Quadrant method, group 11-19

 

General information group 11 - 19

Magic squares of group 10-18 are constructed by means of combining a grid consisting of H- and/or K-quadrants (each quadrant contains 2 times 8 digits), and a grid consisting of A-,B- and C-quadrants (each quadrant contains 4 times 4 digits).

Half of the amount of homogeneous H-grids (HHHH-grids) can be matched with homogeneous A- and with homogeneous C-grids, the other half can be matched

with mixed AC-grids.

Analogously half of the homogeneous K-grids can be matched with homogeneous B- and with homogeneous C-grids, the other half can be matched with mixed BC-grids.

Analogously half of the mixed HK- or KH-grids can be matched with mixed AC- and BC-grids, the other half can be matched with mixed ACC*B- or CABC*-grids.

 

 

Illustration group 11

Magic squares of group 10 are constructed by means of combining H-grids with A-grids.

 

In the example below a row grid with H4 in all four quadrants has been chosen.

 

 

 H4 (row grid)

0

5

2

7

0

5

2

7

6

3

4

1

6

3

4

1

5

0

7

2

5

0

7

2

3

6

1

4

3

6

1

4

0

5

2

7

0

5

2

7

6

3

4

1

6

3

4

1

5

0

7

2

5

0

7

2

3

6

1

4

3

6

1

4

 

 

Now the construction of the matching column grid. Fill the top left quadrant after A1, A2, or A3. In the example A1 has been chosen. (Verify that A1*, A2* and A3* do not not work!).

  

 

 A1 (column grid), step 1

0

7

6

1

 

 

 

 

7

0

1

6

 

 

 

 

1

6

7

0

 

 

 

 

6

1

0

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In the 5th row only 1-6-7-0 is possible. And now you will find out that the third quadrant can only be completed with an A-structure.

 

 

 A1 (column grid), step 2

0

7

6

1

 

 

 

 

7

0

1

6

 

 

 

 

1

6

7

0

 

 

 

 

6

1

0

7

 

 

 

 

1

6

7

0

 

 

 

 

7

0

1

6

 

 

 

 

0

7

6

1

 

 

 

 

6

1

0

7

 

 

 

 

 

 

The right half of the row grid must be filled with the digits 2, 3, 4, and 5. In column 5 it is only possible to fill in 2-5-3-4, or 4-3-5-2 . In the example 4-3-5-2 has been chosen. With both options you can only finish the upper right quadrant successfully when maintaining the A-structure. The down right quadrant follows automatically, and has necessarily also the A-structure.

 

  

 A1 (column grid), step 3

0

7

6

1

4

3

2

5

7

0

1

6

3

4

5

2

1

6

7

0

5

2

3

4

6

1

0

7

2

5

4

3

1

6

7

0

 

 

 

 

6

1

0

7

 

 

 

 

0

7

6

1

 

 

 

 

7

0

1

6

 

 

 

 

 

 

 A1 (column grid), step 4

0

7

6

1

4

3

2

5

7

0

1

6

3

4

5

2

1

6

7

0

5

2

3

4

6

1

0

7

2

5

4

3

1

6

7

0

5

2

3

4

6

1

0

7

2

5

4

3

0

7

6

1

4

3

2

5

7

0

1

6

3

4

5

2

 

 

In total there are 3 (A1, A2, A3) x 2 (options of step 3) = 6 different column grids.

 

Finally you can combine row and column grid to produce the magic square. The square below contains the X magic property (shown in blue).

 

 

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

7

6

1

4

3

2

5

 

 

1

62

51

16

33

30

19

48

6

3

4

1

6

3

4

1

 

 

7

0

1

6

3

4

5

2

 

 

63

4

13

50

31

36

45

18

5

0

7

2

5

0

7

2

 

 

1

6

7

0

5

2

3

4

 

 

14

49

64

3

46

17

32

35

3

6

1

4

3

6

1

4

 

 

6

1

0

7

2

5

4

3

 

 

52

15

2

61

20

47

34

29

0

5

2

7

0

5

2

7

 

 

1

6

7

0

5

2

3

4

 

 

9

54

59

8

41

22

27

40

6

3

4

1

6

3

4

1

 

 

6

1

0

7

2

5

4

3

 

 

55

12

5

58

23

44

37

26

5

0

7

2

5

0

7

2

 

 

0

7

6

1

4

3

2

5

 

 

6

57

56

11

38

25

24

43

3

6

1

4

3

6

1

4

 

 

7

0

1

6

3

4

5

2

 

 

60

7

10

53

28

39

42

21

 

 

The total amount of squares of group 10 is: 48 (row grids H) x 6 (column grids) x 2 (swapping row and column grids) = 576; 24 x 6 = 144 of these squares have the extra magic property X.

 

 

Illustration group 12

Magic squares of group 12 are constructed by means of combining H-grids with C*-grids .

 

In the example below the same row grid as above (H4 repeated in all four quadrants) has been chosen.  

 

 

 H4 (row grid)

0

5

2

7

0

5

2

7

6

3

4

1

6

3

4

1

5

0

7

2

5

0

7

2

3

6

1

4

3

6

1

4

0

5

2

7

0

5

2

7

6

3

4

1

6

3

4

1

5

0

7

2

5

0

7

2

3

6

1

4

3

6

1

4

 

 

Now you put C1*, C3* or C5* in the upper left corner (verify that C2*, C4* and C6* will not work in giving a matching column grid). In the example C1* has been chosen.

 

 

C1* (column grid), step 1

0

7

6

1

 

 

 

 

6

1

0

7

 

 

 

 

1

6

7

0

 

 

 

 

7

0

1

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In the 5th row you have only the matching option 1-6-7-0. And you only can finish the third quadrant successfully when maintaining the C*-structure.

 

   

 C1* (column grid), step 2

0

7

6

1

 

 

 

 

6

1

0

7

 

 

 

 

1

6

7

0

 

 

 

 

7

0

1

6

 

 

 

 

1

6

7

0

 

 

 

 

7

0

1

6

 

 

 

 

0

7

6

1

 

 

 

 

6

1

0

7

 

 

 

 

 

 

The right half of the row grid must be filled in with the digits 2, 3, 4, and 5. Column 5 needs 2-4-3-5 or 4-2-5-3. In the example 2-4-3-5 has been chosen. With both options you can successfully finish the upper right quadrant only when continuing the C*-structure. The down right quadrant follows automatically, and has necessarily the C*-structure.

 

 

 C1* (column grid), step 3 

0

7

6

1

2

5

4

3

6

1

0

7

4

3

2

5

1

6

7

0

3

4

5

2

7

0

1

6

5

2

3

4

1

6

7

0

 

 

 

 

7

0

1

6

 

 

 

 

0

7

6

1

 

 

 

 

6

1

0

7

 

 

 

 

 

 

 C1* (column grid), step 4

0

7

6

1

2

5

4

3

6

1

0

7

4

3

2

5

1

6

7

0

3

4

5

2

7

0

1

6

5

2

3

4

1

6

7

0

3

4

5

2

7

0

1

6

5

2

3

4

0

7

6

1

2

5

4

3

6

1

0

7

4

3

2

5

 

 

In total there are 3 (C1*, C3* or C5*) x 2 (options of step 3) = 6 different column grids.

 

Finally you can combine row and column grid to produce the magic square.

 

 

 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

7

6

1

2

5

4

3

 

 

1

62

51

16

17

46

35

32

6

3

4

1

6

3

4

1

 

 

6

1

0

7

4

3

2

5

 

 

55

12

5

58

39

28

21

42

5

0

7

2

5

0

7

2

 

 

1

6

7

0

3

4

5

2

 

 

14

49

64

3

30

33

48

19

3

6

1

4

3

6

1

4

 

 

7

0

1

6

5

2

3

4

 

 

60

7

10

53

44

23

26

37

0

5

2

7

0

5

2

7

 

 

1

6

7

0

3

4

5

2

 

 

9

54

59

8

25

38

43

24

6

3

4

1

6

3

4

1

 

 

7

0

1

6

5

2

3

4

 

 

63

4

13

50

47

20

29

34

5

0

7

2

5

0

7

2

 

 

0

7

6

1

2

5

4

3

 

 

6

57

56

11

22

41

40

27

3

6

1

4

3

6

1

4

 

 

6

1

0

7

4

3

2

5

 

 

52

15

2

61

36

31

18

45

 

 

The total number of squares of group 12 is: 48 (row grids H) x 6 (column grids) x 2 (swapping row and column grids) = 576; none of these squares can have the extra magic property X.

 

 

Illustration group 13

Magic squares of group 13 are constructed by means of combining 8x8 H-grids with 8x8 AC- or CA-grids.

 

Arbitrary we have constructed the following row grid:

 

 

 H4 (row grid)

0

5

2

7

2

7

0

5

6

3

4

1

4

1

6

3

5

0

7

2

7

2

5

0

3

6

1

4

1

4

3

6

1

4

3

6

3

6

1

4

2

7

0

5

0

5

2

7

4

1

6

3

6

3

4

1

7

2

5

0

5

0

7

2

 

 

Now we must find a matching AC column grid. We start filling the top left quadrant after A1.

 

 

 A1 (column grid), step 1

0

7

6

1

 

 

 

 

7

0

1

6

 

 

 

 

1

6

7

0

 

 

 

 

6

1

0

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The 5th row needs necessarily 1-6-7-0. And you will find out that you can only finish the third quadrant successfully when following the C* structure (if you follow the A-structure you get double pairings when composing the final magic square):

 

 

 A1 (column grid), step 2

0

7

6

1

 

 

 

 

7

0

1

6

 

 

 

 

1

6

7

0

 

 

 

 

6

1

0

7

 

 

 

 

1

6

7

0

 

 

 

 

7

0

1

6

 

 

 

 

0

7

6

1

 

 

 

 

6

1

0

7

 

 

 

 

 

 

Column 5 needs 2-5-3-4 or 4-3-5-2. In the example 2-5-3-4 has been chosen. With both options you can finish the upper right quadrant only successfully when following the

A-structure. The down right quadrant follows automatically, and has necessarily the C*-structure.

 

 

 A1 (column grid), step 3 

0

7

6

1

2

5

4

3

7

0

1

6

5

2

3

4

1

6

7

0

3

4

5

2

6

1

0

7

4

3

2

5

1

6

7

0

 

 

 

 

7

0

1

6

 

 

 

 

0

7

6

1

 

 

 

 

6

1

0

7

 

 

 

 

 

 

 A1 (column grid), step 4

0

7

6

1

2

5

4

3

7

0

1

6

5

2

3

4

1

6

7

0

3

4

5

2

6

1

0

7

4

3

2

5

1

6

7

0

3

4

5

2

7

0

1

6

5

2

3

4

0

7

6

1