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8.9  Intermezzo

In previous sections several procedures were developed for sequential generation of Classes of (Pan) Magic Squares of the 8th 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 4th order (Pan) Magic Squares with magic sum 130 were described.

In section 2.5, procedures were developed to generate 4th 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 8th order Magic Square with magic sum 260 (ref. Attachment 8.7.1).

However we should realize that each Pan Magic Square of the 4th order, based on a set of 16 distinct integers, is a member of a collection of 384 Pan Magic Square of the 4th order (ref. Attachment 8.7.2).

Consequently, based on one single set of Pan Magic Squares of the 4th order as shown above, 24 * 3844 = 0,5 1012 Magic Squares of the 8th order can be constructed.

With procedure MgcSqr4e respectively 5696 , 5696 , 5696 and 7040 Magic Squares of the 4th 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 8th order, which can be constructed based on the distinct integers contained in one single set of 4 Pan Magic Squares of the 4th order as shown above is 24 * 56963 * 7040 = 3,12 1016.

Other sets of Pan Magic Squares of the 4th order will result in other numbers, however of the same order of magnitude.

For the 4 Pan Magic Squares of the 4th order shown in the spread sheet solution CnstrSngl8c, the total number of resulting Magic Squares of the 8th order is 24 * 4736 * 5248 * 4672 * 5568 = 1,55 1016.

Higher order Magic Squares, which can be constructed based on these principles, will be discussed in Section 22.