China is considered the birthplace of magic squares, since it is in this country that there is the Feng Shui teaching, according to which everything matters to influence the flow of Qi: color, shape, and the location of each individual element in space.
The history of the development of magic numbers
A magic square is an n by n table filled with natural numbers from 1 to n2. In this case, the sums for all columns, rows, and diagonals must match. There are magic squares with even and odd order. The numbers that are written in the fields of the table are called the magic cells of the square, and the total value of the numbers that are in any column, row or diagonal is its constant. Sacred, magical, mysterious, perfect magic square. The solution attracts him with apparent simplicity.
Divine Turtle
The magic square Luo Shu was sent to Emperor Yu by the gods to learn the mysteries of the world. According to legend, about four thousand years ago, a large tortoise Shu emerged from the waters of the turbulent Lo River, which was immediately recognized by people as a deity. Andthis tortoise was in fact unusual, as an unusual dotted pattern was applied to its shell. The dots were drawn in such a way that the ancient philosophers came to the conclusion that the square with the dot-numbers on the shell is a map of the world, which was compiled by the mythical founder of the Huang Di civilization in China. If you add the sum of the numbers in each column, row, and both diagonals of the square, you get the number 15, which is equal to the number of days in the 24 cycles of the Chinese solar year.
Emperor Yu decided that the views of the ancient sages were not far from the truth, and ordered to perpetuate the image of a turtle on paper and sealed it with his imperial seal.
Durer's Magic Square
Albrecht Dürer, the famous German artist, immortalized this amazing representative of the imaginary world of numbers in art on the engraving "Melancholia". The Durer square consists of the first 16 true numbers and has a size of 4 by 4. In each column, row and diagonal, the sum of the numbers is 34. The sums of the other quadruples of numbers placed in the corner cells, in the center and on the sides of the central square, are also equal to 34. But the numbers 15 and 14 in the very bottom line of the square imply the date of creation of the engraving - 1514.
Magic square from Khajuraho
In 1838, a young British officer found among the images of goddesses and gods on the temples of Vishvanath a square of the fourth order, which struck the imagination. The sums over the rows, columns and diagonals of this square were the same and equaled 34. They also coincided over the broken linesdiagonals, which were formed if the square was folded into a torus, and in each of the two directions. For such a magical total value of the numbers, the squares are also called devilish.
From any magic square, by rearranging its constituent numbers, you can get a huge number of new magic squares that will have the same properties. As you know, there are no 2 by 2 squares. And 3 by 3 - there is only one. There are already about 800 squares 4 by 4, as in Durer's engraving, and 5 by 5 are already almost 250 thousand. There is a belief that a magic square engraved on silver protects against the plague. And today you can see them among the attributes of soothsayers in Europe, who attribute various mystical properties to them.
