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