8x8 magic square

Quadrant method, Introduction

 

Introduction

You can use the quadrant method of Willem Barink (also creator of the medjig method) to construct an 8x8 most perfect (Franklin pan)magic square. The quadrant method is suited to construct most perfect panmagic 8x8 squares, but in adapted form the method can also be used for the construction of higher order most perfect panmagic squares. See for some panmagic constructions of order 12 and 16 the website http://wba.novaloka.nl/magic-squares.html . This paper deals only with panmagic 8x8 squares, and confines to squares starting with the number 1 top left.

 

Most perfect panmagic 8x8 squares necessarily have also the franklin magic properties, the reverse is not the case (see explanation most perfect).

 

The quadrant method stands for constructing and combining panmagic 4x4 quadrants in order to build two matching 8x8 grids, one for the units and one for the octuples. On page ‘panmagic 5x5 square, explanation’ these grids are called row grid and column grid; to ensure uniformity in terminology on this website, these terms are maintained in this paper.

 

Both row and column grids consist of 8 times the digits 0  to 7. To construct a “most perfect” panmagic 8x8 square, it is essential to realize that the two 8x8 grids both must have the panmagic properties of the square, and also that the quadrants of the grids individually must reflect these properties.

 

This paper deals only with the construction of squares with the number 1 upper-left. Therefore the up left quadrants - the  “0-quadrants” - must have the digit 0 upper-left. According to Willem Barink only 30 such panmagic quadrants exist, diagonally reflections excluded. Based on these 0-quadrants Willem Barink investigated systematically the combinatory possibilities of the quadrants to form panmagic matching grids, found 37 possibilities of combination, and from there he calculated the amounts of possible squares (N.B.: in the  first instalment of this paper the amount was 18; the recent larger amount is not due to new found combinations, but due to an adjusted interpretation and a more precise presentation.).   

 

In this paper you will find successively the table of 0-quadrants, the table of  combination possibilities, and for (nearly) all combinations a more or less detailed illustration. In these illustrations will be shown how to construct and combine the quadrants to form matching 8x8 grids, included a resulting magic square. Based on the method is calculated how many different magic squares the combinations can produce.

 

 

The 30 0-quadrants 

Here they are, categorized according to their structural characteristics:

 

 

0

6

7

1

 

 

0

1

7

6

 

 

0

5

7

2

 

 

0

2

7

5

 

 

0

4

7

3

 

 

0

3

7

4

7

1

0

6

 

 

7

6

0

1

 

 

7

2

0

5

 

 

7

5

0

2

 

 

7

3

0

4

 

 

7

4

0

3

0

6

7

1

 

 

0

1

7

6

 

 

0

5

7

2

 

 

0

2

7

5

 

 

0

4

7

3

 

 

0

3

7

4

7

1

0

6

 

 

7

6

0

1

 

 

7

2

0

5

 

 

7

5

0

2

 

 

7

3

0

4

 

 

7

4

0

3

 G1                     G2                      G3                     G4                      G5                     G6

 

 

0

6

1

7

 

 

0

5

2

7

 

 

0

3

4

7

 

 

0

7

6

1

 

 

0

7

5

2

 

 

0

7

3

4

1

7

0

6

 

 

2

7

0

5

 

 

4

7

0

3

 

 

7

0

1

6

 

 

7

0

2

5

 

 

7

0

4

3

6

0

7

1

 

 

5

0

7

2

 

 

3

0

7

4

 

 

1

6

7

0

 

 

2

5

7

0

 

 

4

3

7

0

7

1

6

0

 

 

7

2

5

0

 

 

7

4

3

0

 

 

6

1

0

7

 

 

5

2

0

7

 

 

3

4

0

7

 B1                  B2                  B3                   A1                  A2                 A3

 

 

0

6

1

7

 

 

0

1

6

7

 

 

0

5

2

7

 

 

0

2

5

7

 

 

0

3

4

7

 

 

0

4

3

7

7

1

6

0

 

 

7

6

1

0

 

 

7

2

5

0

 

 

7

5

2

0

 

 

7

4

3

0

 

 

7

3

4

0

6

0

7

1

 

 

1

0

7

6

 

 

5

0

7

2

 

 

2

0

7

5

 

 

3

0

7

4

 

 

4

0

7

3

1

7

0

6

 

 

6

7

0

1

 

 

2

7

0

5

 

 

5

7

0

2

 

 

4

7

0

3

 

 

3

7

0

4

 C1                      C2                     C3                      C4                     C5                      C6

 

 

 

0

6

1

7

 

 

0

6

1

7

 

 

0

5

2

7

 

 

0

5

2

7

 

 

0

3

4

7

 

 

0

3

4

7

3

5

2

4

 

 

5

3

4

2

 

 

3

6

1

4

 

 

6

3

4

1

 

 

6

5

2

1

 

 

5

6

1

2

6

0

7

1

 

 

6

0

7

1

 

 

5

0

7

2

 

 

5

0

7

2

 

 

3

0

7

4

 

 

3

0

7

4

5

3

4

2

 

 

3

5

2

4

 

 

6

3

4

1

 

 

3

6

1

4

 

 

5

6

1

2

 

 

6

5

2

1

 H1                  H2                  H3                 H4                  H5                  H6

 

 

0

7

1

6

 

 

0

7

1

6

 

 

0

7

2

5

 

 

0

7

2

5

 

 

0

7

4

3

 

 

0

7

4

3

5

2

4

3

 

 

3

4

2

5

 

 

3

4

1

6

 

 

6

1

4

3

 

 

6

1

2

5

 

 

5

2

1

6

6

1

7

0

 

 

6

1

7

0

 

 

5

2

7

0

 

 

5

2

7

0

 

 

3

4

7

0

 

 

3

4

7

0

3

4

2

5

 

 

5

2

4

3

 

 

6

1

4

3

 

 

3

4

1

6

 

 

5

2

1

6

 

 

6

1

2

5

 K1                     K2                      K3                     K4                      K5                     K6

 

 

The quadrants are divided into six different structures, called G, A, B, C, H and K. Playing with these structures in order to construct an 8x8 grid, you will find out that the combination possibilities are limited. Below is shown the scheme of the 14 (different) possibilities to combine the 0-quadrants and its derivatives (= quadrants not starting with 0 upper-left) to form a grid:

 

 

G

G

 

 

A

A

 

 

B

B

 

 

C

C

 

 

C

C

 

 

H

H

 

 

K

K

G

G

 

 

A

A

 

 

B

B

 

 

C

C

 

 

C

C

 

 

H

H

 

 

K

K

 

  

A

A

 

 

A

C

 

 

B

B

 

 

B

C*

 

 

C

A*

 

 

C

A

 

 

H

K

C*

C*

 

 

A

C

 

 

C

C

 

 

B

C*

 

 

B

C*

 

 

B*

C*

 

 

H

K

 

 

With the brown and black C is indicated that the C-structure forms two groups of

combinations of grids: one with C1, C3 or C5 (and derivatives), the other with C2, C4, or C6. When not being 0-quadrants the distinction is tricky, and for the construction of squares not important. However, the distinction is important in order to fix the amount of (different) combinations and from there the calculation of the amounts of squares.

 

With * is meant: diagonally reflected (structure of the) quadrant. When not being 0-quadrants, the distinction between A- and A*-structures, and B- and B*-structures is also tricky, however important in order to fix the amount of combinations and squares.

 

When trying to construct a matching second grid, you will find out that the possibilities of matching the structures are limited. The G-structure behaves very exclusive: G-quadrants not only combine with G-quadrants to form a grid (see the scheme above), they also match only with their own (reflected) structure in the other grid. Below is summarized how the structures in the two grids for each quadrant match upon each other (are

“orthogonal” with each other).

 

G <> G*

A <> B, H

B <> A, K*,

C <> C*, H*

C <> C*, K
 H <> H*, K*, C*, A,

K <> K*, H*, C, B*

 

 

The combinations

See below the combination possibilities, based on the 30 0-quadrants,  and the calculated amount of squares, number 1 upper-left,  the combination can produce.

For the calculation it is crucial to distinguish between:

  • reflexive combinations, which means: concerning structure the grids are diagonal reflections of each other (e.g. combination 1, 3a, 3b, 4a, 4b, etc..); when swapping the grids they give only rise to diagonal reflections of already made squares;
  • non-reflexive combinations (e.g. combination 2, 4c, etc..) ; they give rise to new squares when swapping the grids.

The number in the X-column stands for the amount of squares in the group concerned with a special magic property, viz. that in each row and column not only the sum of the numbers on position 1 to 4 and 5 to 8, but also the sum of the numbers on position 3 to 6 gives the magic sum of 130.

 

 

 

 

 

 Row/column grid

 

 

 

 r x k

  

 

 k/r-switch

 

 Total 

 

 

 X

1

G

G

 

G*

G*

G

G

 

G*

G*

 

 48x48

= 2304

        -

=    2304

 

 36

 

 

 

 

 

 

 

 

 

 

2

A

A

 

B

B

A

A

 

B

B

 

 12x12

=   144

+   144    

=      288

 

 

 

 

3a

C

C

 

C*

C*

C

C

 

C*

C*

 

 12x12

=   144

         -

=      144

C1, C3, C5     

 36

 

 

3b

C

C

 

C*

C*

C

C

 

C*

C*

 

 12x12

=   144

         -

=      144

 

  

 

4a

C

A*

 

C*

B*

B

C*

 

A

C

 

 12x12

=   144

         -

=      144

 

 

 

 

4b

C

A

 

C*

B

B*

C*

 

A*

C

 

 12x12

=   144

         -

=      144

 

 

 

 

4c

A

C

 

B

C*

C*

B*

 

C

A*

 

 12x12

=   144

+   144

=      288

 

 

 

 

5a

C

A*

 

C*

B*

C

A*

 

C*

B*

 

 12x12

=   144

+   144

=      288

 

 

 

 

5b

C

C

 

C*

C*

B*

B*

 

A*

A*

 

 12x12

=   144

+   144

=      288

 

 

 

 

5c

B

C*

 

A

C

B

C*

 

A

C

 

 12x12

=   144

+   144

=      288

 

 

 

 

5d

A

A

 

B

B

C*

C*

 

C

C

 

 12x12

=   144

+   144

=      288

 

 

 

 

 

 

 

 

 

 

 

 

6

H

H

 

H*

H*

H

H

 

H*

H*

 

 48x12

= 576

         -

=     576

 

 

 

 

7

K

K

 

K*

K*

K

K

 

K*

K*

 

 48x12

= 576

         -

=     576

 

144

 

 

8

H

H

 

K*

K*

H

H

 

K*

K*

 

 48x12

= 576

+ 576

=    1152

 

 

 

 

9a

H

H

 

H*

H*

H

H

 

K*

K*

 

 48x12

= 576

+ 576

=    1152

 

 

 

 

 

 

 

 

 

 

 

 

9b

H

H

 

K*

K*

H

H

 

H*

H*

 

 48x12

=   576

+ 576

=    1152

 

 

 

 

9c

K

K

 

K*

K*

K

K

 

H*

H*

 

 48x12

= 576

+ 576

=    1152

 

 

 

 

9d

K

K

 

H*

H*

K

K

 

K*

K*

 

 48x12

=   576

+ 576

=    1152

 

 

 

 

10a

H

K

 

H*

H*

H

K

 

K*

K*

 

 48x12

= 576

       -

=     576

 

 

 

 

10b

K

H

 

K*

K*

K

H

 

H*

H*

 

 48x12

= 576

       -

=     576

 

 

 

 

10c

H

K

 

K*

K*

H

K

 

H*

H*

 

 48x12

= 576

+ 576

=    1152

 

 

 

 

 

 

 

 

 

 

 

 

11

H

H

 

A

A

H

H

 

A

A

 

 48 x 6

=   288

+ 288

=      576

 

144

 

 

12

H

H

 

C*

C*

H

H

 

C*

C*

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

13a

H

H

 

C*

C*

H

H

 

A

A

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

 

 

 

 

 

 

 

 

13b

H

H

 

A

A

H

H

 

C*

C*

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

14

K

K

 

B*

B*

K

K

 

B*

B*

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

15

K

K

 

C

C

K

K

 

C

C

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

16a

K

K

 

C

C

K

K

 

B*

B*

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

16b

K

K

 

B*

B*

K

K

 

C

C

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

17a

H

K

 

C*

B*

H

K

 

C*

B*

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

17b

K

H

 

B*

C*

K

H

 

B*

C*

 

 48 x 6

=   288

+   288

=      576

 

 

 

 

18a

H

K

 

A

C

H

K

 

A

C

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

18b

K

H

 

C

A

K

H

 

C

A

 

 48 x 6

=   288

+ 288

=      576

 

 

 

 

19a

H

K

 

A

C

H

K

 

C*

B*

 

 48 x 6           

=   288

+ 288

=      576

 

 

 

 

19b

H

K

 

C*

B*

H

K

 

A

C

 

 48 x 6           

=   288

+ 288

=      576

 

 

 

 

19c

K

H

 

B*

C*

K

H

 

C

A

 

 48 x 6

=   288

+   288

=      576

 

 

 

 

19d

K

H

 

C

A*

K

H

 

B*

C*

 

 48 x 6

=   288

+   288

=      576

 

 

 

 

 

 

 

   14112

+   8928

    = 23040

 

 360

 

 

The table of combinations shows that there are 4 classes of combinations:

  • combination1, the G-squares, 1 x 2304 squares; 36 with X
  • combination 2-5d, the A,B,C-squares, 16 x 144 = 2304 squares; 72 with X
  • combination 6-10c, the H,K-squares, 16 x 576 = 9216 squares, 144 with X
  • combination 11-19d, the H,K/A,B,C-squares, 16 x 576 = 9216 squares, 144 with X

Apart from the discriminative behaviour of the G-structure, the exclusiveness of the G-squares is manifested in the calculated amounts. There are 2304 G-squares, which is the square number of 48. The totals of the other classes of combinations are the square numbers of respectively 48, 96 and 96. The total amount of all squares is 23040, which is not a square number. But the G-squares excluded, the amount is 20736 squares, which is the square of 144. Moreover, the (one) G-combination excluded, the amount of combinations is 10 + 10 +16 = 36, a square number.

 

The amount of 23.040 (“most perfect”) panmagic 8x8 squares with the number 1 upper-left implies that there are 16 x 23.040 = 368.640 panmagic 8x8 squares, regardless the position of the number 1. This amount is consistent with the results of Loly c.s, and Amela, 2007.

 

 

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