Efficient detection of UA sets is very important in exhaustive search projects. But not so many methods of their detection are known. Method of using "blueprints" of UA sets, to my mind, can be quite efficient only for small UA sets. Several classes of UA sets are still known, for which efficient methods of detection do exist.

1. Bidigital UA sets.

2. Two-row-plus-two-columns UA sets (see blue's post (March 13, 2013) in the thread Investigation of one-band-free patterns on this forum).

I propose a method for "two-box" UA sets detection. Such UA set participate 2 boxes exactly. I'll consider minimal UA sets only, i.e. UA sets not having other UA sets as their subsets.

Here are patterns of all possible 2-boxes (minimal) two-box UA sets.

- Code: Select all
`+-----+-----+ # U4`

|. . x|x . .|

|. . x|x . .|

|. . .|. . .|

+-----+-----+

+-----+-----+ # U6

|. . x|x . .|

|. . x|x . .|

|. . x|x . .|

+-----+-----+

+-----+-----+ # U10

|. . x|x . .|

|. x x|x x .|

|. x x|x x .|

+-----+-----+

+-----+-----+ # U12 (variant 1)

|. x x|x x .|

|. x x|x x .|

|. x x|x x .|

+-----+-----+

+-----+-----+ # U12 (variant 2)

|. x x|x x .|

|. x x|x . x|

|. x x|. x x|

+-----+-----+

+-----+-----+ # U14

|. x x|x x .|

|x x x|x x x|

|x x .|. x x|

+-----+-----+

+-----+-----+ # U18

|x x x|x x x|

|x x x|x x x|

|x x x|x x x|

+-----+-----+

Two-box U6 is well known. Let's consider it in details.

Both its minicolumns must contain the same set of digits. Let's denote this set by A = {a1,a2,a3}.

Denotation

- Code: Select all
`+-----+-----+`

|. . A|A . .|

|. . A|A . .|

|. . A|A . .|

+-----+-----+

shows that all cells marked by "A" letter contains a digit belonging to A set.

Minicolumn in the second box of U6 must be cyclically shifted minicolumn from the first box - cyclically shifted up or down.

Let's consider now more complicated UA set - U12 (variant1).

- Code: Select all
`+-----+-----+`

|. A B|? ? .|

|. A B|? ? .|

|. A B|? ? .|

+-----+-----+

What can we say about 2 minicolumns in the second box? Their top cells must be "A B" or "B A", because these cells must exchange their digits with top cells of the first box minicolumns during UA permutation. So, 6 cells of U12 in the second box must contain the same set of digits as 6 cells in the first box. Here is their possible arrangement.

- Code: Select all
`+-----+-----+`

|. A B|A B .|

|. A B|A B .|

|. A B|B A .|

+-----+-----+

Arrangement

- Code: Select all
`+-----+-----+`

|. A B|A B .|

|. A B|A B .|

|. A B|A B .|

+-----+-----+

is not permitted, because in this case U12 would contain U6 as subset, i.e. it would not be minimal.

Configuration

- Code: Select all
`+-----+-----+`

|. A B|A B .|

|. A B|A B .|

|. A B|B A .|

+-----+-----+

of the first permutation of U12 is the only possible one up to column/row permutations.

Let's find now the second permutation of U12. 3 cells marked by "A" in the second box must go up or down alltogether to "compensate" shift of minicolumn AAA in the first box. That shift can be possible only when minicolimn BBA goes cyclically in the same direction as minicolumn AAB! So, both minicolumns of the second box must move synchronically up or down to form second permutation of this UA set. This implies that BBB values in the second box must be shifted in the same manner as AAA values ralative to AAA and BBB minicolumns in the first box.

Let's consider an example of U12.

- Code: Select all
`+-----+-----+`

|. 1 4|2 5 .|

|. 2 5|3 6 .|

|. 3 6|4 1 .|

+-----+-----+

This is the second permutation (state) of this UA:

- Code: Select all
`+-----+-----+`

|. 2 5|4 1 .|

|. 3 6|2 5 .|

|. 1 4|3 6 .|

+-----+-----+

Continuation follows...

Serg