Logic games. How to solve magic squares? What is this riddle

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

This article will talk about one of these mysteries, and to be more precise - about the magic square. We will analyze in detail how to solve the magic square. Grade 3 of the general education program, of course, it passes, but perhaps not everyone understood or does not remember at all.

What is this riddle?

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 in the sum vertically, horizontally and diagonally give the same value.

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

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

It is also important to know that squares can be 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 a single number.

So, we got acquainted with the definition, now let's talk about how to solve the magic square. Grade 3 of the school curriculum is unlikely to explain everything in such 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 the fourth solution cannot be ignored, namely “at random” . After all, to some extent there is a possibility that an unknowing person will still be able to solve this problem. But we will discard this method in a long box and go directly to the 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 will result from the sum of the digits 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. 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 cell to the right diagonally.

But here the question immediately arises, how to solve the magic square? Grade 3 is unlikely to use this method, and most will have a problem, how to do it in this way, if this cell does not exist? To do everything right, you need to turn on 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 a deuce 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 4 cannot be entered according to this principle, since there is already a unit in its place. In this case, we place the number 4 under 3, and continue. Five - in the center of the square, 6 - in the upper right corner, 7 - under 6, 8 - in the upper left and 9 - in the center of the bottom line.

You now know how to solve the magic square. Grade 3 of Demidov passed, but this author had a little simpler tasks, however, knowing this method, it will be possible to solve any such problem. But this is if the number of columns is odd. But what if we have, for example, a 4 by 4 square? More on this later in the text.

The second way. For a square of double parity

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 the magic square (Grade 3, Demidov, Kozlov, Tonkikh - a task in a mathematics textbook) when the number of its columns is 4? And 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 line up the numbers so that the sum of the vertical, horizontal and diagonal 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, lower left and lower right. If the square were 8 by 8, then it is necessary to paint over not one cell in the corner, but four, 2 by 2 in size.

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

Let's start filling. 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 filled cells. It turns out that we enter 1 in the upper left corner, 4 in the right. Then we fill in the central one 6, 7 and then 10, 11. The lower left 13 and the right one - 16. We think the filling order is clear.

The remaining cells are filled in exactly the same way, only in descending order. That is, since the last number entered was 16, we write 15 at the top of the square. Then 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 square of double parity is much easier to solve than others. Well, we move on to the last method.

The third way. For a single parity square

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's a 6x6 square.

So, we calculate the magic constant. It is equal to 111.

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

Now you need to solve each small square using the very first method that is 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 select three cells visually or with a pencil in square A, namely: the top left, center and bottom left. It turns out that the selected numbers are 8, 5 and 4. In the same way, it is necessary to select the square D (35, 33, 31). All that remains to be done is to swap the highlighted numbers from square D to A.

Now you know the last way to solve the magic square. 3rd class squared single parity dislikes the most. And this is not surprising, of all presented it is the most difficult.

Conclusion

After reading this article, you have learned how to solve the magic square. Grade 3 (Moro - the author of the textbook) offers similar tasks with only a few filled cells. It makes no sense to consider his examples, since knowing all three methods, you can easily solve all the proposed tasks.

Hello!

Children - preschoolers very quickly accumulate more and more new knowledge, skills and experience. They develop speech. They master different ways of 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 rather simplistic and obviously wrong. After all, simply putting a large amount of knowledge into the head of a preschooler child is clearly not enough to mentally develop a child.

Much more important in preparing for school in the process of mental education of a preschooler is the need to develop common ways 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 obtain more and more new knowledge.

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

Puzzle.

Puzzles, they are also called logic games. Such games are very useful for the development of logical thinking and ingenuity in children.
The rate of mental development and growth of children in our time is very high, so parents should pay much attention to the development of the thinking of their children. It is necessary to teach children to think independently, reason, analyze, compare objects and phenomena. Puzzles are logical tasks that will help develop logical thinking and quick wit in children.

The magic square is one of the forms of puzzles. The magic square happens both with drawings and with numbers. With drawings, magic squares can even be for children, starting from 4-5 years old, the simplest. And to the most difficult, for children school age, where there are a lot of different elements that need to be analyzed and only then make an appropriate conclusion.

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

Magic square with drawings for children preschool age. In this magic square, in each row, vertically and horizontally, three different objects should be located. It is necessary to determine what object should be in an 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, in these cells there are three objects: the sun, a mushroom and a flower.
2. Please note that in each row vertically and horizontally there are three different objects (sun, mushroom and flower).

And now we combine everything that we have analyzed into a single whole and we see that in the first vertical row there is a mushroom and the sun, but there is not enough flower in the empty cell.

And now logical tasks - puzzles:

Determine what shape should be instead of the question mark?

Solve the following numerical magic squares. What number in the following squares should be obtained when added along rows, columns and from corner to corner, that is, along diagonals, it is easy to find out by the numbers that are placed in the cells. When you recognize this number, you can easily calculate what numbers you need to put in five empty cells.

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I love games where you have to think. Therefore, our series of articles "top 10" smoothly flows into puzzles. Today I will talk about ten puzzles with numbers. When I rushed to compile this rating, I ran into the problem of finding ten good games, despite the fact that there are a lot of digital puzzles in the App Store! The bad thing is that there are a lot of clones, repetitions and low-quality crafts ... But when the top was compiled, I realized that everyone would find something new in it! Even I met three great games. Go!

threes!

There are numbers on the playing field. The player can move all the numbers to any of the 4 sides. At the same time, if the wall interferes with the movement of any row or column and stand nearby:

a) identical numbers greater than or equal to 3
b) 1 and 2

then they add up and instead of two digits a third appears - the sum. The goal is to score as many points as possible. The game is endless, but it is very difficult to score many points.

After the release of Threes! in the App Store went the dominance of clones under the name "2048".

Shikaku

A simple and not pop puzzle from the creators of Sudoku. The goal in this game is to divide the field with numbers into rectangles so that the area of ​​the rectangles is equal to the number inside it. For the iPad, there is only one implementation of this game.

Numtris: A game of logic and numbers

This is an original adventure game. Tetris with numbers. Numbers fall from above and you need to either collect them according to the Threes principle (1 and 2 will give 3), or remove them by collecting several identical ones (for example, four identical fours). Numtris has a full campaign with many missions. The missions are varied: from holding out for 40 seconds to killing the monster... You can compete with your friends both online and on the same iPad.

The game is very stylish with nice graphics. I recommend trying it, since it's free.

Download Numtris for free (with in-app purchases)

GREG - A mathematical puzzle game

An interesting game for speed and the ability to quickly add numbers. On the field 4 by 4 there are numbers. It is necessary to dial the sum from these numbers so that the number in the circle on top is obtained. As soon as the number is collected, it changes and it is necessary to select the numbers again. The less you use some numbers on the field, the more they heat up ... After 5 such “heatings”, the game may end. Reset occurs after each level. At the end, the game rewards you with some title. Can you beat Math Genius?

Few people loved mathematics in childhood, but mathematical puzzles on the Internet always become hits, because their solution usually does not require in-depth knowledge, but it does require ingenuity and innovative thinking. We invite you to test yourself on the five main logical puzzles of this year.

Task #1

Kumar Ankit asked Facebook users to count how many triangles are shown in his picture. Practically none of the users coped with a seemingly simple task to count the figures. Many are close to the correct answer, but most lack a little attentiveness.

Answer:

There are 24 triangles inside the large triangle, it's easy to calculate, but most users did not pay attention to another triangle hidden in the author's signature. Thus, there are 25 triangles in the picture.

Task #2

An unusual puzzle with two solutions was offered to Internet users by the creators of the site gotumble.com. According to them, one solution to the puzzle is simpler, about 10% of people can find it, but one person in a thousand can reach the second solution. Try it yourself.

Answer:

First decision consists in adding to each following example the result of the previous one. So, adding 5 to the sum of 2 and 5, we get 12. Adding 12 to the sum of 3 and 6, we get 21. And so on. In this case, the correct answer to the puzzle would be 40.

And here second solution, which only one person in a thousand reaches, is to add the first digit of the example with the product of two digits:

2 + 2*5 = 12, 3 + 3*6 = 21, 8 + 8*11 = 96.

Task #3

We have a triangle with four parts, but if we rearrange the parts, an empty square appears in it. How can this be?

Answer:

This is not an optical illusion at all. It's all about the different angles of inclination of the hypotenuse of the red and turquoise triangle - hence the different sizes of the figures.

Task #4

Columnist The Guardian Alex Bellos invited readers to solve a problem that is part of the math final exam in some countries. According to statistics, only one person out of 10 solves it.

We have a cylinder around which a thread is wrapped symmetrically four times. The circumference of the cylinder is 4 cm, and its length is 12 cm. You need to find the length of the thread.

Answer:

The task seems too complicated for most schoolchildren, but in fact, you just need to understand that by turning the cylinder onto a plane, we get an ordinary rectangle with sides - 4 and 12 cm, which can be divided into four smaller rectangles with sides - 4 and 3 cm. Thread in this case it will be the hypotenuse of a right triangle and its length in each of the four figures can be calculated using a simple school formula, it is 5 cm. As a result, the total length of the thread is 20 centimeters.

Task #5

And finally, the last math puzzle that blew up social media. According to the author of the post, it depicts a riddle that is given as a bonus question to students in Singapore. The compilers of the riddle propose to study the numerical sequence and fill in the four free boxes with the missing numbers.

Answer:

Netizens puzzled over this problem for a long time, but even serious mathematicians could not cope with it. And the Ministry of Education of Singapore disowned this task, saying that it had nothing to do with it. So most likely the puzzle was just someone's cruel joke.

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