8x8 magic square

Quadrant method, group 6-10

 

General information group 6-10

In the foregoing groups the quadrants consisted of 4 times 4 digits. In group 6-10 the quadrants consist of 2 times 8 digits.

 

How many 8x8 H-, K-, and combined HK-grids are possible?

 

Fill the top left quadrant after H1, H2, H3, H4, H5 or H6. In the example H4 has been chosen.

 

 

 H4 

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In the upper-right right quadrant there are 4 options, 2 with a H-structure, and 2 with a K-structure.

 

 

 H4                     K4                     H                       K 

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1

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1

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3

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Independent of the above options, for the down-left quadrant 8 options are possible, all with H-structure:

 

 

 H4                     H                      H                      H  

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 H3                     H                      H                      H

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7

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0

 

 

The down-right quadrant follows automatically, and has necessarily the structure of the upper- right quadrant. Below an example of both a HHHH- and a HKHK-grid is given:

 

 

 H4                                H

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2

 

  

 H4                                K 

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From above reasoning it will be clear that there are 16 H4HHH and 16 H4KHK-grids.

 

If you start with a K-quadrant in the upper-left, then you get an analogous reasoning. For example starting with K4 as the upper-left quadrant, there are the following options for the upper-right quadrant:

 

 

 K4                     H                       K                       H4 

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3

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1

4

 

 

Independent of the above options, for the down-left quadrant there are 8 options, all with K- structure:

 

 

 K4                     K                       K                        K 

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6

 K3                      K                       K                       K

 

 

And again there is the possibility to compose 16 K4KKK- and 16 K4HKH-grids.

 

As there are 6 H- or K-quadrants to start with, there are 6 x 16 = 96 (homogeneous) HHHH grids, 96 (homogeneous) KKKK grids, 96 (mixed) HKHK grids and 96 (mixed) KHKH

grids.

 

The table of combinations shows that the grids can be combined in 5 different ways:

 

Group 6 : HHHH/H*H*H*H*,

Group 7 : KKKK/K*K*K*K*,

Group 8 : HHHH/K*K*K*K*,

Group 9 : 4 variants of H/M where H stands for homogeneous, and M for mixed grid,

Group10: 3 variants of M/M.

 

 

Illustration group 6

In the preceding general information group 6-10 the following rowgrid has been constructed:

 

 
 H4 (row grid)

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Now we are going to construct a matching columngrid with H-structure.

Fill the top left quadrant after H1*, H2*, H3*, H4*, H5* or H6*. In the example H5* has been chosen.

 

 

H5* (column grid), step 1 

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The 5th row can only be filled with 4-2-7-1 (the alternative 0-6-3-5 would immediately lead to double pairings when composing the final magic square, just try!). Maintaining the

H*-structure the quadrant can only be filled as follows:

 

 

 H5* (column grid), step 2 

0

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3

5

 

 

 

 

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Important: Note that filling the down-left quadrant after the K*-structure would lead immediately lead to doubling when composing the final magic square (just try!).

 

In the 5th column only 0-3-4-7 and 2-1-6-5 are possible. In the example 2-1-6-5 has been chosen. With both options you can finish the upper-right quadrant, but only when maintaining the H*-structure. The down right quadrant follows automatically, and has

necessarily also the H*-structure

 

 

 H5* (column grid), step 3

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 H5* (column grid), step 4

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Starting the upper-left quadrant with the other H*-fillings leads also to matching column grids. So, in total there are 6 (H1* …..H6*) x 2 (see options of step 3) = 12 different column grids.

 

Below you see the final magic square composed with the above grids:

 

 

  1x digit from row grid +1          +     8x digit from column grid         =     most perfect 8x8 magic square 

0

5

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2

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0

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2

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1

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1

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48

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40

9

62

6

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1

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3

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38

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56

3

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25

3

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7

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60

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52

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37

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63

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55

4

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59

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2

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32

43

6

49

14

57

24

35

 

 

 The total amount of different squares that can be produced in group 6 is: 48 (row grids) x 12 (column grids) = 576. As the combination is reflexive, swapping gives no new squares.

 

 

Illustration group 7

The construction of K-squares goes completely analogous with the construction of

the H-squares.

 

For detailed explanation of the construction, see general information group 6-10. See below two examples of group 7. Note that in both examples the constructed magic square has the extra magic property X (shown in blue).

 

 

 1x digit from row grid +1  +   8x digit from column grid =   most perfect 8x8 magic square

0

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1

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63

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4

  

 

 1x digit from row grid +1  +  8x from column grid           =  most perfect 8x8 magic square

0

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1

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62

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60

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22

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61

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59

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25

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2

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48

49

6

27

40

57

14

19

 

 

The total number of squares of group 7 is: 48 (row grids) x 12 (column grids) = 576; a quarter of them = 144 ( half of the row grids x half of the column grids) shows the magic property X.

 

 

Illustration group 8

Magic squares of group 8 can be constructed by means of combining row grids consisting of H-quadrants with column grids consisting of diagonally reflected K-quadrants, and vice versa.

 

For detailed explanation of the construction of the grids, see general information group 6-10. Below two examples are shown.

 

 

 1x digit from row grid +1  +  8x digit from column grid   =  most perfect 8x8 magic square

0

5

2

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0

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0

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1

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7

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63

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5

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56

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20

13

58

5

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6

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27

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40

19

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1

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7

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60

15

18

37

52

7

26

45

 

 

 1x digit from row grid +1  +   8x digit from column grid  = most perfect 8x8 magic square

0

5

2

7

0

5

2

7

 

 

0

6

5

3

5

3

0

6

 

 

1

54

43

32

41

30

3

56

6

3

4

1

6

3

4

1

 

 

7

1

2

4

2

4

7

1

 

 

63

12

21

34

23

36

61

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5

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2

4

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1

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22

33

64

11

62

9

24

35

3

6

1

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1

4

 

 

5

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44

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2

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4

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42

29

0

5

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0

5

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2

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4

 

 

17

38

59

16

57

14

19

40

6

3

4

1

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1

 

 

5

3

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6

0

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3

 

 

47

28

5

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7

52

45

26

5

0

7

2

5

0

7

2

 

 

0

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3

5

3

0

6

 

 

6

49

48

27

46

25

8

51

3

6

1

4

3

6

1

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7

1

2

4

2

4

7

1

 

 

60

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18

37

20

39

58

13

 

 

The total number of squares of group 8 is: 48 (row grids) x 12 (column grids) x 2 (swapping row and column grids) = 1152.

 

 

Illustration group 9

In magic squares of group 9 one of the grids is of the mixed HK type. An

illustrative example has already been given when treating group 6. The table of

combinations shows 4 different, non-reflexive combinations, all of them generate 1152 squares.

 

Below you find some new examples. First a KKKK/K*K*H*H* example. Developed from

swapping the 6th and 8th row of the first KKKK/K*K*K*K* example of group 7. The swapping makes the extra magic disappear.

 

 

1x digit HK row grid +1 + 8x digit H*K* column grid      =  most perfect 8x8 magic square 

0

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0

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1

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1

4

3

 

 

5

6

0

3

4

7

1

2

 

 

47

50

5

28

39

58

13

20

0

7

2

5

0

7

2

5

 

 

2

1

7

4

3

0

6

5

 

 

17

16

59

38

25

8

51

46

6

1

4

3

6

1

4

3

 

 

7

4

2

1

6

5

3

0

 

 

63

34

21

12

55

42

29

4

5

2

7

0

5

2

7

0

 

 

0

3

5

6

1

2

4

7

 

 

6

27

48

49

14

19

40

57

3

4

1

6

3

4

1

6

 

 

5

6

0

3

4

7

1

2

 

 

44

53

2

31

36

61

10

23

 

 

1x digit HK row grid +1 + 8x digit H*K* column grid      =  most perfect 8x8 magic square

0

5

2

7

0

5

2

7

 

 

0

6

5

3

1

7

4

2

 

 

1

54

43

32

9

62

35

24

6

3

4

1

6

3

4

1

 

 

7

1

2

4

6

0

3

5

 

 

63

12

21

34

55

4

29

42

5

0

7

2

5

0

7

2

 

 

2

4

7

1

3

5

6

0

 

 

22

33

64

11

30

41

56

3

3

6

1

4

3

6

1

4

 

 

5

3

0

6

4

2

1

7

 

 

44

31

2

53

36

23

10

61

0

5

2

7

0

5

2

7

 

 

2

4

7

1

3

5

6

0

 

 

17

38

59

16

25

46

51

8

3

6

1

4

3

6

1

4

 

 

7

1

2

4

6

0

3

5

 

 

60

15

18

37

52

7

26

45

5

0

7

2

5

0

7

2

 

 

0

6

5

3

1

7

4

2

 

 

6

49

48

27

14

57

40

19

6

3

4

1

6

3

4

1

 

 

5

3

0

6

4

2

1

7

 

 

47

28

5

50

39

20

13

58

 

 

All examples stand for an amount of 48 x 12 x 2 = 1152 squares.

 

Illustration group 10

Both grids consist of H- and K-quadrants. There are 3 different combinations, two of them reflexive, the third non-reflexive.

 

For detailed explanation of the construction of HK- and KH-grids, see the general information group 6-10. See below two examples, one for combination 10a (reflexive) and one for combination 10c (non reflexive).

 

 

1x digit HK row grid +1 + 8x digit H*K* column grid      =   most perfect 8x8 magic square

0

5

2

7

0

7

2

5

 

 

0

6

5

3

1

2

4

7

 

 

1

54

43

32

9

24

35

62

6

3

4

1

6

1

4

3

 

 

5

3

0

6

4

7

1

2

 

 

47

28

5

50

39

58

13

20

5

0

7

2

5

2

7

0

 

 

2

4

7

1

3

0

6

5

 

 

22

33

64

11

30

3

56

41

3

6

1

4

3

4

1

6

 

 

7

1

2

4

6

5

3

0

 

 

60

15

18

37

52

45

26

7

4

1

6

3

4

3

6

1

 

 

2

4

7

1

3

0

6

5

 

 

21

34

63

12

29

4

55

42

7

2

5

0

7

0

5

2

 

 

5

3

0

6

4

7

1

2

 

 

48

27

6

49

40

57

14

19

1

4

3

6

1

6

3

4

 

 

0

6

5

3

1

2

4

7

 

 

2

53

44

31

10

23

36

61

2

7

0

5

2

5

0

7

 

 

7

1

2

4

6

5

3

0

 

 

59

16

17

38

51

46

25

8

 

          

The total number of squares of combination 10a is: 48 (row grids) x 12 (column grids) = 576. Combination 10b behaves completely analogous, and generates also 576 squares.

 

And finally an example of combination 10c:

 

 

1x grid HK row grid +1 + 8x digit K*H* column grid       =  most perfect 8x8 magic square

0

5

2

7

0

7

2

5

 

 

0

6

5

3

1

2

4

7

 

 

1

54

43

32

9

24

35

62

6

3

4

1

6

1

4

3

 

 

7

1

2

4

6

5

3

0

 

 

63

12

21

34

55

42

29

4

5

0

7

2

5

2

7

0

 

 

2

4

7

1

3

0

6

5

 

 

22

33

64

11

30

3

56

41

3

6

1

4

3

4

1

6

 

 

5

3

0

6

4

7

1

2

 

 

44

31

2

53

36

61

10

23

4

1

6

3

4

3

6

1

 

 

0

6

5

3

1

2

4

7

 

 

5

50

47

28

13

20

39

58

7

2

5

0

7

0

5

2

 

 

5

3

0

6

4

7

1

2

 

 

48

27

6

49

40

57

14

19

1

4

3

6

1

6

3

4

 

 

2

4

7

1

3

0

6

5

 

 

18

37

60

15

26

7

52

45

2

7

0

5

2

5

0

7

 

 

7

1

2

4

6

5

3

0

 

 

59

16

17

38

51

46

25

8

 

   

The total number of squares of combination 10c is: 48 (row grids) x 12 (column grids) x 2 (swapping row and column grids) = 1152.

 

 

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8x8, Quadrant method, group 6 up to 10.d
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8x8 magic square