Khajuraho method

 

Use the famous Khajuraho 4x4 panmagic square to construct larger magic squares which are a multiple of 4 (= 8x8, 12x12, 16x16, 20x20, … magic square).

 

Rewrite the Khajuraho magic square as follows:

 

 

Khajuraho magic square                Basic magic square 

7

12

1

14

   

7

h-4

1

h-2

2

13

8

11

   

2

h-3

8

h-5

16

3

10

5

   

h

3

h-6

5

9

6

15

4

   

h-7

6

h-1

4

  

 

To construct an 8x8 panmagic square, you need the basic square and 3 extending magic squares:

 

 

7

h-4

1

h-2

+8

-8

+8

-8

2

h-3

8

h-5

+8

-8

+8

-8

h

3

h-6

5

-8

+8

-8

+8

h-7

6

h-1

4

-8

+8

-8

+8

+16

-16

+16

-16

+24

-24

+24

-24

+16

-16

+16

-16

+24

-24

+24

-24

-16

+16

-16

+16

-24

+24

-24

+24

-16

+16

-16

+16

-24

+24

-24

+24

  

 

The highest number in the 8x8 square is 64. Fill in 64 for h and calculate all the numbers. You get the following 8x8 panmagic square. 

 

 

 Panmagic 8x8 magic square

7

60

1

62

15

52

9

54

2

61

8

59

10

53

16

51

64

3

58

5

56

11

50

13

57

6

63

4

49

14

55

12

23

44

17

46

31

36

25

38

18

45

24

43

26

37

32

35

48

19

42

21

40

27

34

29

41

22

47

20

33

30

39

28

  

 

This magic square is almost Franklin panmagic. Only four 2x2 sub-squares in the middle two columns give not 1/2 of the magic sum (1/2 x 260 = 130). If you swap the colours you get the following Franklin panmagic 8x8 square:

 

 

Franklin panmagic 8x8 square 

15

60

1

54

7

52

9

62

2

53

16

59

10

61

8

51

64

11

50

5

56

3

58

13

49

6

63

12

57

14

55

4

31

44

17

38

23

36

25

46

18

37

32

43

26

45

24

35

48

27

34

21

40

19

42

29

33

22

47

28

41

30

39

20

 

 

Use the Khajuraho method to construct magic squares of order is multiple of 4 from 8x8 to infinity. See 8x8, 12x12, 16x16, 20x20, 24x24, 28x28 and 32x32

 

Download
8x8, Khajuraho method original.xls
Microsoft Excel werkblad 80.5 KB

 

It is possible to use each 4x4 panmagic square to construct a 8x8 Franklin panmagic square.

 

See above how to construct the almost perfect 8x8 Franklin panmagic square (replace the numbers 9 up to 16 of the 4x4 panmagic square by 57 up to 64 to create the first 4x4 sub-square and add each time 8 to the eight low numbers and -/- 8 to the eight high numbers to create the three other 4x4 sub-squares).

 

You must swap half of the numbers to get a perfect 8x8 Franklin panmagic square. Which numbers you must swap and how to swap the numbers, depends on the place of the 1 and the 8 in the 4x4 panmagic square.

 

 

1 2
3 4

 

 

If the 1 and the 8 are in the same column, than you must swap half of the numbers of sub-square 1 with 2 and 3 with 4 (= horizontally).

 

If the 1 and the 8 are in the same row, than you must swap half of the numbers of sub-square 1 with 3 and 2 with 4 (= vertically).

 

 

Correction sheet 1                                Correction sheet 2

                                   
                                   
                                   
                                   
                                   
                                   
                                   
                                   

 

 

If the 1 and the 8 are in position 1 & 2 or 3 & 4 of the row/column, than you must use correction sheet 1.

 

If the 1 and the 8 are in position 2 & 3 or 1 & 4 of the row/column, than you must use correction sheet 2.

 

Download
8x8, Khajuraho method.xls
Microsoft Excel werkblad 172.0 KB