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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 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 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:

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) * 32 (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.

9.9.2 Method 2 (Kronecker)

Alternatively it is possible to compose 9th order Magic Squares out of 9 Magic Squares of order 3, each with 9 non-consecutive integers and corresponding Magic Sum.

The corresponding construction method can be summarised as follows:

1. Construct a 3 x 3 Magic Square A1 with elements a1j (j = 1 ... 9);
2. Construct a 3 x 3 Magic Square B with elements bj (j = 1 ... 9);
3. Construct a 3 x 3 Magic Square A2 with elements a2j = 32 * (a1j - 1) (j = 1 ... 9);
4. Construct a 9 x 9 Magic Square C composed out of 9 identical 3 x 3 Magic Sub Squares a2j (j = 1 ... 9);
5. Replace each element cij of Sub Square Cj by cij = cij + bj (i = 1 ... 9);
6. The corresponding Magic Sums of the Sub Squares will be 111, 114, 117, 120, 123, 126, 129, 132 and 135;
7. 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:

A1
 6 1 8 7 5 3 2 9 4
A2
 45 0 63 54 36 18 9 72 27
B
 4 9 2 3 5 7 8 1 6
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 Cj (j = 1 ... 9), the resulting number of 9th order Magic Squares with Magic Sum 369 will be 8 * 89 = 1,07 109.