Introduction
Any normal magic square may be transformed into another magic square by
subtracting each number in turn from n^{2} + 1. This
process is referred to as complementing the magic square.
It is also referred to as 'complementary pair interchange' (CPI for short)
because in effect you are interchanging the two numbers that together sum
to n^{2} + 1 ((R. S. Sery).
Under certain conditions the resulting magic square will be a reflected
copy of the original magic square.
To illustrate using the Loshu magic square

each number subtracted from 10 transforms to 

A horizontal and a vertical reflection will make the
transformed square identical to the original. 
This type of magic square was introduced on Mutsumi Suzuki’s Magic Squares
page when he originally showed 6 order5 magic squares of this type. He coined
the name Selfsimilar magic squares.
Recently I revisited this page and discovered he had greatly expanded it,
including 352 order4 magic squares of this type.
Visit his large, comprehensive magic squares site from my
links page.
On studying his page, I realized that the first group (his group A) of 48
order4 magic squares are the only 48 associated magic squares of order4.
On examining his order5 selfsimilar magic squares, I find that they also are
associated magic squares.
When I checked associated magic squares of other orders I found that in each
case they were self similar.
Contents
Order4 Associated
1 
8 
12 
13 
14 
11 
7 
2 
15 
10 
6 
3 
4 
5 
9 
16 

This magic square is associated and
semipandiagonal.
It is #112 in Frénicle’s ordered list of the 880 order4 magic
squares and the first one that is associated.
All order4 associated magic squares are Dudeney type III .

16 
9 
5 
4 
3 
6 
10 
15 
2 
7 
11 
14 
13 
12 
8 
1 

This is the complementary copy of the above magic
square. It is obtained by subtracting each number of the above
square from 17. It is reflected horizontally and vertically from
the original
or to put it another way, it is rotated 180 degrees from the
original.


This pattern is constructed by joining integer pairs
that sum to n^{2} + 1. The selfsimilar
(complementary) magic square is constructed by simply exchanging the
two numbers of each pair. Because it is completely symmetrical, the
complementary square is selfsimilar and is reflected horizontally
and vertically from the original. 
Higher orders of associated magic squares have similar patterns, only with
more lines. Note that every line passes through the center of the square. For
this reason associated magic squares are also called symmetrical magic squares.
Order4 not
associated
The previous section mentioned that all associated magic squares,
regardless of the order, have the selfsimilar property.
However, there is another type of order4, and almost surely, other orders,
that also have this property.
They are Dudeney’s type VI of which 96 are semipandiagonal and 208 are simple
magic squares.
This is Frénicle’s index #1 , it’s complement and it's Dudeney pattern. It is
a simple magic square.
1 
2 
15 
16 
12 
14 
3 
5 
13 
7 
10 
4 
8 
11 
6 
9 

16 
15 
2 
1 
5 
3 
14 
12 
4 
10 
7 
13 
9 
6 
11 
8 


Note that this complementary square needs only
a horizontal reflection to make it identical to the original 
1 
12 
14 
7 
13 
8 
2 
11 
4 
9 
15 
6 
16 
5 
3 
10 

16 
5 
3 
10 
4 
9 
15 
6 
13 
8 
2 
11 
1 
12 
14 
7 


This one is Frénicle # 182, it's complement and
Dudeney pattern. It is semipandiagonal. 
This time the complementary square needs a
vertical reflection to make it identical to the original.
Comparing the complement pair patterns for the two squares makes it obvious
why this is so.
If magic squares of higher orders have complementary pair patterns
equivalent to these, then those magic squares will also be selfsimilar.
Whether these magic squares are semipandiagonal or simple seems to have no
bearing on whether a horizontal or vertical reflection is required to match
the complement to the original. The complements of both magic squares below
require a horizontal reflection to match the original even though the
squares are of different types. 
Frénicle #53 simple
1 
6 
11 
16 
14 
15 
2 
3 
12 
9 
8 
5 
7 
4 
13 
10 

Frénicle #54 semipandiagonal
1 
6 
11 
16 
15 
12 
5 
2 
8 
3 
14 
9 
10 
13 
4 
7 


Credits regarding order4
Mutsumi Suzuki lists 352
selfcomplementary magic squares of order4.
His group A shows 48 magic squares that when complemented require a horizontal
and a vertical reflection to match the original. These are the 48 Dudeney Type
III associated semipandiagonal magic squares.
His group B lists 304 magic squares that when complemented require only a
horizontal or a vertical reflection to match the original. These are the 96
Dudeney Type VI semipandiagonal magic squares and the 208 Dudeney Type VI
simple magic squares.
Visit Mutsumi Suzuki’s large, comprehensive magic squares page from my
links page.
Bernard Frénicle de Bessy published the 880 basic
solutions for the order4 magic squares in an indexed order in Des Quarrez
Magiques. Acad. R. des Sciences 1693.
H. E. Dudeney published a classification of these 880 magic squares
with the enumeration of each type in Amusements in Mathematics, Dover
Publ., 1970, 0486204731. This is a reprint of a book first published in 1917.
Order4 not
selfsimilar
1 
12 
13 
8 
16 
9 
4 
5 
2 
7 
14 
11 
15 
6 
3 
10 


This is Frénicle #181. Dudeney group XI.
A look at the Dudeney pattern for this magic square confirms that
the complementary pairs are not symmetrical across either the
horizontal or the vertical center lines. This is the condition
required for a magic square to be selfsimilar.
For Order4, groups III and VI are the only two types that have this
characteristic. 
Order5 Associated
14 
10 
1 
22 
18 
20 
11 
7 
3 
24 
21 
17 
13 
9 
5 
2 
23 
19 
15 
6 
8 
4 
25 
16 
12 

12 
16 
25 
4 
8 
6 
15 
19 
23 
2 
5 
9 
13 
17 
21 
24 
3 
7 
11 
20 
18 
22 
1 
10 
14 

Because this is an associated magic square, the
complementary square must be both horizontally and vertically
reflected to match the original. This particular square is also a
lozenge magic square. Notice how the odd numbers are grouped. 
Order6 not
associated
1 
28 
27 
10 
9 
36 
35 
26 
25 
12 
11 
2 
3 
22 
21 
16 
15 
34 
33 
24 
23 
14 
13 
4 
20 
6 
8 
29 
31 
17 
19 
5 
7 
30 
32 
18 


This order6 magic square is not associated but is
symmetric across the vertical center line so produces a selfsimilar
copy of itself. There are no associated pure magic squares of order
6.

Order7 associated
Because this is an associated magic square, the
complementary square must be both horizontally and vertically
reflected to match the original.

42 
18 
29 
9 
45 
26 
6 
20 
35 
11 
43 
23 
3 
40 
4 
36 
16 
31 
12 
48 
28 
33 
13 
49 
25 
1 
37 
17 
22 
2 
38 
19 
34 
14 
46 
10 
47 
27 
7 
39 
15 
30 
44 
24 
5 
41 
21 
32 
8 

8 
32 
21 
41 
5 
24 
44 
30 
15 
39 
7 
27 
47 
10 
46 
14 
34 
19 
38 
2 
22 
17 
37 
1 
25 
49 
13 
33 
28 
48 
12 
31 
16 
36 
4 
40 
3 
23 
43 
11 
35 
20 
6 
26 
45 
9 
29 
18 
42 

Order8 associated
This square is pandiagonal as well as being associated.
7 
42 
55 
26 
31 
50 
47 
2 
62 
19 
14 
35 
38 
11 
22 
59 
1 
48 
49 
32 
25 
56 
41 
8 
60 
21 
12 
37 
36 
13 
20 
61 
4 
45 
52 
29 
28 
53 
44 
5 
57 
24 
9 
40 
33 
16 
17 
64 
6 
43 
54 
27 
30 
51 
46 
3 
63 
18 
15 
34 
39 
10 
23 
58 

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

Order8 not
associated
17 
23 
9 
53 
12 
56 
42 
48 
5 
50 
22 
46 
19 
43 
15 
60 
13 
64 
38 
36 
29 
27 
1 
52 
47 
3 
35 
31 
34 
30 
62 
18 
7 
4 
25 
37 
28 
40 
61 
58 
63 
59 
32 
26 
39 
33 
6 
2 
57 
16 
44 
20 
45 
21 
49 
8 
51 
41 
55 
11 
54 
10 
24 
14 


This magic square is not associated but symmetric
across the vertical center, so produces a horizontally reflected
version of itself. It is also a bordered magic square. A double
row of cells surrounds a 4 x 4 center magic square. Complementing
only the center magic square or only the border cells will produce 2
variations. Rotating the center 4 x 4 either 90 or 270 degrees
produces other variations. 
Order9 associated
71 
64 
69 
8 
1 
6 
53 
46 
51 
66 
68 
70 
3 
5 
7 
48 
50 
52 
67 
72 
65 
4 
9 
2 
49 
54 
47 
26 
19 
24 
44 
37 
42 
62 
55 
60 
21 
23 
25 
39 
41 
43 
57 
59 
61 
22 
27 
20 
40 
45 
38 
58 
63 
56 
35 
28 
33 
80 
73 
78 
17 
10 
15 
30 
32 
34 
75 
77 
79 
12 
14 
16 
31 
36 
29 
76 
81 
74 
13 
18 
11 


65 
72 
67 
2 
9 
4 
47 
54 
49 
70 
68 
66 
7 
5 
3 
52 
50 
48 
69 
64 
71 
6 
1 
8 
51 
46 
53 
20 
27 
22 
38 
45 
40 
56 
63 
58 
25 
23 
21 
43 
41 
39 
61 
59 
57 
24 
19 
26 
42 
37 
44 
60 
55 
62 
29 
36 
31 
74 
81 
76 
11 
18 
13 
34 
32 
30 
79 
77 
75 
16 
14 
12 
33 
28 
35 
78 
73 
80 
15 
10 
17 

This associated magic square is also composite. It
consists of nine order3 associated magic squares themselves
arranged as an order3 magic square. If each number in this order9
square is exchanged with it's complement as per the Dudeney pattern,
the result is the same magic square rotated 180 degrees. In
addition, each of the nine order3 magic squares can be converted to
its complement (itself but rotated 180 degrees) by subtracting each
number from the sum of the first and last number in that magic
square. This in turn will produce another order9 associated magic
square (shown at left) which is also selfsimilar.
Any order3 or combinations of order3 may be rotated to get
variations in the order9 magic square.
Notice that for simplicity, some lines in the pattern (above) are
covered up by others. The center horizontal line, for example,
actually consists of four pairs. 
Higher dimensions
The selfsimilar feature also works for associated magic cubes and
tesseracts. This works for the higher dimensions as well.
Shown here is an order3 cube and an order3 tesseract, both are
complements of those shown on my John Hendricks
page. Refer to that page for more information on these figures.
Just as the 1 order3 magic square is associated, so also are the 4 basic
magic cubes and the 58 basic magic tesseracts.

The magic cube (left) has 13 complementary pairs
with the two members on opposite sides of the central number 14.
This selfsimilar figure is one of the 47 equivalents to the cube on
Hendricks page.
The magic tesseract (right) has 40 complementary
pairs around the central number 41. Two such pairs are 32, 50 and
18, 64. This selfsimilar figure is one of the 383 equivalents to
the tesseract on Hendricks page. 

Summery
All associated magic squares (and higher dimensions) have the
selfsimilar property. i.e. if each number in the square is subtracted
from the sum of the first and last number in the series, the resulting
magic square is a duplicate of the original but rotated 180 degrees.
This process of complementing all numbers may be referred to as
complementary pair interchange (CPI).
Any nonassociated magic squares that have complementary pairs that are
symmetric across either the horizontal or the vertical center line produce
selfsimilar copies that are either horizontally or vertically reflections of
themselves.
I have shown such a pattern (above) for orders 4 and 6.
All order4 associated magic squares are semipandiagonal.
Are ALL associated magic squares semipandiagonal?
