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.

12345678
910111213141516
1718192021222324
2526272829303132
3334353637383940
4142434445464748
4950515253545556
5758596061626364

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.

  • 87654321r
    910111213141516l
    2423222120191817r
    2526272829303132l
    3334353637383940l
    4847464544434241r
    4950515253545556l
    6463626160595857r
    tbtbbtbt
  • 863661603581r
    950115253145516l
    2447224544194217r
    2534273637303932l
    3326352829383140l
    4823462120431841r
    4910511213541556l
    647625459257r
    tbtbbtbt

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.

llrrrrlllrrllrrlrlrllrlr
lrlrrlrlrllrrllrrrllllrr

And for n=12 there exist already 20 patterns.

lllrrrrrrllllrlrlrrlrlrlrlllrrrrlllrrlrrllllrrlr
llrlrrrrlrlllrlrrllrrlrlrllrlrrlrllrrrlllrrlllrr
llrrlrrlrrlllrrllrrllrrlrllrrllrrllrrrllrllrllrr
llrrrllrrrlllrrlrllrlrrlrlrllrrllrlrrrlrllllrlrr
lrllrrrrllrllrrrllllrrrlrlrlrllrlrlrrrrllllllrrr

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:

llrrrrlllrrlrrllllrrlrrllrrllrlrrlrllrrlrlrllrlr
rrllllrrrllrrllrllrrrrllrllrrlrllrlrlrrllrrlrllr
lrrlllrrrrlllrrlrrllllrrlrlrrlrlrllrrllrrllrlrrl
rllrrrllllrrlrlrrlrllrrlrlrllrlrlrrlrllrlrrllrrl
llrrrrllrllrrlrllrlrrllrrllrlrlrrlrllrrlrllrrllr

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

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

123456789101112l
242322212019181716151413r
252627282930313233343536l
373839404142434445464748l
605958575655545352515049r
727170696867666564636261r
848382818079787776757473r
969594939291908988878685r
979899100101102103104105106107108l
109110111112113114115116117118119120l
132131130129128127126125124123122121r
133134135136137138139140141142143144l
btbttbtbbttb

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

13321354513871401411011144l
132231302120127181251241514121r
109261112829114311161173435120l
9738994041102431041054647108l
965994575691548988515085r
847182696879667776636273r
728370818067786564757461r
609558939255905352878649r
37983910010142103444510610748l
251102711211330115323311811936l
241312212912819126171612312213r
1134313613761398914214312l
btbttbtbbttb