8.9 Intermezzo
In previous sections several procedures were developed for sequential generation of Classes of (Pan) Magic Squares of the
8^{th} order, based on couples of linear equations describing subject Classes.
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.
Magic Squares of order 8  with magic sum 260  being composed out of 4^{th} order (Pan) Magic Squares with magic sum 130 were described.
In section 2.5, procedures were developed to generate 4^{th} order (Pan) Magic Squares with magic sum 130.
With some minor modifications subject procedures can be used to find a set of 4 Pan Magic Squares with magic sum 130  each containing 16 different integers  as shown below:
A
4 
5 
59 
62 
57 
64 
2 
7 
6 
3 
61 
60 
63 
58 
8 
1 

B
12 
13 
51 
54 
49 
56 
10 
15 
14 
11 
53 
52 
55 
50 
16 
9 

C
20 
21 
43 
46 
41 
48 
18 
23 
22 
19 
45 
44 
47 
42 
24 
17 

D
28 
29 
35 
38 
33 
40 
26 
31 
30 
27 
37 
36 
39 
34 
32 
25 

These four squares can be arranged in 24 ways as an 8^{th} order Magic Square with magic sum 260 (ref. Attachment 8.7.1).
However we should realize that each Pan Magic Square of the 4^{th} order, based on a set of 16 distinct integers, is a member of a collection of 384 Pan Magic Square of the 4^{th} order (ref. Attachment 8.7.2).
Consequently, based on one single set of Pan Magic Squares of the 4^{th} order as shown above, 24 * 384^{4} = 0,5 10^{12} Magic Squares of the 8^{th} order can be constructed.
With procedure MgcSqr4e respectively 5696 , 5696 , 5696 and 7040 Magic Squares of the 4^{th} order can be generated, based on the distinct integers contained in the 4 Pan Magic Squares shown above.
So the total number of Magic Squares of the 8^{th} order, which can be constructed based on the distinct integers contained in one single set of 4 Pan Magic Squares of the 4^{th} order as shown above is 24 * 5696^{3} * 7040 = 3,12 10^{16}.
Other sets of Pan Magic Squares of the 4^{th} order will result in other numbers, however of the same order of magnitude.
For the 4 Pan Magic Squares of the 4^{th} order shown in the spread sheet solution CnstrSngl8c, the total number of resulting Magic Squares of the 8^{th} order is 24 * 4736 * 5248 * 4672 * 5568 = 1,55 10^{16}.
Higher order Magic Squares, which can be constructed based on these principles, will be discussed in Section 22.
