9.9 Intermezzo
In previous sections several procedures were developed for sequential generation of Classes of (Pan) Magic Squares of order 9, based on couples of linear equations, matrix operation and/or automatic filtering methods.
Due to the vast amount of independent variables of the applied equations, the solutions could only be obtained by keeping a number of these variables constant.
Different clusters of solutions could be obtained by keeping other sets of independent variables constant.
It is not possible to compose Magic Squares of order 9  with Magic Constant 369  out of 3^{th} order Magic Squares with Magic Constant 123.
However it is possible to compose a 9^{th} order Magic Square out of 9 Magic Squares of order 3 with different Magic Sums.
9.9.1 Method 1 (Conseciutive Integers)
Magic Squares of order 9 can be composed out of 9 Magic Sub Squares,
each with 9 consecutive integers and corresponding Magic Sum.
The construction method can be summarised as follows:

Construct a 9 x 9 Magic Square C composed out of 9 identical 3 x 3 Magic Sub Squares C_{j} (j = 1 ... 9);

Construct a 3 x 3 Magic Square B with elements b_{j} (j = 1 ... 9);

Replace each element c_{ij} of Sub Square C_{j} by c_{ij} =
c_{ij} + (b_{j}  1) * 3^{2} (i = 1 ... 9);

The corresponding Magic Sums of the Sub Squares will be 15, 42, 69, 96, 123, 150, 177, 204 and 231;

The result will be a 9 x 9 Magic Square with 81 consecutive integers and resulting Magic Sum 369.
An example obtained by subject method is shown below:
B

C
33 
28 
35 
34 
32 
30 
29 
36 
31 

78 
73 
80 
79 
77 
75 
74 
81 
76 

15 
10 
17 
16 
14 
12 
11 
18 
13 

24 
19 
26 
25 
23 
21 
20 
27 
22 

42 
37 
44 
43 
41 
39 
38 
45 
40 

60 
55 
62 
61 
59 
57 
56 
63 
58 

69 
64 
71 
70 
68 
66 
65 
72 
67 


51 
46 
53 
52 
50 
48 
47 
54 
49 


MC's
96 
231 
42 
69 
123 
177 
204 
15 
150 

It can be noticed that the resulting Composed Magic Square C is symmetric (associated).
With 8 possible squares for both B and C_{j}
(j = 1 ... 9), the resulting number of 9^{th} order Magic Squares with Magic Sum 369 will be
8 * 8^{9} = 1,07 10^{9}.
9.9.2 Method 2 (Kronecker)
Alternatively
it is possible to compose 9^{th} order Magic Squares out of 9 Magic Squares of order 3, each with 9 nonconsecutive integers and corresponding Magic Sum.
The corresponding construction method can be summarised as follows:

Construct a 3 x 3 Magic Square A1 with elements a_{1j} (j = 1 ... 9);

Construct a 3 x 3 Magic Square B with elements b_{j} (j = 1 ... 9);

Construct a 3 x 3 Magic Square A2 with elements
a_{2j} = 3^{2} * (a_{1j}  1) (j = 1 ... 9);

Construct a 9 x 9 Magic Square C composed out of 9 identical 3 x 3 Magic Sub Squares a_{2j} (j = 1 ... 9);

Replace each element c_{ij} of Sub Square C_{j} by c_{ij} = c_{ij} + b_{j} (i = 1 ... 9);

The corresponding Magic Sums of the Sub Squares will be
111,
114,
117,
120,
123,
126,
129,
132 and
135;

The result will be a 9 x 9 Magic Square with 81 consecutive integers and resulting Magic Sum 369.
An example obtained by subject method is shown below:
B

C
49 
4 
67 
58 
40 
22 
13 
76 
31 

54 
9 
72 
63 
45 
27 
18 
81 
36 

47 
2 
65 
56 
38 
20 
11 
74 
29 

48 
3 
66 
57 
39 
21 
12 
75 
30 

50 
5 
68 
59 
41 
23 
14 
77 
32 

52 
7 
70 
61 
43 
25 
16 
79 
34 

53 
8 
71 
62 
44 
26 
17 
80 
35 

46 
1 
64 
55 
37 
19 
10 
73 
28 

51 
6 
69 
60 
42 
24 
15 
78 
33 


MC's
120 
135 
114 
117 
123 
129 
132 
111 
126 

It can be noticed that the resulting Composed Magic Square C is symmetric (associated).
With 8 possible squares for both B and C_{j}
(j = 1 ... 9), the resulting number of 9^{th} order Magic Squares with Magic Sum 369 will be
8 * 8^{9} = 1,07 10^{9}.
