# 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.

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

•  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
•  1 34 33 6 24 15 16 19 18 21 22 13 31 4 3 36
•  2 5 7 8 9 10 11 12 14 17 20 23 25 26 27 28 29 30 32 35

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:

• Swap the numbers in the right column with respect to the horizontal axis.
• Swap the numbers in the center rows. The row that lies directly below the horizontal central axis is excluded from the swap.

Then the two marked rows are changed.

• All numbers are exchanged symmetrically with respect to the vertical central axis.
• The numbers in the center block are swapped symmetrically with respect to the horizontal axis.
• The two numbers from the first column are swapped.

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

• We choose a row from block B or H and invert the numbers with respect to the vertical axis.
• Select a row from block D and swap the numbers with the numbers of the corresponding row from block F.

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

 1 2 33 34 35 6 30 8 28 27 11 7 19 23 15 16 14 24 18 20 21 22 17 13 12 26 10 9 29 25 31 32 4 3 5 36

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.

 1 2 3 94 95 96 97 98 9 10 11 12 13 87 86 85 84 88 19 20 80 29 23 77 76 75 74 28 22 21 70 69 68 34 35 36 37 33 62 61 60 59 58 44 45 46 47 43 52 51 41 42 53 54 55 56 57 48 49 50 40 39 38 64 65 66 67 63 32 31 30 79 73 27 26 25 24 78 72 71 81 82 83 17 16 15 14 18 89 90 91 92 93 7 6 5 4 8 99 100