<HOME> <<PREVIOUS] [NEXT>>
How to construct most perfect panmagic 8x8 squares with the quadrant
method
See page ‘most perfect magic squares, explanation’ for a summary of the up till now known methods to
construct most perfect panmagic squares. One of the methods is the quadrant method of Willem Barink
(see “medjig method”, page 6x6 magic-squares). 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.
N.B.: Most perfect panmagic 8x8 squares necessarily have also the Franklin magic properties, the reverse is
not the case.
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 pan-
magic 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 possi-
bilities of combination and from there he calculated the amounts of possible squares (N.B.: in the first instal-
ment of this paper this 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
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 derivates (= 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 struc-
tures 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 dia-
gonal reflections of already made squares;
- and 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 c
|
|
c/r-switch
|
Total
|
|
X
|
|
1
|
|
48x48
|
= 2304
|
-
|
= 2304
|
|
36
|
|
|
|
|
|
|
|
|
|
|
2
|
|
12x12
|
= 144
|
+ 144
|
= 288
|
|
|
|
3a
|
|
12x12
|
= 144
|
-
|
= 144
|
C1, C3, C5
|
36
|
|
3b
|
|
12x12
|
= 144
|
-
|
= 144
|
|
|
|
4a
|
|
12x12
|
= 144
|
-
|
= 144
|
|
|
|
4b
|
|
12x12
|
= 144
|
-
|
= 144
|
|
|
|
4c
|
|
12x12
|
= 144
|
+ 144
|
= 288
|
|
|
|
5a
|
|
12x12
|
= 144
|
+ 144
|
= 288
|
|
|
|
5b
|
|
12x12
|
= 144
|
+ 144
|
= 288
|
|
|
|
5c
|
|
12x12
|
= 144
|
+ 144
|
= 288
|
|
|
|
5d
|
|
12x12
|
= 144
|
+ 144
|
= 288
|
|
|
|
|
|
|
|
|
|
|
|
|
6
|
|
48x12
|
= 576
|
-
|
= 576
|
|
|
|
7
|
|
48x12
|
= 576
|
-
|
= 576
|
|
144
|
|
8
|
|
48x12
|
= 576
|
+ 576
|
= 1152
|
|
|
|
9a
|
|
48x12
|
= 576
|
+ 576
|
= 1152
|
|
|
|
9b
|
|
48x12
|
= 576
|
+ 576
|
= 1152
|
|
|
|
9c
|
|
48x12
|
= 576
|
+ 576
|
= 1152
|
|
|
|
9d
|
|
48x12
|
= 576
|
+ 576
|
= 1152
|
|
|
|
10a
|
|
48x12
|
= 576
|
-
|
= 576
|
|
|
|
10b
|
|
48x12
|
= 576
|
-
|
= 576
|
|
|
|
10c
|
|
48x12
|
= 576
|
+ 576
|
= 1152
|
|
|
|
|
|
|
|
|
|
|
|
|
11
|
|
48 x 6
|
= 288
|
+ 288
|
= 576
|
|
144
|
|
12
|
|
48 x 6
|
= 288
|
+ 288
|
= 576
|
|
|
|
13a
|
|
48 x 6
|
= 288
|
+ 288
|
= 576
|
|
|
|
13b
|
| |
