Magic square: how does it work?

Magic square: how does it work?

Magic squares were used in entertaining mathematics from antiquity. In modern culture these tables have gained popularity thanks to the Japanese crossword puzzles of sudoku.

In magic square the integers are distributed in such a way that their sum across, to vertical and diagonal is equal to the same number, so-called magic constant.

Magic square in the cultures of the world

Example of magic square is Lo Shiu representing table 3 on 3. In it are entered figure from 1 to 9 in such a way that in the sum each of lines and diagonal gives number 15.

One Chinese legend narrates as once during flood the king tried to build the channel which would dewater in the sea. Suddenly from the Lo River the turtle with the strange drawing on armor has appeared. It was the grid with figures from 1 to 9 entered in squares. The sum of numbers on each party of square and also on diagonal was 15. This number corresponded to the number of days in each of 24 cycles of the Chinese solar year.

Lo Shiu's square is also called magic square of Saturn. In the lower line of this square in the middle there is number 1, and in the right upper cage number 2.

The magic square is present also at other cultures: Persian, Arab, Indian, European. He was depicted in the engraving Melancholy in 1514 by the German artist Albrecht Duerer.

The magic square on Duerer's engraving is considered the first that ever appeared in the European art culture.

How to solve magic square

It is necessary to solve magic square, filling cells with numbers so that on each line in the sum the magic constant has turned out. Whether the party of magic square can consist of even odd quantity of cells. The most popular magic squares consist of nine (3х3) or sixteen (4х4) cells. There is big variety of magic squares and versions of their decision.

How to solve square with even number of cells

You need the sheet of paper with the square 4х4 drawn on them, simple pencil and eraser. Enter in cells of square of number from 1 to 16, since upper left cage. 1 2 3 45 6 7 89 10 11 1213 14 15 16 Magic constant of this square – 34. Trade places of number on the diagonal line from 1 to 16. For simplicity trade places 16 and 1, and then 6 and 11. As a result on diagonal will stand figure 16, 11, 6, 1.16 2 3 4 5 11 7 8 9 10 6 1213 14 15 1pomenyayte places of number on the second diagonal line. This line begins with figure 4 and comes to an end with figure 13. Trade their places. Now trade places two other numbers – 7 and 10. From top to down on the line of number will be located in such order: 13, 10, 7, 4.16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1esli you will calculate the sum on every line, 34 will turn out. This method works with other squares with even quantity of cells.

Author: «MirrorInfo» Dream Team