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

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2

4

0

6

7

1

2

4

5

3

7

1

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6

5

3

2

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6

7

1

2

4

5

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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

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6

1

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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

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6

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7

 

 

 

 

0

7

6

1

 

 

 

 

7

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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

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3

 

 

 

6

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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

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6

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7

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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

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5

 

 

 

 

5

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7

2

5

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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

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5

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2

7

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5

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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

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2

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7

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7

2

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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

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2

5

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7

2

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4

1

6

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1

6

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1

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4

1

1

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6

1

4

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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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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