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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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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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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 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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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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 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 

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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 

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