Explanation most perfect (Franklin pan)magic 8x8 square

 

The most perfect (Franklin pan)magic 8x8 square consist of 4 proportional 4x4 panmagic squares. 

 

 

4x4 panmagic square                                                4x4 sub-square of 8x8

1

8

13

12

 

 

 

 

 

 

 

1

54

12

63

15

10

3

6

 

 

 

 

 

 

 

16

59

5

50

4

5

16

9

 

 

 

 

 

 

 

53

2

64

11

14

11

2

7

 

 

 

 

 

 

 

60

15

49

6

 

 

In both squares the sum of two digits of a colour always totals to the lowest plus the highest digit of the magic square (1+16=17 respectively 1+64=65). With each time two colours you can get all eight (pan)diagonals (see 4x4 magic square, explanation).

 

Look below at the patterns of the panmagic 4x4 square and the most perfect 8x8 square.

 

 

1

8

13

12

 

 

 

 

 

 

 

1

8

13

12

15

10

3

6

 

 

 

 

 

 

 

15

10

3

6

4

5

16

9

 

 

 

 

 

 

 

4

5

16

9

14

11

2

7

 

 

 

 

 

 

 

14

11

2

7

 

  

 9 + 25 = 34                                                           16 + 18 = 34

  

 

1

54

12

63

3

56

10

61

 

 

 

1

54

12

63

3

56

10

61

16

59

5

50

14

57

7

52

 

 

 

16

59

5

50

14

57

7

52

53

2

64

11

55

4

62

9

 

 

 

53

2

64

11

55

4

62

9

60

15

49

6

58

13

51

8

 

 

 

60

15

49

6

58

13

51

8

17

38

28

47

19

40

26

45

 

 

 

17

38

28

47

19

40

26

45

32

43

21

34

30

41

23

36

 

 

 

32

43

21

34

30

41

23

36

37

18

48

27

39

20

46

25

 

 

 

37

18

48

27

39

20

46

25

44

31

33

22

42

29

35

24

 

 

 

44

31

33

22

42

29

35

24

 

  

55 + 75 + 59 + 71 = 130 + 130 = 160                  17 + 113 + 49 + 91 = 130 + 130 = 160

 

 

Because of the structure the sum of the digits of each 1/2 row/column/(pan)diagonal and of each 2x2 sub-square is allways (half of the magic sum: 1/2 x 260 =) 130.

  

There are the 3 following swap possibilities:

[1th] You can swap row 1&3 and/or row 2&4 and or row 5&7 and/or row 6&8 and/or column 1&3 and/or column 2&4 and/or column 5&7 and/or column 6&8.

 

[2nd] You can swap the upper half with the down half and/or the right half with the left half.

 

[3rd] You can swap row 1&2 and row 3&4 and row 5&6 and row 7&8 and/or column 1&2 and column 3&4 and column 5&6 and column 7&8.

 

If you combine the 3 swap possibilities you can get each digit out of 1 up to 64 in the top left corner. Try it!!!

 

From Willem Barink we learn that a small part of the most perfect magic squares has an extra magic feature. See the following most perfect magic 8x8 square:

 

1

32

43

54

9

24

35

62

 

 

1

32

43

54

9

24

35

62

60

37

18

15

52

45

26

7

 

 

60

37

18

15

52

45

26

7

22

11

64

33

30

3

56

41

 

 

22

11

64

33

30

3

56

41

47

50

5

28

39

58

13

20

 

 

47

50

5

28

39

58

13

20

17

16

59

38

25

8

51

46

 

 

17

16

59

38

25

8

51

46

44

53

2

31

36

61

10

23

 

 

44

53

2

31

36

61

10

23

6

27

48

49

14

19

40

57

 

 

6

27

48

49

14

19

40

57

63

34

21

12

55

42

29

4

 

 

63

34

21

12

55

42

29

4

   

 

33 + 97 = 130                                                61 + 69 = 130

 

 

The extra feature is that in each row and each column (not only ) adding the digits from position (1 up to 4 and 5 up to 8, but also from) 3 up to 6 gives the magic sum of 130.