are arranged according to the same rules. (slide 6)

You can notice a number of patterns that make it easier to fill in the cells of the square or make it possible to solve the problem with a smaller number of data in the condition.

For example, in the conditions of problems similar to the previous one, it is not necessary to indicate what amount should be obtained in any direction.

Task 2. Find a way to calculate the sum over the rows, columns, and diagonals from the previous problem.

You can argue as follows: the sum of the numbers in each line is the same, there are 3 such lines, which means the sum of the numbers in each line is three times less than the sum of all the numbers. Therefore, in our example, the sum in each row is 15 (45:3). But this number can be found in other ways: add the three central numbers 4, 5 and 6, or multiply the central number 5 by 3.

Task 3. Numbers are given: 2, 3, 4, 5, 6, 7, 8, 9, 10. It is required to enter them in the cells of the square so that in any direction the sum is the same number. Some of the numbers are already inscribed in the square. (slide 7)

Task 4. Numbers 5, 6, 7, 8, 9, 10, 11, 12, 13 are given. Two of them are inscribed in the cells of the square. Write the rest so that in any direction the sum is the same number. (slide 9)

Let's look at all three filled squares and try to find a number of patterns that will help fill the square with even fewer numbers inscribed in the square. (slide 11)

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

2, 3, 4, 5, 6, 7, 8, 9, 10

5, 6, 7, 8, 9, 10, 11, 12, 13

See what number is in the center of the square? How is it located in the series of given numbers? (slide 12) (In the center of the square, the number that is in the fifth place in our sequence is always written, i.e., equally removed from its left and right edges.)

You can notice a number of other features: in the square on opposite sides of the central number there are numbers that are equally distant from the left and right edges of the sequence. Let's show pairs of corresponding numbers using the example of filling a square with numbers from 1 to 9: (slide 13)

Knowing this, you can fill the square, almost without counting.

See how the numbers next to the central one are located in the square, as well as the numbers written from them through one number. They are connected by lines at the top. (They are located along the diagonals of the square.) And where are the rest of the numbers, which are connected by lines from below? (They are arranged vertically and horizontally.)

Let's check whether such patterns are observed in other squares. (slide 14)

(Yes, such patterns hold.)

So let's sum it up. What properties of magic squares have we found out?

1) To find the sum of the numbers in each column or row, you can multiply the central number by 3.

2) In the center of the square is the number written in the fifth row.

3) In the square on opposite sides of the central number are numbers that are equally distant from the left and right edges of the sequence.

4) The numbers next to the central one and one from it are located along the diagonals of the square. The numbers standing on the edge and through one from it are located in a square vertically and horizontally.

Task 5. Numbers are given: 3, 4, 5, 6, 7, 8, 9, 10, 11. Write them in the cells of the square so that the same number is obtained in any direction. (slide 15)

(Let's find what amount should be obtained in each direction. To do this, multiply the central number 7 by 3. As a result, we get 21. Put the number 7 in the center of the square, on one diagonal of the numbers 6 and 8, on the other - 4 and 10. It remains to arrange the missing numbers: the sum of the numbers written in the first line is 10, 11 is missing before 21, which means that in the empty cell of the top line we write the number 11 (first on the right).Then in the bottom line we write the number 3 (first on the left).In the left column we write the number 5 ( 21 - (6 + 10)), then it remains to write the number 9 in the right column. Thus, we placed all 9 numbers in the cells of the magic square, while not a single number was placed in the square according to the condition of the problem.)

The problem has several solutions, but all squares are obtained from others by symmetry about the midlines or diagonal. (slide 16)

Task 6. Given the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18. Write them in the cells of the square so that in any direction you get the same number in total.

One of the solutions on the slide. (slide 17)

Task 7. Compare the conditions of problems 1 and 6 and think about how you could solve the problem, knowing the solution to problem 1.

(The numbers from Problem 6 are twice the corresponding numbers from Problem 1. Therefore, you can simply double each number of the square from Problem 1 and get the desired square.)

There are various ways to construct magic squares. Consider the method of terraces, which was invented by the ancient Chinese. Following this method, it is necessary to rotate the “natural” number square about the center by half a right angle (slide 19) and separate the table 3´3 with a square frame. (slide 20) With numbers written outside the frame, and forming ledges ("terraces"), we fill in the empty cells at opposite side tables. (slide 21)

Similarly, any square of odd order can be constructed. Let's fill the cells of the 5´5 magic square with numbers from 1 to 25. (slides 22, 23, 24)

To construct a 4´4 magic square, the simplest and most accessible method is the following: in a "natural" square, additional numbers on the main diagonals are swapped, while the rest remain unchanged. (slides 25, 26)

Summing up the lesson

What secret of magic squares did you discover today in class? What helped you in this?

There are several different classifications of magic squares.

fifth order, designed to somehow systematize them. In the book

Martin Gardner [GM90, pp. 244-345] describes one of these methods -

according to the number in the central square. The method is curious, but nothing more.

How many squares of the sixth order exist is still unknown, but there are approximately 1.77 x 1019. The number is huge, so there is no hope of counting them using exhaustive search, but no one could come up with a formula for calculating magic squares.

How to make a magic square?

There are many ways to construct magic squares. The easiest way to make magic squares odd order. We will use the method proposed by the French scientist of the 17th century A. de la Louber (De La Loubère). It is based on five rules, the operation of which we will consider on the simplest magic square 3 x 3 cells.

Rule 1. Put 1 in the middle column of the first row (Fig. 5.7).

Rice. 5.7. First number

Rule 2. Put the next number, if possible, in the cell adjacent to the current one diagonally to the right and above (Fig. 5.8).

Rice. 5.8. Trying to put the second number

Rule 3. If the new cell goes beyond the square above, then write the number in the very bottom line and in the next column (Fig. 5.9).

Rice. 5.9. We put the second number

Rule 4. If the cell goes beyond the square on the right, then write the number in the very first column and in the previous line (Fig. 5.10).

Rice. 5.10. We put the third number

Rule 5. If the cell is already occupied, then write down the next number under the current cell (Fig. 5.11).

Rice. 5.11. We put the fourth number

Rice. 5.12. We put the fifth and sixth number

Follow Rules 3, 4, 5 again until you complete the entire square (Fig.

Isn't it true, the rules are very simple and clear, but it's still quite tedious to arrange even 9 numbers. However, knowing the algorithm for constructing magic squares, we can easily entrust the computer with all routine work, leaving ourselves only creative work, that is, writing a program.

Rice. 5.13. Fill in the square with the following numbers

Project Magic squares (Magic)

Field set for the program magic squares quite obvious:

// PROGRAM FOR GENERATION

// ODD MAGIC SQUARE

// BY THE DE LA LOUBERT METHOD

public partial class Form1 : Form

//Max. square dimensions: const int MAX_SIZE = 27; //var

intn=0; // square order int [,] mq; // magic square

int number=0; // current number to square

intcol=0; // current column int row=0; // current line

The de la Louber method is suitable for making odd squares of any size, so we can let the user choose the order of the square, while reasonably limiting the freedom of choice to 27 cells.

After the user presses the coveted button btnGen Generate! , the btnGen_Click method creates an array to store numbers and passes into the generate method:

// PRESS THE "GENERATE" BUTTON

private void btnGen_Click(object sender, EventArgs e)

//order of the square:

n = (int)udNum.Value;

//create an array:

mq = new int ;

//generate magic square: generate();

lstRes.TopIndex = lstRes.Items.Count-27;

Here we begin to act according to the rules of de la Louber and write the first number - one - in the middle cell of the first row of the square (or array, if you like):

//Generate the magic square void generate()(

//first number: number=1;

//column for the first number - middle: col = n / 2 + 1;

//line for the first number - the first one: row=1;

//square it: mq= number;

Now we sequentially add the rest of the cells in cells - from two to n * n:

// move on to the next number:

We remember, just in case, the coordinates of the actual cell

int tc=col; int tr = row;

and move to the next cell diagonally:

We check the implementation of the third rule:

if (row< 1) row= n;

And then the fourth:

if (col > n) ( col=1;

goto rule3;

And fifth:

if (mq != 0) ( col=tc;

row=tr+1; goto rule3;

How do we know that there is already a number in the cell of the square? - Very simple: we prudently wrote zeros in all cells, and the numbers in the finished square are greater than zero. So, by the value of the array element, we will immediately determine whether the cell is empty or already with a number! Please note that here we need those cell coordinates that we remembered before searching for the cell for the next number.

Sooner or later, we will find a suitable cell for the number and write it to the corresponding array cell:

//square it: mq = number;

Try another way to organize the check of the admissibility of the transition to the

wow cell!

If this number was the last, then the program has fulfilled its obligations, otherwise it voluntarily proceeds to provide the cell with the following number:

//if not all numbers are set, then if (number< n*n)

//go to the next number: goto nextNumber;

And now the square is ready! We calculate its magic sum and print it on the screen:

) //generate()

Printing the elements of an array is very simple, but it is important to take into account the alignment of numbers of different "lengths", because a square can contain one-, two-, and three-digit numbers:

//Print the magic square void writeMQ()

lstRes.ForeColor = Color .Black;

string s = "Magic sum = " + (n*n*n+n)/2; lstRes.Items.Add(s);

lstRes.Items.Add("" );

// print the magic square: for (int i= 1; i<= n; ++i){

s="" ;

for (int j= 1; j<= n; ++j){

if (n*n > 10 && mq< 10) s += " " ; if (n*n >100 && mq< 100) s += " " ; s= s + mq + " " ;

lstRes.Items.Add(s);

lstRes.Items.Add("" ); )//writeMQ()

We launch the program - the squares are obtained quickly and feast for the eyes (Fig.

Rice. 5.14. Quite a square!

In the book by S. Goodman, S. Hidetniemi Introduction to the development and analysis of algorithms

mov , on pages 297-299 we will find the same algorithm, but in a "reduced" presentation. It is not as "transparent" as our version, but it works correctly.

Add a button btnGen2 Generate 2! and write the algorithm in the language

C-sharp to the btnGen2_Click method:

//Algorithm ODDMS

private void btnGen2_Click(object sender, EventArgs e)

//order of the square: n = (int )udNum.Value;

//create an array:

mq = new int ;

//generate magic square: int row = 1;

int col = (n+1)/2;

for (int i = 1; i<= n * n; ++i)

mq = i; if (i % n == 0)

if (row == 1) row = n;

if (col == n) col = 1;

//square completed: writeMQ();

lstRes.TopIndex = lstRes.Items.Count - 27;

We click the button and make sure that “our” squares are generated (Fig.

Rice. 5.15. Old algorithm in a new guise

Testing with Chaturanga Shorin Alexander

5.2.1 About the magic of numbers. What are magic squares

A lot can be said about the magic of numbers. As an example, at the beginning of this study, we already mentioned the number 4. A lot can be said in this way about any number.

For example, the number 1 is one, the beginning of everything. Number 2 - separation, the opposite of the two sexes. 3 - triangle ... And so on. This is a very fertile topic, which you can delve into endlessly.

Therefore, let's leave it and move on to the magic squares, which are directly related to Chaturanga.

Magic squares are square tables of integers that have unique properties: for example, the sums of numbers along any row, any column, and any of the two main diagonals are equal to the same number.

It is believed that magic squares were invented in ancient China, and were also known in ancient India, where Chaturanga originates from. In particular, this is proved by N. M. Rudin in his book “From the Magic Square to Chess”.

According to legend, during the reign of Emperor Yu (c. 2200 BC), a sacred tortoise surfaced from the waters of the Yellow River, on the shell of which mysterious hieroglyphs were inscribed. These signs are known as lo-shu and are equivalent to a magic square. In the 11th century they learned about magic squares in India, and then in Japan, where in the 16th century. Magic squares have been the subject of an extensive literature. He introduced Europeans to magic squares in the 15th century. Byzantine writer E. Moshopoulos. The first square invented by a European is A. Dürer's square depicted on his famous engraving "Melancholy 1". The date of the engraving (1514) is indicated by 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 built squares of the 3rd, 4th, 5th, 6th, 7th, 8th and 9th orders, which were associated with the astrology of the 7 planets. There was a belief that a magic square engraved on silver protected from the plague. Even today, among the attributes of European soothsayers, one can see magic squares.

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

Each element of the magic square is called a cell. A square whose side is 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 constant of the square and is equal to S= n(n 2 + 1)/2. Proved that n– 3. For a square of the 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. Cells that are symmetrical about the center of the square are called skew-symmetric.

Magic squares can be constructed, for example, using the method of a 17th-century French geometer. A. de la Lubera.

According to the method of A. de la Loubert, the magic square 5 × 5 can be constructed as follows:

The number 1 is placed in the central cell of the top row. All natural numbers are arranged in a natural order cyclically from bottom to top in the cells of the diagonals from right to left. Having reached the upper edge of the square (as in the case of the number 1), we continue to fill in 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 in the diagonal coming from the left cell with the line above. Having reached a filled cell (number 5) or a corner (number 15), the trajectory descends one cell down, after which the filling process continues.

It turns out such a magic square:

You can also use the method of F. de la Hire (1640-1718), which is based on two original squares. 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 occurs twice in one row or in one 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. The cell-by-cell sum of these two squares forms a magic square. This method is also used in the construction of squares of even order.