What kind of application does the magic square have in life?

It's good and makes the mind active.

Nth order magic square. It is to fill in the square of n n with the number of n 2 [n squared], rows, columns, and diagonals.

The sum of values is a perfect magic square, and the rows, columns, and values are equal to the imperfect magic square. This sum value is called the magic sum value.

The formula for the sum value of an nth-order magic square is:

nn=1/2xn(n2+1)

Note: n2 is the square of n].

Merzirac method to generate odd-order magic squares.

Place 1 in the square centered in the first row, and fill in the top right with ....If there is already a number in the upper right, move it down one square to continue filling. As shown in the figure below, the fifth-order magic square generated by the merziral method is as follows

Merzirac method, some people also call it stair method, I call it diagonal step, that is, take x+y diagonal step (the numbers are filled in the order of the upper right), -y jump step (if there are numbers in the upper right or out of the diagonal, then move down one square to continue to fill in).

In fact, the diagonal step method can be filled in the numbers in 4 directions in turn, that is, the upper right, the lower right, the upper left, and the lower left 4 directions, and each oblique step can have 2 kinds of jumping steps, namely left (right) skipping steps, up (down) skipping steps.

For x+y ramps, the corresponding skips can be -x, -y. [Remember, a skip step is the opposite of the x (or y) of the x+y oblique step.] If you take a diagonal step in the upper right direction, skipping a step is a left (or down) step; Take a diagonal step in the lower left direction, and a jump step is a step to the right (or up); And so on and so forth

Categories: Education, Science, >> Science & Technology.

Problem description: The minimum day is in the head**, don't forget to fill in obliquely, write on the top and write on the bottom, and write on the left on the right.

Explanation: Fill in the smallest number in the first line of the positive **, and then fill in the upper right corner diagonally, if it is out of the box, write the bottom of the corresponding column, if the right is out of the box, write the left side of the corresponding line. What does "out of line" mean?

I hope you can give an example, thank you.

Analysis: The magic square, also known as the Rubik's Cube, the square or hall square, it first originated in China. Yang Hui, a mathematician of the Song Dynasty, called it a vertical and horizontal diagram.

The so-called vertical and horizontal diagram is a square matrix from 1 to n 2, and these n 2 natural numbers are arranged into n rows and n columns according to the law of a hui. It has the property of arranging the appropriate numbers on a table of various geometric shapes, and if a simple logical operation is performed on these numbers, the final sum or product is exactly the same, regardless of which route is taken. Regarding the origin of the magic square, there are "river maps" and "Luo Shu" in China.

Legend has it that in ancient times, Fu Xi took the world, the country was well organized, moved by the flowers so the Yellow River jumped out of a dragon horse, the back of the wax oak with a picture, but as a gift to him, this is the "river map", is the earliest magic square Fu Xi rented by the "river map" and deduced the gossip, later Yu flood, a big turtle floated out of the water, its back has a picture and words, people call it "Luoshu". There are 45 black and white circles drawn by "Luo Shu". Represent these small circles and numbers that are joined together to get nine.

These nine numbers can form a vertical and horizontal chart, and people call the magic square with nine numbers, 3 rows and 3 columns called the 3rd order magic square, in addition, there are 4th and 5th orders.

Later, after research, people came up with the formula for calculating the sum of all the numbers in each row, column, and diagonal of any order magic square as:

nn=1/2n(n 2+1)

where n is the order of the magic square, and the number sought is nn

The magic square was first recorded in the Spring and Autumn Period of China in 500 B.C. in the "Da Dai Li", which shows that the Chinese people already knew the arrangement of the magic square as early as 2500 years ago. Abroad, in 130 A.D., the Greek Seon first mentioned the magic square.

China not only has the right to invent the magic square, but also is a country that conducts in-depth research on the magic square. The mathematician Yang Hui of the 13th century A.D. had already compiled the magic square of the 3 10th order, which was recorded in his book "The Algorithm of the Continuation of the Ancient Extraction Hall" written in 1275. In Europe, it was not until 574 that the famous German painter Dio Gong drew a complete 4th magic square.

Out of the box is the next number of the number just filled in should be filled in the upper right corner, but the upper right corner is no longer in this 3 * 3 side box, it should be filled in the lower box of the previous number.

The magic square is very fun, you can draw it yourself and try it to know.

The fourth-order magic square is the simplest double-even magic square, and its composition method is two sentences:

sequential fill-in; Symmetrically swap numbers with a central point]. Take the fourth-order magic square composed of 1-16 as an example:

1. First put 1 in any of the 4 corners of the fourth-order Lu Eye Magic Square, and fill in the remaining numbers in order in the same direction.

2. Symmetrically swap numbers with a center point. (There are two methods of symmetrical exchange).

1) Symmetrically exchange the number of diagonal mu or on the center point (i.e., -10 interchange) to complete the magic square, and the magic sum value = 34.

2) Symmetrically exchange the number on the non-diagonal with the center point of Xunwu (i.e., -9 swap) to complete the magic square, and the magic sum value = 34.

What kind of 16 numbers make up a fourth-order magic square? [4 groups of numbers in a group of 4 numbers (a total of 16 numbers), the symmetry between groups is equal, and the number of each group is symmetrical and equal, so that 16 numbers can form a fourth-order magic square.

To put it simply, it is to fill in n n numbers into the square of n n in a certain way to form a number matrix, so that the sum (or product) of n numbers on each row, each column and two diagonals is equal, we call this number matrix n order sum magic square (or n order product magic square), and this sum (or product) is called the magic sum value (or magic product value).

Generally speaking, when we talk about nth order magic squares, we mean n order and magic squares.

The figure below shows a 3rd order magic square completed with 1-9, and the magic sum value = 15

Here's a 4th order magic square:

Spring method to generate double even magic square formulas:

Sequential fill-in, symmetrically swapping numbers at the center point. 】

The 4th order of the magic square is the simplest double dual magic square, and its method:

The first step is to fill in the numbers in order, first put 1 in any of the 4 corners of the 4th order magic square, and fill in the remaining numbers in order in the same direction.

In the second step, the numbers are symmetrically swapped at the center point. (There are two methods of symmetrical exchange).

Method 1: Symmetrically swap the numbers on the diagonal with the center point (i.e., -10 swap) to complete the magic square, and the magic sum value is 34.

Method 2: Symmetrically swap the numbers on the non-diagonal with the center point (i.e., -9 swap) to complete the magic square, and the magic sum value is 34.

There are 880 methods for the 4th order of the magic square.

Below is a 5th order magic square that is done using a vaulting method. (This vaulting method is only applicable to nth-order fantasy squares where n is not a multiple of 3).

Put the smallest number 1 in any square, take 1 step to the right, take 2 steps up and jump step, and fill in ....If there is already a number in the falling square, go back one square and continue to fill in, and you can complete the magic square.

The magic square done with this method is called a perfect magic square. To put it simply, it is to spread out any of the above-mentioned 5th order magic squares as tiles, and then take any 5 or 5 squares to form a 5th order magic square. If you are interested, you may want to give it a try

Now let's talk about the product of the magic square, where the product of the numbers in each row, column, and two diagonals is equal.

The figure below shows a set of 3rd order product magic squares.

The product of the numbers in each row, each column, and two diagonals is equal to 216. This is the magic square with the smallest product of different 9 natural numbers.

The product of the numbers in each row, column, and two diagonals is equal to 1000.

A magic square of order n is a square matrix of n order consisting of the first n 2 (n to the 2nd power) natural numbers, and the sum of n numbers contained in each row, column and two diagonals is equal.

2. Magic Square is also a traditional Chinese game. In the old days, it was more common in government offices and schools. It is a natural number from one to several numbers arranged in a square of vertical and horizontal numbers, so that the sum of several numbers in the same row, in the same column, and on the same diagonal is equal.

3. The number of magic squares (number series A006052 in OEIS) has not been solved.

Fifth-order magic square, will you fill it in?