Office Applications and Entertainment, Magic Squares

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 3^th order Magic Squares with Magic Constant 123.

However it is possible to compose a 9^th 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 C_j (j = 1 ... 9);
2. Construct a 3 x 3 Magic Square B with elements b_j (j = 1 ... 9);
3. Replace each element c_ij of Sub Square C_j by c_ij = c_ij + (b_j - 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 C_j (j = 1 ... 9), the resulting number of 9^th 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 9^th 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 a_1j (j = 1 ... 9);
2. Construct a 3 x 3 Magic Square B with elements b_j (j = 1 ... 9);
3. Construct a 3 x 3 Magic Square A2 with elements a_2j = 3² * (a_1j - 1) (j = 1 ... 9);
4. Construct a 9 x 9 Magic Square C composed out of 9 identical 3 x 3 Magic Sub Squares a_2j (j = 1 ... 9);
5. Replace each element c_ij of Sub Square C_j by c_ij = c_ij + b_j (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 C_j (j = 1 ... 9), the resulting number of 9^th order Magic Squares with Magic Sum 369 will be 8 * 8⁹ = 1,07 10⁹.