Two dimensional

Famous magic squares

 

Lo Shu

The oldest known magic square is the Lo Shu dated 2800 before Christ. This 3x3 magic square got it's name because of a Chinese legend (= 'the story of the turtle'). Lo Shu means book of the river.

 

8

1

6

3

5

7

4

9

2

  

Khajuraho

The first pure magic square of order 4 is an inscription at the Jain Parsvanatha Temple, Khajuraho, India.

  

 7

12

 1

14

 2

13

 8

11

16

 3

10

 5

 9

 6

15

 4

 

 

The first special feature of the magic square is that the sum of the numbers of all broken diagonals (e.g. 16+13+1+4=34) give the magic sum of 34. So, this magic square is called a pan(diagonal)magic or diabolic (= devil's) magic square.

 

The second special feature of the magic square is that the sum of the numbers of each random chosen 2x2 subsquare (e.g. 2+13+16+3=34) is 34.

 

Because of the above mentioned features you can make a carpet of 2x2, 3x3, 4x4, ... the Khajuraho magic square and each random chosen 4x4 sub-square is a pure panmagic (and 2x2 compact) square.

 

  

Dürer

In 1514 the German Renaissance artist/mathematician Albrecht Dürer (1471-1528) published in an engraving entitled Melancholia I the following (symmetric) 4x4 magic square:

 

  

16

 3

 2

13

  5

10

11

  8

  9

  6

  7

12

  4

15

14

  1

 

 

If you replace 4 and 1 by the fourth and first letters of the alphabet you can read the last row as D1514A, that are the initials of Albrecht Dürer and the year, in which he published the magic square. If you replace the letters of ALBRECHT DURER bij it's numbers in the alphabet and you add all the numbers, you get 135. Add 135 and 1 (= the symbol of God; the 1 is bigger than all the others numbers in the engraving) and you get 136 and that is exactly the sum of all numbers from 1 up to 16 in the magic square!!!

 

Euler

A latin square of order n is an n by n square with n different numbers and you find all the numbers from 1 up to n in all rows and all columns. To keep it simple a latin square of order 4 is an 4x4 Sudoku.

 

The famous mathematician Leonhard Euler (born on 15 April 1707 in Basel) discovered: Take 2 Latin squares of the same order (n). An extra feature of squares (N and M) is that all the numbers from 1 up to n are not only in the rows and columns but also in the diagonals. Create the square V = n x [N -/- 1] + M. This new square is always magic, but you don't always find all the numbers from 1 up to n x n in the square, so you don't always get a valid magic square.

 

Sator Arepo

See the following Latin square with letters instead of numbers. This famous 2000 year old “magic” square is called the Sator Arepo.

  

 

                 
S
A  T  O R
                  A R  E  P O
                  T E  N  E T
                  O P  E  R A
                  R O  T  A S

  

 

The Latin sentence “SATOR AREPO TENET OPERA ROTAS” (which means something like 'The sower Arepo keeps the world turning') is a palindrome. A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward as forward. You can read the square in different directions, from left to right, from right to left, and vertically from top to bottom or from bottom to top.

  

Franklin

Benjamin Franklin was born on 17 January 1706 in Boston as the son of a chandler. When he was 12 years old Franklin started to learn the job of newspaper editor and printer. 18 years later Franklin was the owner of a famous newspaper. Because Franklin was very wealthy, he decided to become a scientist and discoverded the lightning rod. Franklin was also interested in magic squares and created magic squares with special magic features, the so-called Franklin features. See for example Franklin's magic square of order 8:

  

 

               

52

61

4

13

20

29

36

45

               

14

3

62

51

46

35

30

19

               

53

60

5

12

21

28

37

44

               

11

6

59

54

43

38

27

22

               

55

58

7

10

23

26

39

42

               

9

8

57

56

41

40

25

24

               

50

63

2

15

18

31

34

47

               

16

1

64

49

48

33

32

17

  

  

Franklin’s magic square has the following special magic features:

 

  1.  The sum of the numbers in each 1/2 row and 1/2 column is 130 (= 1/2 of the magic sum of 260).

  2. The sum of the numbers of each of the 4 bended diagonals and each of the 4 parallel bent diagonals (see below) gives the magic sum of 260.

  

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 3. The sum of the numbers in each random chosen 2x2 sub-square gives 130 (= 1/2 of the magic sum of 260).

Two dimensional