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

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

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

 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 5 0 7 2 4 1 6 3 1 4 3 6 3 6 1 4 6 3 4 1 7 2 5 0 2 7 0 5 5 0 7 2 0 5 2 7 1 4 3 6 4 1 6 3 6 3 4 1 3 6 1 4 2 7 0 5 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 upper-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 preceding general information group 6-10 the following rowgrid 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 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

 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 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 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

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