Office Applications and Entertainment, Magic Squares

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 * 384⁴ = 0,5 10¹² 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 * 5696³ * 7040 = 3,12 10¹⁶.

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 10¹⁶.

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