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