Office Applications and Entertaiment, Magic Squares

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9.8   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 3th order Magic Squares with Magic Constant 123.

However it is possible to compose a 9th order Magic Square out of 9 Magic Squares of order 3, each with 9 consecutive integers and corresponding Magic Sum.

The construction method can be summarised as follows:

  1. Construct a 9 x 9 Magic Square C composed out of 9 identical 3 x 3 Magic Sub Squares Cj (j = 1 ... 9);
  2. Construct a 3 x 3 Magic Square B with elements bj (j = 1 ... 9);
  3. Replace each element cij of Sub Square Cj by cij = cij + (bj - 1) * 9 (i = 1 ... 9);
  4. The corresponding Magic Sums of the Sub Squares will be 15, 42, 69, 96, 123, 150, 177, 204 and 231;
  5. 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
4 9 2
3 5 7
8 1 6
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
6 1 8
7 5 3
2 9 4
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 Cj (j = 1 ... 9), the resulting number of 9th order Magic Squares with Magic Sum 369 will be 8 * 89 = 1,07 109.


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