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 numbers of a colour always totals to the lowest plus the highest number 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 = 260                  17 + 113 + 49 + 91 = 130 + 130 = 260

 

 

Because of the structure the sum of the numbers 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 number out of 1 up to 64 in the top left corner. Try it!!!

 

From Willem Barink we learn that a small part (1/64) 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 numbers from position (1 up to 4 and 5 up to 8, but also from) 3 up to 6 gives the magic sum of 130.

 

Most perfect on this website is different from most perfect on other websites. In my opinion the 8x8 Franklin panmagic square is most perfect. But according to Kathleen Ollerenshaw, the famous mathematician, the 8x8 complete magic square is most perfect. See below how you can transform a 8x8 Franklin panmagic square into a 8x8 complete magic square (by swapping rows and columns systematically).

 

 

 

Franklin panmagic 8x8 (= according to me most perfect) square

 
                             
     

130

130

130

130

130

130

130

130

       
     

130

130

130

130

130

130

130

130

       
   

130

               

130

     

130

130

 

1

32

38

59

5

28

34

63

 

260

260

 

130

130

 

46

51

9

24

42

55

13

20

 

260

260

 

130

130

 

27

6

64

33

31

2

60

37

 

260

260

 

130

130

 

56

41

19

14

52

45

23

10

 

260

260

 

130

130

 

11

22

48

49

15

18

44

53

 

260

260

 

130

130

 

40

57

3

30

36

61

7

26

 

260

260

 

130

130

 

17

16

54

43

21

12

50

47

 

260

260

 

130

130

 

62

35

25

8

58

39

29

4

       
   

130

               

130

     
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
                             
                             
     

1

32

38

59

5

28

34

63

       
     

46

51

9

24

42

55

13

20

       
     

11

22

48

49

15

18

44

53

       
     

40

57

3

30

36

61

7

26

       
     

27

6

64

33

31

2

60

37

       
     

56

41

19

14

52

45

23

10

       
     

17

16

54

43

21

12

50

47

       
     

62

35

25

8

58

39

29

4

       
                             
                             

Complete (= K. Ollerenshaw's most perfect) magic 8x8 square

                             
     

98

162

98

162

98

162

98

162

       
     

162

98

162

98

162

98

162

98

       
   

128

               

128

     

66

194

 

1

32

5

28

38

59

34

63

 

260

260

 

194

66

 

46

51

42

55

9

24

13

20

 

260

260

 

66

194

 

11

22

15

18

48

49

44

53

 

260

260

 

194

66

 

40

57

36

61

3

30

7

26

 

260

260

 

66

194

 

27

6

31

2

64

33

60

37

 

260

260

 

194

66

 

56

41

52

45

19

14

23

10

 

260

260

 

66

194

 

17

16

21

12

54

43

50

47

 

260

260

 

194

66

 

62

35

58

39

25

8

29

4

       
   

132

               

132

     
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         
     

130

130

130

130

130

130

130

         

 

 

Addition of the numbers in half rows/columns/diagonals of the complete magic square don't give half of the magic sum. That is why I think the Franklin panmagic 8x8 square is most perfect and the complete magic square is not.

 

You can transform all most perfect magic squares which are a multiple of 4 from 8x8 (= 8x8, 12x12, 16x16, 20x20, ...) into complete magic squares. In all downloads of most perfect squares on this website you find the transformation from most perfect into complete.

 

Download
most perfect 8x8 magic square.xls
Microsoft Excel werkblad 507.5 KB