Sudoku: Using Wing Pincers as a Link in a Skyscraper or Kite

For years now, some of us (Marty, Keith, and Helmut, et al) have been using “useless” or “flightless” wings as a link to make eliminations in more difficult puzzles.  We have all remarked on how useful and powerful the technique might be.

More recently, Helmut Saueregger (Nataraj) engaged me in a new discussion on this, and he has extended his Helmut’s Sudoku Helper (HSH) to find these things.  HSH is here:

http://www.saueregger.at/sudoku/HSH/HSH.htm (Note 1)

You should be looking at HSH V4.1 or greater.

For the following, we will use XY-wings as the example.  However, the same logic applies to W- and M-wings.

There are two requirements:

1. You know how to find Kites and Skyscrapers, which are collectively known as Turbot Fish (Note 2).

2. You know how to find an XY-wing.

(We also imply that you use pencil and paper to solve Sudoku for personal enjoyment. If you see Sudoku solving as a contest in computer software, please do not read further.)

The simple point here is that the pincers of a Wing can be one link of the two links that make up a Turbot pattern.

Here is an example (from Helmut):

+-------+-------+-------+
| 2 8 . | 1 . . | . . . |
| . 1 5 | 7 4 . | 6 8 . |
| . . . | . . . | 9 . . |
+-------+-------+-------+
| 5 . . | . 9 . | . . 8 |
| . . . | 4 . 2 | . . . |
| 1 . . | . 8 . | . . 4 |
+-------+-------+-------+
| . . 7 | . . . | . . . |
| . 9 2 | . 3 6 | 7 5 . |
| . . . | . . 4 | . 6 9 |
+-------+-------+-------+

After basics:

+-------------------+-------------------+-------------------+
| 2     8     69    | 1     56    59    | 34    34    7     |
| 39    1     5     | 7     4     39    | 6     8     2     |
| 367   3467  346   | 23    26    8     | 9     1     5     |
+-------------------+-------------------+-------------------+
| 5     23467 346   | 36    9     137   | 123   237   8     |
| 379   37    8     | 4     15    2     | 15    379   6     |
| 1     2367  369   | 356   8     357   | 235   2379  4     |
+-------------------+-------------------+-------------------+
| 68    56    7     | 9     125   15    | 248   24    3     |
| 4     9     2     | 8     3     6     | 7     5     1     |
| 38    35    1     | 25    7     4     | 28    6     9     |
+-------------------+-------------------+-------------------+

There is an XY-wing 2-35 in R9C4.  After that:

+-------------------+-------------------+-------------------+
| 2     8     69    | 1     56    59    | 34    34    7     |
| 39#   1     5     | 7     4     39#   | 6     8     2     |
| 367   467   346   | 23@   26    8     | 9     1     5     |
+-------------------+-------------------+-------------------+
| 5     23467 346   | 36    9     137   | 123   237   8     |
| 379   37    8     | 4     15    2     | 15    379   6     |
| 1     2367  369   | 356   8     357   | 235   2379  4     |
+-------------------+-------------------+-------------------+
| 68    56    7     | 9     125   15    | 248   24    3     |
| 4     9     2     | 8     3     6     | 7     5     1     |
|-38    35@   1     | 25    7     4     | 28    6     9     |
+-------------------+-------------------+-------------------+

It seems you now have to join the chain gang (Note 3). But, let’s look closer:
@@ are the Wing pincers.  One or both is 3.
## is a strong link on 3.  Together @@ and ## make a sort of Kite.
R9C1 is not 3, puzzle solved!
Note that there is also a flightless XY-wing 23-6 in R3C4 that takes out 6 in R4C3:

+-------------------+-------------------+-------------------+
| 2     8     69#   | 1     56#   59    | 34    34    7     |
| 39    1     5     | 7     4     39    | 6     8     2     |
| 367   467   346   | 23    26@   8     | 9     1     5     |
+-------------------+-------------------+-------------------+
| 5     23467 34-6  | 36@   9     137   | 123   237   8     |
| 379   37    8     | 4     15    2     | 15    379   6     |
| 1     2367  369   | 356   8     357   | 235   2379  4     |
+-------------------+-------------------+-------------------+
| 68    56    7     | 9     125   15    | 248   24    3     |
| 4     9     2     | 8     3     6     | 7     5     1     |
| 38    35    1     | 25    7     4     | 28    6     9     |
+-------------------+-------------------+-------------------+

How difficult is this?  I think it is quite easy.
Personally, after doing the basics, I check for strong links to find single-digit eliminations (usually Turbots, which are Skyscrapers and Kites). That is explained here:
http://forum.enjoysudoku.com/post247168.html#p247168
As a side benefit, this Turbot search is a check that you have done the basics correctly.
After that, I look for XY-, W- and M-wings (note 4). Even if these wings do not make eliminations by themselves (they are “flightless”), I now have all the information to find the “Turbot Wings” described here (Note 5).
So, I think this is not a very “advanced” technique. Actually, I think it simplifies the issue, by taking links found by multi-digit techniques back into the single-digit world.

Keith

1. Notes:
   Helmut’s Sudoku Helper (HSH) is an interesting tool that allows you to explore paths through puzzles. It is not a Sudoku solver per se.
2. If this is new to you, start with Havard’s excellent explanation: http://forum.enjoysudoku.com/strong-links-for-beginners-t3326.html
3. See also http://hodoku.sourceforge.net/en/tech_sdp.php for more on Skyscrapers, Kites, and Turbots.
4. https://www.youtube.com/watch?v=zBn5aIfZElE
   For more on M- and W-wings see: http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=2143
5. Names are an aid to communication, and I think the name “Turbot Wing” is appropriately descriptive of a Turbot where one of the two links comes from a Wing.