Are perfect squares for natural numbers only

Magic squares


A natural (normal) magic square of order (edge ​​length) n is a square arrangement of the numbers 1,2, ..., n2, where the sums of each row, each column and each of the two diagonals have the same value. This value is called the magic sum.

If the smallest number that occurs is greater than or equal to 1, one generally speaks of a magic square if each number occurs only once.

Magic squares of order n are considered equivalent if they can be converted into one another by rotation or reflection.

Often times, an n-order magic square is referred to as an n x n magic square.



1) Rotation by 90 °, 180 ° and 270 ° around the center of the square as well as mirroring the main axes and diagonals of the square turns a magic square into a magic square. These eight magic squares are equivalent; it is enough to examine one of them.
It has become common to use the standard Frénicle form:

  • The element in the upper left corner [1,1] is the smallest of the four elements in the corners.

  • The element to the right of it [1,2] is smaller than the element below [2,1].

2) The following applies to the row, column or diagonal sum (magic number) of a natural magic square:
The row, column or diagonal sum of a natural magic square from n = 3 to n = 10 is: 15, 34, 65, 111, 175, 260, 369 and 505.

3) A magic square remains magical if you add, subtract, multiply or divide any number with a constant C, e.g .:

o stands in place of + - * :  


Number of different magic squares

There is a trivial magic square with edge length 1, but none with edge length 2. In the standard Frénicle form or another basic form there is:

1 magical 3 x 3 square
880 magical 4x4 squares (found by Frénicle de Bessy in 1693)
275305224 magic 5 x 5 squares (calculated by Richard Schroeppel in 1973)
1,78 ·1019       magic 6 x 6 squares (estimated in 1998 by K. Pinn and C. Wieczerkowski)
3,80·1034 magic 7 x 7 squares (", confirmed by W. Trump)

In addition, the number of magic squares up to order 10 is known approximately.

What makes working with normal magic squares with increasing order n so difficult?

The probabilities (Ws) that by randomly setting the n2 Numbers from 1 to n2 Obtaining a normal magic square of order n decreases with increasing n, where: n! = 1 2 3 ... n.

Without known structures or laws of formation for magic squares, it becomes increasingly futile to generate a magic square purely by chance.

Variations of magic squares

Semi-magical (semi-magical)Squares

Only the row sums and column sums are the same.

The magic sum here is 34, the diagonal sums are not 34.


Pandiagonal magic (panmagic) squares

At a pandiagonal magic square not only the sum of the diagonals but also that of the broken diagonals must be the same. The broken diagonals run parallel to the main or secondary diagonal, with elements outside the square being shifted by an edge length.
The magic sum here is 65. Along the lines of the same color, the dashed lines also result in a total of 65. The corresponding parallel lines to the secondary diagonal also result in a total of 65.

Symmetrical magic squares

If a magic square also fulfills the condition that the sum of two elements that are point-symmetrical to the center of the square is equal, it is called a symmetric magic square.
The magic total here is 34. In each of the cells with the same color, the total is 34.


Ultra magic squares

Magic squares that are both symmetrical and pandiagonal are called ultra magical.

Concentric magic squares

A concentric or edged magic square is a magical square that remains magical even if you remove its edge.
The magic total here is 65. The inner magic square has the magic total 39.

Embedded magic squares

A embedded magic square is a square in which one or more magic squares are embedded.
The magic total here is 260. The four embedded magic squares each have a magic total of 130.

Self-complementary magic squares

If every number z of a natural magic square of order n is replaced by its complement n2+ 1-z is replaced, a new magic square is created. If the new magic square is equivalent to the starting square, i.e. can be mapped onto it through rotations and reflections, one speaks of a self-complementary magic square. Sometimes it will self-similar called.
The magic sum here is 65. The magic square on the right is complementary to the left one. The two magic squares are point-symmetrical to each other.


Perfectly perfect magic squares

One speaks of a perfectly perfect magic square if it has the following three properties:

1.    Any 2x2 sub-square always has the same sum

2.    The sum of two elements of a diagonal whose distance is n / 2 always has the value ½ S = 2 (n2 + 1)

3.    It is a square of the order n = 4k, k = 1, 2, 3, ....

The magic total here is 34.

To 1) Examples: 9 + 6 + 3 + 16 = 34, 1 + 14 + 4 + 15 = 34
To 2) 16 + 1 = 17, 6 + 11 = 17, 5 + 12 = 17, 15 + 2 = 17


Compound magic squares

Compound magic squares are composed of multiple magic squares.
The magic total here is 369.


Bimagic squares

A magic square is called bimagicEven if it remains magical after all the numbers have been squared.
G. Pfeffermann discovered the following bimagic square in 1890.
The left magic square has the magic total 360, the right bimagic square the magic total 11180.


Prime squares

All numbers in the magic square are prime numbers.



The oldest known magic square probably goes to the emperor Loh-Shu back, which dates back to around 2800 BC. lived in China. The odd numbers are shown as white dots and the even numbers as black dots.
Due to the color assignment, this representation also has a philosophical and mystical meaning.
White is the color of the symbol Yang. It thus represents the properties of the symbol Yang. The odd numbers are assigned to the male.
black is the color of the symbol Yin. The even numbers are thus assigned to the feminine. (see number mysticism)