9 - Block - Method   (unknown author)

As with the 9-block method for double-even orders, the square is divided into nine blocks, which are separated by a column or row.

9-Block-Method (single-even picture 1)

First, the numbers are written in their natural order, where the numbers in blocks B, D, F and H are replaced by their complements.

  • 123456
    789101112
    131415161718
    192021222324
    252627282930
    313233343536
  • 134336
    24151619
    18212213
    314336
  • 25
    789101112
    1417
    2023
    252627282930
    3235

The numbers in the rows and columns highlighted in green are specially processed so that they are shown specifically in the following right figure. Four individual cells lie on the diagonals and are fixed and no longer changed. All other numbers in these columns and rows are now swapped according to a fixed pattern.

First we take the two marked columns and start with the following steps:

9-Block-Method (single-even picture 2)

Then the two marked rows are changed.

9-Block-Method (single-even picture 3)

Finally, some of the nine blocks have to be changed.

9-Block-Method (single-even picture 4)

If you now enter the remaining numbers, you get a single-even magic square of size n=6.

123334356
3082827117
192315161424
182021221713
12261092925
313243536

This algorithm is not so easy to implement due to the special mix-ups. Nevertheless, it can be applied to any single-even order. The following figure shows a square of order 10, which was generated with this algorithm. In the PDF document all intermediate steps are described and shown.

1239495969798910
11121387868584881920
80292377767574282221
70696834353637336261
60595844454647435251
41425354555657484950
40393864656667633231
30797327262524787271
81828317161514188990
9192937654899100