Как составить магический квадрат онлайн

A magic square of size N is an array of N rows and N columns containing numbers (usually positive integers) so that the sum of each row, each column and each diagonal are all equal to the same value, called magic sum (or magic constant).

Formula: Set number 1 on the left of the median row, the other numbers are written following the rule: if the cell is empty, in the cell in the bottom right of the previous one, else directly to the left of the occupied cell. When the cell does not exist (after the last column or the last row), go to the other side of the square/matrix.

The magic way to solve magic squares is to set the equations that govern each row and column, with unknowns. The constraints being that the unknowns are all different positive integers (distinct including numbers already placed).

The equations obtained sometimes have several solutions.

The constant values $ M $ of the sums of the magic squares have a minimum value (for non-zero integer positive values).

$$ M = n(n^2+1)/2 $$

For a size 3×3, the minimum constant is 15, for 4×4 it is 34, for 5×5 it is 65, 6×6 it is 111, then 175, 260, …

Any lower sum will force the use of either negative numbers or fractions (not whole numbers) to solve the magic square.

The values can be as large as you want, so the maximum magic sum/value is infinity.

Franklin’s square is a panmagic square with a magic constant of 260.

This is a 3×3 magic square used in Feng Shui which is represented as well

Kaldor’s magic square is a square used in economics, which has nothing to do with digits or numbers of mathematics but rather with concepts from economic policy.

Yes, there are magic cubes, their magic value is $$ M = n(n^3+1)/2 $$ (which may or may not have magic diagonals)