Symmetric transformation (Liki)

 

Paulus Gerdes introduced the Liki magic square (see http://plus.maths.org/content/new-designs-africa). He showed that it is possible to transform a square with consecutive numbers into a magic square by swapping half of the numbers symmetrically. You can use this method to construct magic squares which are a multiple of 4 (= 4x4, 8x8, 12x12, 16x16, ... magic square).

The 4x4 magic squares contains the numbers 1 up to 16. Each number has an inverse number. The inverse number is the largest number of the square plus 1 minus the original number. See below the results:

1   --> Inverse number is 16 + 1 -/- 1 =
16
2   --> Inverse number is 16 + 1 -/- 2 =
15
3   --> Inverse number is 16 + 1 -/- 3 =
14
4   --> Inverse number is 16 + 1 -/- 4 =
13
5   --> Inverse number is 16 + 1 -/- 5 =
12
6   --> Inverse number is 16 + 1 -/- 6 =
11
7   --> Inverse number is 16 + 1 -/- 7 =
10
8   --> Inverse number is 16 + 1 -/-   8 =
9
9   --> Inverse number is 16 + 1 -/-   9 =
8
10 --> Inverse number is 16 + 1 -/- 10 =
7
11 --> Inverse number is 16 + 1 -/- 11 =
6
12 --> Inverse number is 16 + 1 -/- 12 =
5
13 --> Inverse number is 16 + 1 -/- 13 =
4
14 --> Inverse number is 16 + 1 -/- 14 =
3
15 --> Inverse number is 16 + 1 -/- 15 =
2
16 --> Inverse number is 16 + 1 -/- 16 =
1

 

Transform the 4x4 square with consecutive numbers in a symmetric 4x4 magic square:

 


4x4 square with consecutive numbers

 

   

28

32

36

40

 
 

34

       

34

10

 

1

2

3

4

 

26

 

5

6

7

8

 

42

 

9

10

11

12

 

58

 

13

14

15

16

 

 

 

Notify that the square with consecutive numbers is already symmetric and the (main) diagonals give already the magic sum of 34.

 


Symmetric 4x4 magic square (1)

 

   

34

34

34

34

 
 

34

       

34

34

 

1

15

14

4

 

34

 

12

6

7

9

 

34

 

8

10

11

5

 

34

 

13

3

2

16

 

 


ór
 
Symmetric 4x4 magic square (2)

 

   

34

34

34

34

 
 

34

       

34

34

 

16

2

3

13

 

34

 

5

11

10

8

 

34

 

9

7

6

12

 

34

 

4

14

15

1

 

 

Download
4x4, symmetric transformation (Liki).xls
Microsoft Excel werkblad 38.5 KB

 

Take one step extra and you can transform a 4x4 square with consecutive numbers into the famous square of Albrecht Dürer:

  

 

Dürer’s magic square:

 

   

28

32

36

40

         

28

36

32

40

         

34

34

34

34

 
 

34

       

34

     

34

       

34

     

34

       

34

10

 

1

2

3

4

     

10

 

1

3

2

4

     

34

 

16

3

2

13

 

26

 

5

6

7

8

     

26

 

5

7

6

8

     

34

 

5

10

11

8

 

42

 

9

10

11

12

     

42

 

9

11

10

12

     

34

 

9

6

7

12

 

58

 

13

14

15

16

     

58

 

13

15

14

16

     

34

 

4

15

14

1

 

 

Download
4x4, Dürer transformation.xls
Microsoft Excel werkblad 42.5 KB