Magic square: magical, tangential, perfect

The homeland of magic squares is considered China, since it is in this country that there is a teaching of feng shui, according to which everything: the color, the shape, and the location of each individual element in space, is important for influencing the Qi stream.

History of the development of magic numbers

A magic square is a table of size n on n, which is filled with natural numbers from 1 to n2. In this case, the sums for all columns, rows, and diagonals must coincide. There are magic squares with an even and odd order. The numbers that are written in the fields of the table are called the magic squares of the square, and the total value of the numbers that stand in any column, row or diagonal is its constant. A sacred, magical, mysterious, perfect magic square. The solution attracts him with apparent simplicity.

The Divine Turtle

The magic square of Luo Shu was sent to the Emperor Yu gods for the knowledge of the secrets of the world. According to legend, about four thousand years ago from the waters of the stormy Luo River came a large turtle Shu, which was immediately recognized by people as a deity. And this turtle was really unusual, as an unusual dot pattern was applied to its shell. The points were drawn in such a way that the ancient philosophers came to the idea that the square with the numbers-points on the shell is a map of the world, which was 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 equals the number of days in the 24 cycles of the Chinese solar year.

Emperor Yu decided that the considerations of the ancient wise men are not far from the truth, and he ordered to immortalise on the paper the image of the tortoise and bound it with his imperial seal.

Durer's magic square

Albrecht Durer, the famous German artist, immortalized this amazing representative of the imaginary world of numbers in art on the engraving of "Melancholy". The Durer square consists of 16 first true numbers and has a size of 4 by 4. In each column, row and on the diagonal, the sum of the numbers is 34. The sums of the other quadruples of numbers located in the corner cells, at the center and at the sides of the central square, are also 34. But the numbers 15 and 14 in the bottom line of the square mean the date of the engraving - 1514 year.

The Magic Square of Khajuraho

In 1838, a young British officer found among the images of goddesses and gods on the Viswanath temples a square of the fourth order that impressed the imagination. The sums over the rows, columns, and diagonals of this square were the same and equaled 34. They also coincided in broken diagonals that formed if the square was turned into a torus, and in each of the two directions. For such a magical total value of figures, squares are also called diabolical.

From any magic square, you can, by permuting its constituent numbers, get a huge number of new magic squares that will have the same properties. As you know, there are no squares 2 to 2. A 3 to 3 - there is only one. Squares of 4 to 4, as in the engraving of Durer, have already drawn up about 800, and 5 by 5 already almost 250 thousand. There is a belief that a magic square engraved on silver protects against plague. And today you can see them among the attributes of soothsayers in Europe, which attribute to them various mystical properties.