Quadrant method, group 1-5

Illustration group 1

Magic squares of group 1 can be constructed by means of combining G-grids (i.e. all quadrants have a G-structure) with G*-grids (i.e. all quadrants have a G*-structure).

First you construct the row grid. Fill the upper left quadrant with the digits after G1, G2, G3, G4, G5 or G6. In the example G1 has been chosen.

G1 (row grid), step 1

0	6	7	1
7	1	0	6
0	6	7	1
7	1	0	6

Necessarily the same G-quadrant must be repeated in the down left corner (as is shown with the purple digits).

G1 (row grid), step 2

0	6	7	1
7	1	0	6
0	6	7	1
7	1	0	6
0	6	7	1
7	1	0	6
0	6	7	1
7	1	0	6

The right half of the row grid must be filled with the digits 2, 3, 4, and 5. In column 5 and 7 it is only possible to continue with an alteration of 2-5, 5-2, 3-4, or 4-3, that is four options. In the example 2-5 has been chosen.

G1 (row grid), step 3

0	6	7	1	2	5
7	1	0	6	5	2
0	6	7	1	2	5
7	1	0	6	5	2
0	6	7	1	2	5
7	1	0	6	5	2
0	6	7	1	2	5
7	1	0	6	5	2

Now in column 6 and 8 it is only possible to continue with an alteration of 3-4 or 4-3, that is two options. In the example 4-3 has been chosen.

G1 (row grid), step 4

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

In total there are 6 (G1 to G6) x 4 (options of step 3) x 2 (options of step 4) = 48 possible row grids.

By means of diagonally reflecting the 48 row grids you can produce 48 column grids. In the G-group it is possible to match all 48 row grids with all 48 column grids. See below one of the 64 possible squares of G1/G1*. Note that in the example the column grid is the reflection of the row grid.

1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square

0	6	7	1	2	4	5	3	0	7	0	7	0	7	0	7	1	63	8	58	3	61	6	60
7	1	0	6	5	3	2	4	6	1	6	1	6	1	6	1	56	10	49	15	54	12	51	13
0	6	7	1	2	4	5	3	7	0	7	0	7	0	7	0	57	7	64	2	59	5	62	4
7	1	0	6	5	3	2	4	1	6	1	6	1	6	1	6	16	50	9	55	14	52	11	53
0	6	7	1	2	4	5	3	2	5	2	5	2	5	2	5	17	47	24	42	19	45	22	44
7	1	0	6	5	3	2	4	4	3	4	3	4	3	4	3	40	26	33	31	38	28	35	29
0	6	7	1	2	4	5	3	5	2	5	2	5	2	5	2	41	23	48	18	43	21	46	20
7	1	0	6	5	3	2	4	3	4	3	4	3	4	3	4	32	34	25	39	30	36	27	37

The combination G/G* gives in total 48x48 = 2304 squares.

The first square we made contains the extra magic property X. Make sure that this

property can only arise in case of the following 6 sequences of digits in the first row or column: 0-6-7-1-2-4-5-3, 0-6-7-1-4-2-3-5, 0-5-7-2-1-4-6-3, 0-5-7-2-4-1-3-6, 0-3-7-4-1-2-6-5 and 0-3-7-4-2-1-5-6. Consequently the amount of squares in group 1 containing this property is 6 x 6 = 36.

Illustration group 2

Magic squares of group 2 can be produced by means of combining A-grids with B-grids.

First you construct the row grid. Fill the upper left quadrant after A1, A2 or A3. In the example A1 has been chosen.

A1 (row grid), step 1

0	7	6	1
7	0	1	6
1	6	7	0
6	1	0	7

In the 5th row you can only continue with 0-7-6-1 or 1-6-7-0. With both options you can finish the down left quadrant, continuing the A-structure (N.B.: that is a choice, you are

going to construct an A-grid!). In the example 0-7-6-1 has been chosen, which

means repeating the first quadrant.

A1 (row grid), step 2

0	7	6	1
7	0	1	6
1	6	7	0
6	1	0	7
0	7	6	1
7	0	1	6
1	6	7	0
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 continue with 2-5-3-4, or 4-3-5-2. In the example 2-5-3-4 has been

chosen. With both options you can finish the upper right quadrant, and

continuing the A-structure. The down right quadrant follows automatically, and

shows necessarily also the A-structure.

A1 (row grid), step 3

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

A1 (row grid), step 4

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

Recaputilating, we started with A1 in the upper left corner. There are 2 options to finish the down left quadrant in A. Then there are two options to finish the top right quadrant in A. Finally there is only one option to finish the down right quadrant. So, putting A1 in the upper left corner gives 4 different options to produce the AAAA-grid. However, we could have started by putting A2 or A3 in the upper left corner. Conclusion: there are 3 x 4 = 12 AAAA grids.

Now we construct a matching column grid. Fill the upper left quadrant after B1, B2, or B3. In the example B2 has been chosen.

B2 (column grid), step 1

0	5	2	7
2	7	0	5
5	0	7	2
7	2	5	0

Consider the 5th column. To maintain the panmagic properties position 1 and 2 must be filled with the digits 0 and 2, so the sequence becomes 0-2-5-7 or 2-0-7-5. In the example

0-2-5-7 has been chosen, which means repeating the first quadrant.

B2 (column grid), step 2

0	5	2	7	0	5	2	7
2	7	0	5	2	7	0	5
5	0	7	2	5	0	7	2
7	2	5	0	7	2	5	0

The down half of the column grid must be filled with the digits 1, 3, 4, and 6. For the 5th row there are two options: 4-1-6-3 of 1-4-3-6. In the example 4-1-6-3 has been chosen. With both options you can finish the down left and right quadrants, and maintaining the B-structure.

B2 (column grid), step 3

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

B2 (column grid), step 4

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

In total there are 3 (B1, B2, B3) x 2 (options of step 3) x 2 (options of step 4) = 12

different BBBB grids.

Finally the magic square can be composed:

1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square

0	7	6	1	2	5	4	3	0	5	2	7	0	5	2	7	1	48	23	58	3	46	21	60
7	0	1	6	5	2	3	4	2	7	0	5	2	7	0	5	24	57	2	47	22	59	4	45
1	6	7	0	3	4	5	2	5	0	7	2	5	0	7	2	42	7	64	17	44	5	62	19
6	1	0	7	4	3	2	5	7	2	5	0	7	2	5	0	63	18	41	8	61	20	43	6
0	7	6	1	2	5	4	3	4	1	6	3	4	1	6	3	33	16	55	26	35	14	53	28
7	0	1	6	5	2	3	4	6	3	4	1	6	3	4	1	56	25	34	15	54	27	36	13
1	6	7	0	3	4	5	2	1	4	3	6	1	4	3	6	10	39	32	49	12	37	30	51
6	1	0	7	4	3	2	5	3	6	1	4	3	6	1	4	31	50	9	40	29	52	11	38

The combination A/B gives in total 12x12 = 144 possibilities. Structure A and B are

no reflections of each other; consequently the combination B/A (= swapping row

and column grids) gives another 144 squares, and the total amount of squares of

combination 2 is 144 + 144 = 288.

See also quadrant method applied to construct a most perfect 12x12 A/B panmagic square on the website of Willem Barink: http://wba.novaloka.nl/magic-squares.html

Illustration group 3

Magic squares of group 3 can be produced by means of combining C-grids with C*-grids combination 3a), and C-grids with C*-grids (combination 3b).

First you construct the row grid. Fill the upper left quadrant after C1, C3, or C5. In the example C1 (combination 3a) has been chosen.

C1 (row grid), step 1

0	6	1	7
7	1	6	0
6	0	7	1
1	7	0	6

Consider the 5th column. To maintain the panmagic properties position 2 and 3 must be filled with the digits 7 and 6, so the sequence becomes 0-7-6-1 or 1-6-7-0. With both options you can finish the upper-right quadrant, maintaining the C-structure (N.B.: Make sure that when you would have chosen C2 (combination 3b) upper-left, you would have been forced to fill the down-left quadrant with the digits 0,7,1,6.). In the example the sequence 0-7-6-1 has been chosen.

C1 (row grid), step 2

0	6	1	7	0	6	1	7
7	1	6	0	7	1	6	0
6	0	7	1	6	0	7	1
1	7	0	6	1	7	0	6

The down half of the grid must be filled in with the digits 2, 3, 4, and 5. Row 5 can only be filled with 2-4-3-5, or 4-2-5-3. In the example 2-4-3-5 has been chosen. With both options you can finish the down-left and down-right quadrants, maintaining the C-structure.

C1 (row grid), step 3

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

C1 (row grid), step 4

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

In total there are 3 (C1, C3, C5) x 2 (options of step 3) x 2 (options of step 4) = 12 different row grids.

Construction of a matching column grid goes the same way. By experiment you can find out that you have to start the grid with C1*, C3* or C5*. If you start the grid with C2*, C4* or C6* there is no solution to complete a matching column grid. Keep in mind that when having started the row grid with C1, C3 or C5, then you have to start the column grid with C1*, C3*, or C5*. The same is valid for C2, C4, and C6.

Started the column grid with C1*, C3* or C5*, there are 2 options to continue in the upper right quadrant, and two options to continue in the down left quadrant, so the amount of matching grids is 4. So, the above row grid has 3 x 4 = 12 matching column grids.

See below one of the the combinations C1/C1*.

1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square

0	6	1	7	0	6	1	7	0	7	6	1	2	5	4	3	1	63	50	16	17	47	34	32
7	1	6	0	7	1	6	0	6	1	0	7	4	3	2	5	56	10	7	57	40	26	23	41
6	0	7	1	6	0	7	1	1	6	7	0	3	4	5	2	15	49	64	2	31	33	48	18
1	7	0	6	1	7	0	6	7	0	1	6	5	2	3	4	58	8	9	55	42	24	25	39
2	4	3	5	2	4	3	5	0	7	6	1	2	5	4	3	3	61	52	14	19	45	36	30
5	3	4	2	5	3	4	2	6	1	0	7	4	3	2	5	54	12	5	59	38	28	21	43
4	2	5	3	4	2	5	3	1	6	7	0	3	4	5	2	13	51	62	4	29	35	46	20
3	5	2	4	3	5	2	4	7	0	1	6	5	2	3	4	60	6	11	53	44	22	27	37

The combination C/C* gives in total 12x12 = 144 possibilities. The combination C*/C

(= swapping row and column grids) gives no new squares. So the total number of squares of group 3a is 144. Analogously group 3b (combination C/C*) gives also 144 squares., which makes in total 288 squares for group 3.

In the C/C*- combination a quarter ( = 36) of the produced squares contains the extra magic property X. This can be deduced from above example, combined with the following arguments:

Half of the rowgrids have rows with a suited sequence of digits, e.g. 0-6-1-7-0-6-1-7 is suited, 0-6-1-7-1-7-0-6 is not, see above example;
Likewise half of the columngrids have colums with a suited sequence of digits.

See quadrant method applied to construct a most perfect panmagic 12x12 and 16x16 C/C* square (with the extra magic property X) on the website of Willem Barink: http://wba.novaloka.nl/magic-squares.html

Illustration group 4

Magic squares of group 4 can be constructed by means of combining row grids consisting of A-, B-, C- and C-quadrants with column grids of the same, reflected structures. There are the following 3 options to combine the ABC-grids, corresponding with group 4a, 4b and 4c in the table of combinations:

1. 2. 3. .

C

A*

C*

B*

C

A

C*

B

A

C

B

C*

B

C*

A

C

B*

C*

A*

C

C*

B*

C

A*

Make sure that options 1 and 2 are reflexive combinations, but that option 2 is not

(cannot be!) a reflection of option 1. Make also sure that option 3 is a non-reflexive combination. In the below illustration is chosen for option 2 (combination 4b).

First you construct the CA*BC* row grid. Fill the upper left quadrant after C2, C4, or C6. In the example C2 has been chosen.

C2 (row grid), step 1

0	1	6	7
7	6	1	0
1	0	7	6
6	7	0	1

In the 5th row you can only continue with 0-1-6-7 or 1-0-7-6. In the example 0-1-6-7 has been chosen. Because of option 2, the down-left corner must be finished after B1*.

C2 (row grid), step 2

0	1	6	7
7	6	1	0
1	0	7	6
6	7	0	1
0	1	6	7
6	7	0	1
1	0	7	6
7	6	1	0

B1*

The right half of the row grid must be filled with the digits 2, 3, 4, and 5. In column 5 only the sequences 2-5-3-4, and 4-3-5-2 are possible. In the example 2-5-3-4 has been chosen. With both options you can to finish the upper right quadrant with an A-structure (our choice, option 2). Then the down right quadrant follows automatically (and has obviously the C*-structure).

C2 (row grid), step 3 A

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

B1*

C2 (row grid ), step 4 A

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

B1* C1*

In total there are 3 (C2, C4, C6) x 2 (options of step 3) x 2 (options of step 4) = 12 different CABC* row grids.

Now you construct the column grid. Fill the upper left corner with a C*-structure, that means C2*, C4* or C6*. In the example below C6* has been chosen.. Then the top right quadrant has been completed in B (two options). Further the down left quadrant has been completed in A, again two options.

C6* (column grid) B3*

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

A C4

The amount of matching C*BAC column grids is 3 x 2 x 2 = 12.

Next row and column grid can be combined to the final magic square.

1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square

0	1	6	7	2	5	4	3	0	7	4	3	0	3	4	7	1	58	39	32	3	30	37	60
7	6	1	0	5	2	3	4	4	3	0	7	4	7	0	3	40	31	2	57	38	59	4	29
1	0	7	6	3	4	5	2	3	4	7	0	3	0	7	4	26	33	64	7	28	5	62	35
6	7	0	1	4	3	2	5	7	0	3	4	7	4	3	0	63	8	25	34	61	36	27	6
0	1	6	7	2	5	4	3	1	6	5	2	1	2	5	6	9	50	47	24	11	22	45	52
6	7	0	1	4	3	2	5	6	1	2	5	6	5	2	1	55	16	17	42	53	44	19	14
1	0	7	6	3	4	5	2	2	5	6	1	2	1	6	5	18	41	56	15	20	13	54	43
7	6	1	0	5	2	3	4	5	2	1	6	5	6	1	2	48	23	10	49	46	51	12	21

The combination CABC*/C*BAC gives in total 12x12 = 144 squares. Being a reflexive combination, swapping gives no new squares.

The row and column grids of combination 4a are constructed in a similar way as combination 4b, and the combination gives also 144 squares.

Finally an example of the non-reflexive combination 4c, in which an ACC*B- row grid has been combined with a BC*CA-column grid.

1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square

0	7	4	3	0	3	4	7	0	1	6	7	2	5	4	3	1	16	53	60	17	44	37	32
7	0	3	4	7	4	3	0	6	7	0	1	4	3	2	5	56	57	4	13	40	29	20	41
3	4	7	0	3	0	7	4	1	0	7	6	3	4	5	2	12	5	64	49	28	33	48	21
4	3	0	7	4	7	0	3	7	6	1	0	5	2	3	4	61	52	9	8	45	24	25	36
1	6	5	2	1	2	5	6	0	1	6	7	2	5	4	3	2	15	54	59	18	43	38	31
5	2	1	6	5	6	1	2	7	6	1	0	5	2	3	4	62	51	10	7	46	23	26	35
2	5	6	1	2	1	6	5	1	0	7	6	3	4	5	2	11	6	63	50	27	34	47	22
6	1	2	5	6	5	2	1	6	7	0	1	4	3	2	5	55	58	3	14	39	30	19	42

Again there are 3x2x2 row grids and 3x2x2 column grids. The total number of squares that can be generated in group 4c is: 12 (row grids) x 12 (column grids) x 2 (swapping row and column grids) = 288.

In total the combinations of group 4 produce 144 + 144 + 288 = 576 squares.

Illustration group 5

Magic squares of group 5 can be constructed by means of combining AC or BC row grids with BC* or AC* column grids and vice versa. See below for the possible combinations.

1. 2. 3. 4.

A

B

C*

A

C

C*

C

A*

C*

B*

C*

C

B

C*

A

C

B*

A*

C

A*

C*

B*

Note that none of the options is a reflexive combination. Option 1 is combination 5d in the combination table, option 2 is 5c, option 3 is 5b and option 4 is 5a.

In the below illustration combination 5c (option 2) has been chosen. First you construct the BC* row grid. Fill the top left quadrant after B1, B2 or B3. In the example B2 has been chosen.

B2 (row grid), step 1

0	5	2	7
2	7	0	5
5	0	7	2
7	2	5	0

The 5th column necessarily needs 0-2-5-7 or 2-0-7-5. With both options the upper right

quadrant can be finished with a C*-structure. (Note that C*- does not work!). In the example 0-2-5-7 has been chosen.

B2 (row grid), step 2 C*

0	5	2	7	0	7	2	5
2	7	0	5	2	5	0	7
5	0	7	2	5	2	7	0
7	2	5	0	7	0	5	2

The down half of the row grid must be filled with the digits 1, 3, 4, and 6. Row 5 of the down left quadrant needs necessarily 1-4-3-6, or 4-1-6-3. In the example 1-4-3-6 has been

chosen, and the quadrant is finished following the B-structure. The down right quadrant follows automatically and has (obviously) also the C*-structure.

B2 (row grid), step 3

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

B2

B2 (row grid), step 4 C*

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

B2 C*

In total there are 3 (B1, B2, B3) x 2 (options of step 2) x 2 (options of step 3) = 12

different BC*BC* row grids.

Now you construct an ACAC column grid. For example the following one.

A1 (column grid) C

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

A1 C

And the final square. See below.

1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square

0	5	2	7	0	7	2	5	0	7	6	1	2	3	4	5	1	62	51	16	17	32	35	46
2	7	0	5	2	5	0	7	7	0	1	6	5	4	3	2	59	8	9	54	43	38	25	24
5	0	7	2	5	2	7	0	1	6	7	0	3	2	5	4	14	49	64	3	30	19	48	33
7	2	5	0	7	0	5	2	6	1	0	7	4	5	2	3	56	11	6	57	40	41	22	27
1	4	3	6	1	6	3	4	0	7	6	1	2	3	4	5	2	61	52	15	18	31	36	45
3	6	1	4	3	4	1	6	7	0	1	6	5	4	3	2	60	7	10	53	44	37	26	23
4	1	6	3	4	3	6	1	1	6	7	0	3	2	5	4	13	50	63	4	29	20	47	34
6	3	4	1	6	1	4	3	6	1	0	7	4	5	2	3	55	12	5	58	39	42	21	28

There are 3x2x2 BC*BC* row grids and 3x2x2 ACAC column grids. The total amount of squares that can be generated with combination 5c is: 12 (row grids) x 12 (column grids) x 2 (swapping row and column grids) = 288

The row and column grids of the other combinations of group 5 are produced in a similar way as group 5c. See the following example, in which a CCBB- row grid has been combined with a C*C*AA-column grid (combination 5b). Also this combination, being non-reflexive, produces 288 squares.

1x digit from row grid +1 + 8x digit from column grid = most perfect 8x8 magic square

0	4	3	7	1	5	2	6	0	7	1	6	1	6	0	7	1	61	12	56	10	54	3	63
7	3	4	0	6	2	5	1	1	6	0	7	0	7	1	6	16	52	5	57	7	59	14	50
4	0	7	3	5	1	6	2	6	1	7	0	7	0	6	1	53	9	64	4	62	2	55	11
3	7	0	4	2	6	1	5	7	0	6	1	6	1	7	0	60	8	49	13	51	15	58	6
0	4	3	7	1	5	2	6	4	3	5	2	5	2	4	3	33	29	44	24	42	22	35	31
3	7	0	4	2	6	1	5	3	4	2	5	2	5	3	4	28	40	17	45	19	47	26	38
4	0	7	3	5	1	6	2	2	5	3	4	3	4	2	5	21	41	32	36	30	34	23	43
7	3	4	0	6	2	5	1	5	2	4	3	4	3	5	2	48	20	37	25	39	27	46	18

In total the combinations of group 5 produce 4 x 288 = 1152 squares.

8x8, Quadrant method, group 1 up to 5.do

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