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 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, each with 9 consecutive integers and corresponding Magic Sum.
The construction method can be summarised as follows:

Construct a 9 x 9 Magic Square C composed out of 9 identical 3 x 3 Magic Sub Squares C_{j} (j = 1 ... 9);

Construct a 3 x 3 Magic Square B with elements b_{j} (j = 1 ... 9);

Replace each element c_{ij} of Sub Square C_{j} by c_{ij} = c_{ij} + (b_{j}  1) * 9 (i = 1 ... 9);

The corresponding Magic Sums of the Sub Squares will be 15, 42, 69, 96, 123, 150, 177, 204 and 231;

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:
