Shen:   regular and special patterns

Ching Tseng Shen's method is based on a square in a natural order, in which he reverses ⁿ/₄ pairs of horizontally symmetrical columns and an equal number of vertically symmetrical rows. Thus, this process corresponds to the Method of reversions that Planck has already described.

For clarification, Shen marks the unchanged rows with l and the inverted rows with r, as well as the normal columns with t and the inverted with b.

For order n=4, Shen found exactly two of these symmetrical patterns, which he calls regular, namely lrrl and rllr. In contrast, six of these patterns already exist for n=8.

And for n=12 there exist already 20 patterns.

These regular patterns can be transferred accordingly to the columns, where the code lr for the rows is replaced by tb, of course.

Any combination of these regular patterns creates a magic square, regardless of whether the rows are reversed first and then the columns according to their labels or vice versa. The resulting magic squares are always identical, regardless of the order of the exchanges.

Special patterns

Shen goes one step further and also introduces special patterns. These are composed of any regular patterns of lower orders, and are of course no longer arranged symmetrically. For order n=12 there are, for example, the following 20 special patterns for the rows:

With regular and special patterns, additional magic squares can now be created, whereby the order of the exchanges is important here.

• Choose a regular pattern for the rows and a special pattern for the columns, and reverse the rows first and then the columns.
• Choose a special pattern for the columns and a regular pattern for the columns, and reverse the columns first and then the rows.

Such an example with a regular pattern for the rows and a special pattern for the columns is shown in the following figure for a 12th-order square. Here are the rows reversed in the first step

and the columns later. The result is a magic square.