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Quadrant method group 6 up to 10
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General information 8x8 grids group 6-10

In the foregoing groups the quadrants consist 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 
0
5
2
7
 
 
 
 
6
3
4
1
 
 
 
 
5
0
7
2
 
 
 
 
3
6
1
4
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
In the top right quadrant there are 4 options, 2 with a H-structure, and 2 with a K-structure.
 
 
        H4                         K4                           H                           K
0
5
2
7
 
 
0
7
2
5
 
 
2
7
0
5
 
 
2
5
0
7
6
3
4
1
 
 
6
1
4
3
 
 
4
1
6
3
 
 
4
3
6
1
5
0
7
2
 
 
5
2
7
0
 
 
7
2
5
0
 
 
7
0
5
2
3
6
1
4
 
 
3
4
1
6
 
 
1
4
3
6
 
 
1
6
3
4
 

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

 
         H4                         H                            H                           H 
0
5
2
7
 
 
5
0
7
2
 
 
4
1
6
3
 
 
1
4
3
6
6
3
4
1
 
 
3
6
1
4
 
 
2
7
0
5
 
 
7
2
5
0
5
0
7
2
 
 
0
5
2
7
 
 
1
4
3
6
 
 
4
1
6
3
3
6
1
4
 
 
6
3
4
1
 
 
7
2
5
0
 
 
2
7
0
5


           H3                          H                           H                           H         
0
5
2
7
 
 
1
4
3
6
 
 
5
0
7
2
 
 
4
1
6
3
3
6
1
4
 
 
2
7
0
5
 
 
6
3
4
1
 
 
2
7
0
5
5
0
7
2
 
 
4
1
6
3
 
 
0
5
2
7
 
 
1
4
3
6
6
3
4
1
 
 
7
2
5
0
 
 
3
6
1
4
 
 
7
2
5
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
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
4
1
6
3
6
3
4
1
2
7
0
5
0
5
2
7
1
4
3
6
3
6
1
4
7
2
5
0
5
0
7
2
 
 
 
                     H4                                 K
0
5
2
7
0
7
2
5
6
3
4
1
6
1
4
3
5
0
7
2
5
2
7
0
3
6
1
4
3
4
1
6
4
1
6
3
4
3
6
1
2
7
0
5
2
5
0
7
1
4
3
6
1
6
3
4
7
2
5
0
7
0
5
2
 
 
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
top right quadrant:

 
         K4                          H                           K                          H4
0
7
2
5
 
 
2
7
0
5
 
 
2
5
0
7
 
 
0
5
2
7
6
1
4
3
 
 
4
1
6
3
 
 
4
3
6
1
 
 
6
3
4
1
5
2
7
0
 
 
7
2
5
0
 
 
7
0
5
2
 
 
5
0
7
2
3
4
1
6
 
 
1
4
3
6
 
 
1
6
3
4
 
 
3
6
1
4
 
 
Independent of the above options, for the down left quadrant there are 8 options, all with K-
structure:
 
 
        K4                           K                           K                            K
0
7
2
5
 
 
1
6
3
4
 
 
4
3
6
1
 
 
5
2
7
0
6
1
4
3
 
 
7
0
5
2
 
 
2
5
0
7
 
 
3
4
1
6
5
2
7
0
 
 
4
3
6
1
 
 
1
6
3
4
 
 
0
7
2
5
3
4
1
6
 
 
2
5
0
7
 
 
7
0
5
2
 
 
6
1
4
3
 
                 
0
7
2
5
 
 
1
6
3
4
 
 
4
3
6
1
 
 
5
2
7
0
3
4
1
6
 
 
2
5
0
7
 
 
7
0
5
2
 
 
6
1
4
3
5
2
7
0
 
 
4
3
6
1
 
 
1
6
3
4
 
 
0
7
2
5
6
1
4
3
 
 
7
0
5
2
 
 
2
5
0
7
 
 
3
4
1
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 previous general information group 6-10 the following row grid has been constructed:
 
                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
4
1
6
3
6
3
4
1
2
7
0
5
0
5
2
7
1
4
3
6
3
6
1
4
7
2
5
0
5
0
7
2
 

Now we are going to construct a matching column-grid 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
0
6
3
5
 
 
 
 
3
5
0
6
 
 
 
 
4
2
7
1
 
 
 
 
7
1
4
2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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
6
3
5
 
 
 
 
3
5
0
6
 
 
 
 
4
2
7
1
 
 
 
 
7
1
4
2
 
 
 
 
4
2
7
1
 
 
 
 
7
1
4
2
 
 
 
 
0
6
3
5
 
 
 
 
3
5
0
6
 
 
 
 
 
 
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
0
6
3
5
2
4
1
7
3
5
0
6
1
7
2
4
4
2
7
1
6
0
5
3
7
1
4
2
5
3
6
0
4
2
7
1
 
 
 
 
7
1
4
2
 
 
 
 
0
6
3
5
 
 
 
 
3
5
0
6
 
 
 
 
 
 
                      H5* (column grid), step 4
0
6
3
5
2
4
1
7
3
5
0
6
1
7
2
4
4
2
7
1
6
0
5
3
7
1
4
2
5
3
6
0
4
2
7
1
6
0
5
3
7
1
4
2
5
3
6
0
0
6
3
5
2
4
1
7
3
5
0
6
1
7
2
4
 
 
Starting the top 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
2
7
2
7
0
5
 
 
0
6
3
5
2
4
1
7
 
 
1
54
27
48
19
40
9
62
6
3
4
1
4
1
6
3
 
 
3
5
0
6
1
7
2
4
 
 
31
44
5
50
13
58
23
36
5
0
7
2
7
2
5
0
 
 
4
2
7
1
6
0
5
3
 
 
38
17
64
11
56
3
46
25
3
6
1
4
1
4
3
6
 
 
7
1
4
2
5
3
6
0
 
 
60
15
34
21
42
29
52
7
4
1
6
3
6
3
4
1
 
 
4
2
7
1
6
0
5
3
 
 
37
18
63
12
55
4
45
26
2
7
0
5
0
5
2
7
 
 
7
1
4
2
5
3
6
0
 
 
59
16
33
22
41
30
51
8
1
4
3
6
3
6
1
4
 
 
0
6
3
5
2
4
1
7
 
 
2
53
28
47
20
39
10
61
7
2
5
0
5
0
7
2
 
 
3
5
0
6
1
7
2
4
 
 
32
43
6
49
14
57
24
35
                H                         H                                       H*                       H*   
 
 
Important: Note that when we swap the 6th and the 8th row in the above construction the
type of combination changes from HHHH/H*H*H*H* into HHHH/H*H*K*K*, see below :
 
 
  1x digit from row grid +1    +     8x digit from column grid     =     most perfect 8x8 magic square
0
5
2
7
2
7
0
5
 
 
0
6
3
5
2
4
1
7
 
 
1
54
27
48
19
40
9
62
6
3
4
1
4
1
6
3
 
 
3
5
0
6
1
7
2
4
 
 
31
44
5
50
13
58
23
36
5
0
7
2
7
2
5
0
 
 
4
2
7
1
6
0
5
3
 
 
38
17
64
11
56
3