Magic square

A magic square is a square array of numbers or letters, provided that certain requirements are met.

• 5.1 The Lo - Shu
• 5.2 The magic square of Albrecht Dürer
• 5.3 The magic square on the Sagrada Família
• 5.4 Goethe's witches basics

Definition

The usual definition of a magic square is:

" A magic square of the edge length is a square array of numbers such that the sum of the numbers of all rows, columns and the two diagonals is the same. "

There are numerous variants of magic squares, in which not all of these conditions are met, or additional restrictions are required ( see below).

It is also seen that each arithmetic sequence is suitable for a magic square.

Properties

It is obvious that once again a magic square is formed by rotation of 90 °, 180 ° and 270 ° as well as by reflection in the main axes and diagonals of a magic square. These eight magic squares are equivalent; it is enough to examine one of them. It has become common to use the Frénicle standard form here:

• The element in the upper left corner of [1,1] is the smallest of the four elements in the corners.
• Right next to the element [1,2 ] is less than the element including [2,1 ].

The row and column sum is called a magic number. It is easy to see that the magic number times the sum of the numbers from 1 to must be:

The first of these magic numbers are starting with

Special magic squares

Symmetric magic squares

Meets a magic square the additional condition that the sum of two elements ( for straight ) or the central element are point symmetric to the center ( at odd magic squares ), are the same, so it is called symmetric magic square. Sometimes the term associative magic square will be used. As one can easily show the sum of two such elements must be; at odd symmetric magic squares midfield has the value.

Pandiagonal magic squares

In a magic square pandiagonal not only the sum of the diagonal, but also the broken diagonals must be equal. The broken diagonals parallel to the main or secondary diagonal, with elements outside the square are shifted by an edge length. The smallest edge length for a pandiagonal magic square is 4

Magic squares, which are both symmetric pandiagonal, called ultra magical.

Prime squares

There are numerous variants of magic squares, where the requirement is dropped that only the numbers from 1 to happen to, but additional conditions must be met. The best known of these are prime squares in which all elements have to be prime numbers.

The number of magic squares

There is a ( trivial ) magic square with edge length 1, but none with edge length 2 Apart from symmetry operations or specified in the Frénicle standard form, there is only a single magic square with edge length 3 (see Lo - Shu ). All 880 magic squares with side length 4 have been found from 1693 Frénicle de Bessy. With an edge length 5 there are 275 305 224 magic squares; In addition, exact figures are not known, but there are relatively reliable to about assessments. The most extensive calculations were performed by Walter Trump. Also, the number of symmetrical, pandiagonaler and ultra magic squares for smaller is known, for example, there are 48 symmetric magic squares with side length 4 and 16 ultra magic squares with side length 5

Famous examples

The Lo - Shu

One example is the Saturn seal from China with the sum 15:

The magic square of Albrecht Dürer

One of the most famous magic squares can be found in Albrecht Dürer's engraving Melencolia I. The Dürer's square has the following properties:

• It is a symmetrical magic square.
• The sum of the elements of the four quadrants is always the magic number 34
• The sum of the four corner sections and the center of the four fields is always 34
• The sum of the four fields, each of which is of the four corner panels of 1 or 2 is further shifted in the clockwise direction in each field 34 (8 14 9 15 3 12 5 2).
• Also, the sum of which is arranged in the form of a dragon quadrilateral elements ( for example, 2 10 8 14, 3 ​​ 9 7 15) is 34
• In the middle of the last line of the year 1514, the year in which Dürer anfertigte the bite appears.
• At the beginning of the last row is a 4, at the end of a 1 Substituting these numbers with the same letter of the alphabet, we obtain D and A, the monogram of the artist ( Albrecht Dürer )

The magic square on the Sagrada Família

The facade dedicated to the Passion of the Sagrada Familia in Barcelona, ​​the work of sculptor Josep Maria Subirachs, contains a magic square:

It is not a magic square in the strict sense, because not all numbers 1-16 appear (there are no 12 and 16), 10 and 14, however, come twice before. The magic number is 33, an allusion to the age of Christ. It is closely related to the Dürer square, it can be constructed from this by interchanging rows and columns and by subtracting one in four fields (11, 12, 15, 16).

Goethe's witches basics

There are many interpretations of the Witch multiplication tables from Goethe's Faust. In addition to the obvious assumption that it simply is nonsense ( " I think, the old woman speaks in a fever " ), it was interpreted as a construction manual for a magic square - an interpretation that not one hundred percent convinced.

Construction of magic squares

For the construction of magic squares there are three different methods that depend on the edge length. The simplest method works for all magic squares of odd edge length (ie 3 × 3, 5 × 5, 7 × 7, etc. ). It begins at the top of the center 1, and then fills in the other figures in order according to the following rule in the other fields:

Here, the magic square is considered to be repeated periodically, that is, if one goes beyond the top edge (this happens at the first step ), it comes from below in again, and when you go to the right, then one coming from the left back inside. Here a constructed according to this rule 7x7 square:

The two other methods are straight- edge length of squares, the one for all squares whose edge length is divisible by 4, and the other for those in which the radical is 2 through 4 the parts.

A playful method for constructing magic squares even orders > 4 is using Medjig solutions. For this you need the parts of the game Medjig puzzles ( Philos games, Code No. 6343 ). These are distributed into four quadrants squares, after which the points with 0, 1, 2 and 3 are shown in various configurations. The puzzle has 18 pieces, all arrangements there three times. See the figure below. The aim of the puzzle is to arbitrarily remove 9 squares of the meeting and to present this part assembly in a 3 x 3 square, so that in each resulting row, column and diagonal, the sum of 9 ( points ).

The construction of a magic square of order 6 with the help of Medjig puzzles goes as follows: Take a 3 x 3 Medjig solution to this time you can choose unlimited from the total assembly. Then you take the well-known classical magic square of order 3, and distribute all the fields it into four quadrants. Next, you fill in the quadrant with the original number and the three derived numbers modulo 9 to 36, the Medjig solution below. The original box with the number 8 is thus distributed in four fields with the numbers 8 (= 8 0 x 9 ), 17 (= 8 1 x 9 ), 26 (= 8 2 x 9 ) and 35 (= 8 3 x 9 ), the field with the number 3 is 3, 12, 21 and 30, and so on. See the example below.

In the same way, you can create magic squares of order 8. One to producing only a 4 x 4 Medjig solution ( sum of the scores of each row, column, diagonal 12), and scale then for example the one shown above 4 x 4 Dürer magic square modulo 16 to 64 In general, one needs for the construction of magic squares of order ≥ 10 in this way several sets Medjig parts. For 12 you can order a 3 x 3 Medjig solution doubled horizontally and vertically, and then the constructed above 6 x 6 magic square modulo -36 spread by 144 It is similar with order 16

Magic squares of size 4 x 4 with a specific sum of the digits can be constructed using the following scheme, where the variables a and b are arbitrary integers:

The magic sum is in each case (21a b). If this example the value amounted to 88, subtract an integer ( = a) multiples on 21, the rest is then the number b. For example ( as shown in the right square): 88 - 3 x 21 = 25

Magic squares of this type does not generally consist of the numbers 1, 2, 3, ... 16 and an unfavorable choice of the values ​​a and b can have two fields contain the same number. The magic sum (21a b) is for it not only in the rows, columns and diagonals contain, but also in the four quadrants, the four corner squares, as well as in the small square of the four inner fields.

Others

The 4 x 4 magic square wherein the quadrant give the magic sum can - if the property that each of the numbers from 1 to 16 is to occur exactly once dispensed - as a linear combination of the following eight generating mutually congruent squares be shown:

It is noted that these generating eight squares are not linearly independent because

Ie there is a non -trivial linear combination ( a linear combination whose coefficients are not all = 0), which gives the 0 - square. In other words, each of the eight generating squares can be represented as a linear combination of the remaining seven. Seven generating squares but are necessary to generate all the magic 4x4 squares with the additional property " quadrants "; the vector space of 4x4 magic squares generated from these squares is 7- dimensional in this sense.

It is noteworthy that in all eight generating squares A-H as in Albrecht Dürer's magic square not only rows, columns and diagonals always the same amount deliver (1 ), but also each of the four " quadrants ", the four central squares and the four corner squares. This means that all magic squares, we win as a linear combination of these generators have this property.

The magic square of the engraving " Melencolia I" by Albrecht Dürer as a linear combination of the generating squares A- G:

The sum of the coefficients is of course 34 = -4 8 14 6 16 -5 -1.

The fact that the four quadrants give the magic sum, is not necessarily so. The following magic square does not have this property and is therefore linearly independent to the squares A- H:

Taking the square of the squares still 7 A-H, we obtain a basis for the 8- dimensional vector space of all 4 × 4 magic squares. The sum of the four corners and the center of this field is also square (as in all the 4 x 4 magic - squares ) the magic sum.

Letter squares

A magic square is a letter brainteaser, in the rows and columns of the square each have the same words occur. An example of this is the Sator Square: