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).

 

Paulus Gerdes constructed the following symmetric 8x8 magic square:


8x8 square with consecutive numbers

 

   

232

240

248

256

264

272

280

288

 
 

260

               

260

36

 

1

2

3

4

5

6

7

8

 

100

 

9

10

11

12

13

14

15

16

 

164

 

17

18

19

20

21

22

23

24

 

228

 

25

26

27

28

29

30

31

32

 

292

 

33

34

35

36

37

38

39

40

 

356

 

41

42

43

44

45

46

47

48

 

420

 

49

50

51

52

53

54

55

56

 

484

 

57

58

59

60

61

62

63

64

 

 

 

Symmetric 8x8 magic square

 

   

260

260

260

260

260

260

260

260

 
 

260

               

260

260

 

1

63

3

61

60

6

58

8

 

260

 

56

55

11

12

13

14

50

49

 

260

 

17

18

46

45

44

43

23

24

 

260

 

40

26

38

28

29

35

31

33

 

260

 

32

34

30

36

37

27

39

25

 

260

 

41

42

22

21

20

19

47

48

 

260

 

16

15

51

52

53

54

10

9

 

260

 

57

7

59

5

4

62

2

64

 



It is a beautiful result, but it is possible to get a more magic result.

 

 

      52 56 60 64 68 72 76 80  
      180 184 188 192 196 200 204 208  
    260                 260
10 26   1 2 3 4 5 6 7 8  
42 58   9 10 11 12 13 14 15 16  
74 90   17 18 19 20 21 22 23 24  
106 122   25 26 27 28 29 30 31 32  
138 154   33 34 35 36 37 38 39 40  
170 186   41 42 43 44 45 46 47 48  
202 218   49 50 51 52 53 54 55 56  
234 250   57 58 59 60 61 62 63 64  

 

 

      130 130 130 130 130 130 130 130  
      130 130 130 130 130 130 130 130  
    260                 260
130 130   1 63 62 4 5 59 58 8  
130 130   56 10 11 53 52 14 15 49  
130 130   48 18 19 45 44 22 23 41  
130 130   25 39 38 28 29 35 34 32  
130 130   33 31 30 36 37 27 26 40  
130 130   24 42 43 21 20 46 47 17  
130 130   16 50 51 13 12 54 55 9  
130 130   57 7 6 60 61 3 2 64  

 

 

Each 1/2 row/column gives 1/2 of the magic sum.

 

By refining the grids you can construct an ultramagic 8x8 square.

 

 

1x number from first grid                     +       1x [number -/- 1] from second grid   =

1 2 2 1 2 1 1 2     1 1 2 2 3 3 4 4
3 4 4 3 4 3 3 4     1 1 2 2 3 3 4 4
3 4 4 3 4 3 3 4     5 5 6 6 7 7 8 8
1 2 2 1 2 1 1 2     5 5 6 6 7 7 8 8
3 4 4 3 4 3 3 4     9 9 10 10 11 11 12 12
1 2 2 1 2 1 1 2     9 9 10 10 11 11 12 12
1 2 2 1 2 1 1 2     13 13 14 14 15 15 16 16
3 4 4 3 4 3 3 4     13 13 14 14 15 15 16 16

 

 

symmetric transformation                    =      Ultra magic 8x8 square

1 2 6 5 10 9 13 14     1 63 6 60 10 56 13 51
3 4 8 7 12 11 15 16     62 4 57 7 53 11 50 16
19 20 24 23 28 27 31 32     19 45 24 42 28 38 31 33
17 18 22 21 26 25 29 30     48 18 43 21 39 25 36 30
35 36 40 39 44 43 47 48     35 29 40 26 44 22 47 17
33 34 38 37 42 41 45 46     32 34 27 37 23 41 20 46
49 50 54 53 58 57 61 62     49 15 54 12 58 8 61 3
51 52 56 55 60 59 63 64     14 52 9 55 5 59 2 64

 

 

This 8x8 magic square is panmagic, 2x2 compact, symmetric and each 1/2 row/column gives 1/2 of the magic sum.

 

You can use this method to construct magic squares which are a multiple of 4 from 8x8 to infinite (= 8x8, 12x12, 16x16, ... magic square).

 

Download
8x8, Symmetric transformation (Liki).xls
Microsoft Excel werkblad 89.0 KB