Can anyone talk about the magic square

The topic is relatively large.

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

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

<> below is the 4th order of the 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.

Details of the 5th order magic square completed using the vaulting method below.

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

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.

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.

To put it simply, if you are interested in the Magic Quadrant, you can go to my space and have a look.

The so-called magic square (magic square), also called vertical and horizontal graph, is to put n2 natural numbers starting from 1 in the square matrix of n n; In a certain layout, the sum of the numbers in the rows, columns, and two diagonals is exactly the same.

For the specific construction method of the magic square, please refer to "The Construction of the Magic Square".

The magic square is a widely circulated mathematical game, which is said to have been discovered as early as when Dayu controlled the water. The magic square is characterized by an array of n squares composed of natural numbers, called the magic square of order n, and the sum of the numbers in each row, column, and two diagonals is equal.

The specific method of the Robese method is as follows:

Put 1 (or the smallest number) in the middle of the first line; The remaining n2-1 numbers are arranged according to the following rules: 1) Each number is placed in the upper right box of the previous number; 2) If the number is placed in a cell that is beyond the top row, then put it in the bottom row.

It should still be placed in the right column; 3) If the number is placed in the rightmost column, then put it in the leftmost column, still in the previous row; 4)。

If the number is to be placed in a cell that is beyond the top row and beyond the rightmost column, then place it in the same column on the next row of the previous number; 5) If the number to be placed in the box has already been filled in, the processing method is the same as 4).

Categories: Education Science.

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 graph.

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 governed, moved by the flowers so the Yellow River jumped out of a dragon horse, carrying a picture on his back, but as a gift to him, this is the "river map", is the earliest magic square Fu Xi rented through the "river map" and deduced the gossip, later Yu when the flood, a big turtle surfaced in 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.