How to solve magic squares? Logic games Puzzle with numbers in a square of different colors.

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Children - preschoolers very quickly accumulate more and more new knowledge, skills and experience. They are developing speech. They take over in different ways mental activity, improve in all aspects of their mental development.

But very often the mental education of preschool children comes down to providing the child with as much information and knowledge as possible about the world around us. This approach is quite simplistic and obviously incorrect. After all, it’s easy to put it into the head of a preschooler large number knowledge is clearly not enough to mentally develop a child.

Much more important when preparing for school in the process of mental education of a preschooler is the need to develop general methods cognitive activity(this is the ability to compare, analyze, evaluate, generalize). And most importantly, it is also necessary to ensure that the child himself strives to acquire more and more new knowledge.

The ability to compare, analyze, evaluate, and generalize can be developed in a child by solving various puzzles.

Puzzles.

Puzzles, they are also called logic games. Such games are very useful for developing logical thinking and intelligence in children.
The rate of mental development and growth of children in our time is very high, so parents need to pay a lot of attention to the development of their children’s thinking. It is necessary to teach children to think independently, reason, analyze, compare objects and phenomena. Puzzles are logic problems which will help develop logical thinking and intelligence in children.

The magic square is a form of puzzle. The magic square comes with both pictures and numbers. With drawings magic squares may even be the simplest for children starting from 4-5 years old. And to the most difficult ones, for children school age, where there are a lot of different elements that need to be analyzed and only then made the appropriate conclusion.

What is a magic number or magic square - this is a square table, in our case of nine cells, three vertically and three horizontally, in which numbers are written in each cell so that the sum of the numbers in rows, in columns and from corner to corner , that is, the diagonals are the same. This is easy to see in the figure.

Magic square with drawings for children preschool age. In this magic square, three different objects should be located vertically and horizontally in each row. You need to determine which item should be in the empty cell. What needs to be done? It is necessary to analyze the entire square, that is, divide the whole into parts:
1. Please note that there are 9 cells in the magic square; these cells contain three objects: the sun, a mushroom and a flower.
2. Please note that in each row there are three different objects located vertically and horizontally (a sun, a mushroom and a flower).

Now we combine everything that we analyzed into a single whole and see that in the first vertical row there is a mushroom and a sun, but the empty cell is missing a flower.

And now logical tasks - puzzles:

Determine what figure should be instead of the question mark?

Solve the following number magic squares. What number in the following squares should be obtained when adding along rows, columns and from corner to corner, that is, by diagonals, is easy to find out by the numbers that are placed in the cells. Once you know this number, you can easily calculate which numbers need to be placed in the five empty cells.

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There are various techniques for constructing squares of single parity and double parity.

There are an unimaginable number of mathematical riddles. Each of them is unique in its own way, but their beauty lies in the fact that to solve it you inevitably need to come to formulas. Of course, you can try to solve them, as they say, but it will be very long and practically unsuccessful.

This article will talk about one of these riddles, and to be more precise, about the magic square. We will look in detail at how to solve the magic square. 3rd grade of the general education program, of course, this goes through, but perhaps not everyone understood or does not remember at all.

What is this mystery?

Or, as it is also called, magic, is a table in which the number of columns and rows is the same, and they are all filled with different numbers. The main task is that these numbers add up vertically, horizontally and diagonally to the same value.

In addition to the magic square, there is also a semi-magic square. It implies that the sum of numbers is the same only vertically and horizontally. A magic square is “normal” only if one was used to fill it.

There is also such a thing as a symmetrical magic square - this is when the value of the sum of two digits is equal, while they are located symmetrically with respect to the center.

It is also important to know that squares can be of any size other than 2 by 2. A 1 by 1 square is also considered magical, since all conditions are met, although it consists of one single number.

So, we have familiarized ourselves with the definition, now let’s talk about how to solve a magic square. The 3rd grade school curriculum is unlikely to explain everything in as much detail as this article.

What are the solutions?

Those people who know how to solve a magic square (grade 3 knows for sure) will immediately say that there are only three solutions, and each of them is suitable for different squares, but still one cannot ignore the fourth solution, namely “at random” . After all, to some extent there is a possibility that an ignorant person will still be able to solve this problem. But we will throw this method into the long box and move directly to formulas and methods.

First way. When the square is odd

This method is only suitable for solving a square that has an odd number of cells, for example, 3 by 3 or 5 by 5.

So, in any case, it is initially necessary to find the magic constant. This is the number that is obtained by summing the numbers diagonally, vertically and horizontally. It is calculated using the formula:

In this example, we will consider a three by three square, so the formula will look like this (n is the number of columns):

So, we have a square in front of us. The first thing to do is to enter the number one in the center of the first line from the top. All subsequent numbers must be placed one square to the right diagonally.

But here the question immediately arises: how to solve the magic square? 3rd grade is unlikely to use this method, and most will have a problem, how to do it this way if this cell does not exist? To do everything correctly, you need to use your imagination and draw a similar magic square on top and it will turn out that the number 2 will be in it in the lower right cell. This means that in our square we enter the two in the same place. This means that we need to enter the numbers so that they add up to 15.

Subsequent numbers are entered in exactly the same way. That is, 3 will be in the center of the first column. But it will not be possible to enter 4 using this principle, since there is already a unit in its place. In this case, place the number 4 under 3 and continue. The 5 is in the center of the square, the 6 is in the top right corner, the 7 is below the 6, the 8 is in the top left, and the 9 is in the center of the bottom line.

You now know how to solve the magic square. I passed Demidov's 3rd grade, but this author had slightly simpler tasks, however, knowing this method, you will be able to solve any similar problem. But this is if the number of columns is odd. But what should we do if, for example, we have a 4 by 4 square? More on this later in the text.

Second way. For a double parity square

A double parity square is one whose number of columns can be divided by both 2 and 4. Now we will consider a 4 by 4 square.

So, how to solve a magic square (3rd grade, Demidov, Kozlov, Tonkikh - a task in a mathematics textbook) when the number of its columns is 4? It's very simple. Easier than the example before.

First of all, we find the magic constant using the same formula that was given last time. In this example, the number is 34. Now you need to arrange the numbers so that the sum vertically, horizontally and diagonally is the same.

First of all, you need to paint over some cells, you can do this with a pencil or in your imagination. We paint over all the corners, that is, the upper left cell and the upper right, the lower left and the lower right. If the square were 8 by 8, then you need to paint over not one square in the corner, but four, measuring 2 by 2.

Now you need to paint the center of this square, so that its corners touch the corners of the already shaded cells. In this example, we will get a 2 by 2 square in the center.

Let's start filling it out. We will fill in from left to right, in the order in which the cells are located, only we will enter the value in the shaded cells. It turns out that we enter 1 in the upper left corner, 4 in the right corner. Then we fill in the central one with 6, 7 and then 10, 11. The lower left 13 and 16 in the right. We think the order of filling is clear.

We fill out the remaining cells in the same way, only in descending order. That is, since the last number entered was 16, then at the top of the square we write 15. Next is 14. Then 12, 9 and so on, as shown in the picture.

Now you know the second way to solve the magic square. Grade 3 will agree that the double parity square is much easier to solve than the others. Well, we move on to the last method.

Third way. For a square of single parity

A square of single parity is a square whose number of columns can be divided by two, but not by four. In this case it is a 6 by 6 square.

So, let's calculate the magic constant. It is equal to 111.

Now we need to visually divide our square into four different squares 3 by 3. You will get four small squares measuring 3 by 3 in one large 6 by 6. Let's call the top left one A, the bottom right one B, the top right one C, and the bottom left one D.

Now you need to solve each small square using the very first method given in this article. It turns out that in square A there will be numbers from 1 to 9, in B - from 10 to 18, in C - from 19 to 27 and D - from 28 to 36.

Once you have solved all four squares, work will begin on A and D. It is necessary to highlight three cells in square A visually or using a pencil, namely: the upper left, central and lower left. It turns out that the highlighted numbers are 8, 5 and 4. In the same way, you need to select square D (35, 33, 31). All that remains to be done is to swap the selected numbers from square D to A.

Now you know the last way to solve the magic square. Grade 3 doesn't like the square of single parity the most. And this is not surprising, of all those presented it is the most complex.

Conclusion

After reading this article, you learned how to solve a magic square. Grade 3 (Moro is the author of the textbook) offers similar problems with only a few filled cells. There is no point in considering his examples, since knowing all three methods, you can easily solve all the proposed problems.

How to solve magic squares?

A puzzle like Sudoku is commonly called a magic square. This is a square whose cells are filled with numbers so that the sum at the end of any row, column and diagonal is the same. In magic square puzzles, some numbers are missing, and you need to arrange them so as to meet the condition described above equal amount. How to solve magic squares?

Methods for solving magic squares

In order for the solution of magic squares to be correct, you need to know the very magic sum that should be obtained when adding numbers in rows, columns and diagonals. After this, placing the missing numbers becomes much easier. How to find this amount?

Method 1

The simplest version of a magic square is when one of the rows, one of the columns or one of the diagonals is completely filled with numbers. In this case, all that remains is to calculate the sum of these numbers and select solutions.

Method 2

The sum of the numbers at the ends of rows, columns and diagonals can be calculated using special formulas. In this case, the formula for squares with an even number of cells in one row will differ from squares with an odd number of cells.

So, for even squares the following formula is suitable:

• n + ((n+1) * n * (n-1) / 2) , where n is the number of cells in one row.

For odd squares the formula is:

• n * (n 2 +1) / 2, where n is also the number of cells in one row.

Example solution

Let's consider solutions to a magic square of nine cells with numbers from 1 to 9. First, let's calculate the sum that should be obtained at the ends. We have 3 cells in one line, that is, n = 3. Substitute the value into the formula:

• 3 * (3 2 +1) / 2 = 3 * 10 / 2 = 15

Now we select the numbers so that the sum is 15.

Next, the algorithm will require a little spatial imagination. Place the number 1 in the middle of the top line. We place each next number on the right diagonally upward. We try to put 2. But there are no cells there, if we substitute another identical imaginary square above our square, then the number 2 will be in the lower right corner of this
new square. We transfer it to our square and place it in the lower right corner. We also put the number 3 on the right diagonally upward - and again there is no cell there, using an imaginary square we find out that its place is in the middle of the left column. We put the number 4 according to the same principle, but this cell is occupied by one - in this case we put it directly under the number 3. The number 5 diagonally up and to the right of 4 is in the very center, and the number 6 is in the upper right corner. The number 7, with the help of imagination, should have ended up in the lower left corner. But there is already a 4 there, so we put it directly under the number 6. The number 8 appears with the help of an imaginary square in the upper left corner, and the number 9 in the remaining cell in the middle of the right column. The general algorithm is as follows: put the next number at the top right diagonally, if there is no space, use an imaginary square, and if the cell is occupied, then put the number directly below the previous one.