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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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 Posted: Sat Aug 11, 2007 7:42 pm Post subject: The W-wing |
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George Woods seems to be credited with pointing this out. Look here, for example: http://www.dailysudoku.com/sudoku/forums/viewtopic.php?p=6357
A W-wing is a short chain that is easy to recognize, because it includes two cells that have the same two candidates, but are not a pair.
Suppose you have two cells, each with the same two candidates, that cannot "see" each other, so they are not a pair. And, suppose that each of these cells lines up with an end of a strong link in one of the candidates.
(a) <WX> -- <XY> =strong link= <XZ> -- <WX> (A)
W and X are the two candidates, in the two cells (a) and (A). Y and Z represent one or more candidates. The cells <XY> and <XZ> are strongly linked: They are the only two cells in a row, column, or box that have <X> as a candidate.
There are two possibilities for (a):
1. (a) is <W>.
2. (a) is not <W>, it is <X>. Then, we must have
(a) <X> -- <Y> =strong link= <X> -- <W> (A)
In other words, either (a) or (A) must be <W>. <W> can be eliminated from all cells that "see" both (a) and (A).
Here is an example:
| Code: | Puzzle: DB081107 ******
+-------+-------+-------+
| 8 . . | 1 4 . | . 6 . |
| . 6 3 | . . . | 7 4 . |
| . . 4 | . . 2 | . . . |
+-------+-------+-------+
| 6 . . | 4 3 . | . 2 . |
| . . . | . . . | . . . |
| . 8 . | . 5 1 | . . 6 |
+-------+-------+-------+
| . . . | 6 . . | 9 . . |
| . 9 1 | . . . | 6 3 . |
| . 4 . | . 9 5 | . . 7 |
+-------+-------+-------+ |
After the basics, a Unique Rectangle, and an X-wing, you get here:
| Code: | +---------------------+-------------------+-------------------+
| 8 257 2579 | 1 4 37 | 235 6 2359 |
| 12 6 3 | 5 8 9 | 7 4 12 |
| 179 157 4 | 37 6 2 | 1358 189 1358 |
+---------------------+-------------------+-------------------+
| 6 157 579 | 4 3 78 | 158 2 1589 |
| 12479# 12357# 257# | 2789 27a 6 | 1458 1789x 1358 |
| 34 8 27A | 279# 5 1 | 34 79x 6 |
+---------------------+-------------------+-------------------+
| 237 237 8 | 6 1 34 | 9 5 24 |
| 5 9 1 | 278 27 478 | 6 3 248 |
| 23 4 6 | 238 9 5 | 128 18 7 |
+---------------------+-------------------+-------------------+ |
Note the cells a and A that have candidates <27> and the strong link for <7> in C8, labelled x.
One of a or A is <2>, and we can eliminate <2> from the cells marked #.
So, if you have two cells with the same two candidates, see if you can "connect" them with a strong link in either of the candidates.
This pattern seems to be very common, and often quite useful.
(This is a nice W-wing example. But, if you want to quickly solve this particular puzzle, use the Turbot Fish, present in the initial grid, to take out <2> in R9C1 - as pointed out by Tracy in another thread.)
Last edited by keith on Sun Oct 26, 2008 5:07 am; edited 1 time in total |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sun Aug 12, 2007 4:46 pm Post subject: Another example |
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Here is another example. It is Michael Mepham's Diabolical of Sunday, August 12.
| Code: | Puzzle: MM081207 Diabolical
+-------+-------+-------+
| 2 . 3 | . . . | 9 . . |
| . . . | 3 . 2 | 5 . . |
| 7 . . | 8 . 4 | . . 2 |
+-------+-------+-------+
| 8 . . | . . . | . . 9 |
| . 9 4 | . . . | 2 7 . |
| 5 . . | . . . | . . 6 |
+-------+-------+-------+
| 1 . . | 2 . 6 | . . 5 |
| . . 5 | 4 . 1 | . . . |
| . . 6 | . . . | 7 . 4 |
+-------+-------+-------+ |
After the basics, and an X-wing on <8>, you get to:
| Code: | +----------------+----------------+-----------------+
| 2 1568 3 | 157 156 57 | 9 4 18# |
| 4 16 189 | 3 169 2 | 5 68 7 |
| 7 156 19 | 8 1569 4 | 13a 36x 2 |
+----------------+----------------+-----------------+
| 8 137 127 | 6 24 37 | 134# 5 9 |
| 6 9 4 | 15 158 358 | 2 7 13A |
| 5 137 127 | 179 24 379 | 1348# 38x 6 |
+----------------+----------------+-----------------+
| 1 4 78 | 2 37 6 | 38 9 5 |
| 9 78 5 | 4 37 1 | 6 2 38 |
| 3 2 6 | 59 58 589 | 7 1 4 |
+----------------+----------------+-----------------+ |
The cells with two candidates <13> are a and A. The strong link on <3> is x-x. We can eliminate <1> from the cells marked #, to solve the puzzle.
Keith |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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TKiel
Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI
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| Posted: Wed Aug 22, 2007 8:48 pm Post subject: |
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| I haven't looked at any of these links to see if they could answer my question but I wondered if there was any reason why the strong link couldn't be composed of more than two cells as long as it was an even number of cells? All the examples I've seen use a two cell strong link. |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Wed Aug 22, 2007 10:15 pm Post subject: |
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Tracy,
Yes, the external strong link can involve multiple cells. For instance, the "Pincers" of an XY-Chain can do the trick. Or, the ends of some other sort of Alternate Implication Chain (AIC). More important than an even number of cells (which holds for basic coloring or simple chains such as XY-Chains, but not necessarily for more involved structures) is to say that they must have opposite polarity ("+/-" or "red/green" or the like).
An additional note:
While the W-Wing usually involves a standard external strong link on X, we can also make use of the sort of link involved in the pincers of a Color Wing (or two-cluster multi-coloring) to do the trick, though I don't know if this is likely to come up often (or ever).
In the strong link case, one but not both of the "pincers" must be X. This means that one or both of the W-Wing bivalue cells must by Y.
In the Color Wing case, one or both of the "pincers" must be X. (This is not the same as a strong link; I don't know if it has a name.) The effect on the W-Wing cells is the same: one or both must be Y. So, the same eliminations occur. |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Thu Aug 23, 2007 12:09 am Post subject: |
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| TKiel wrote: | | I haven't looked at any of these links to see if they could answer my question but I wondered if there was any reason why the strong link couldn't be composed of more than two cells as long as it was an even number of cells? All the examples I've seen use a two cell strong link. |
Tracy,
Yes. Buried somewhere in the "Other Puzzles" discussion is an example where the connection is four cells (three links).
As Asellus says, any chain with opposite polarity on the ends will theoretically do. If the chain is not simple coloring, I find it hard to conceive how one would recognize it, or why the fact that the ends of the (extended) chain have the same two candidates is remarkable.
Keith |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Fri Aug 24, 2007 6:48 pm Post subject: |
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As it happens, I didn't need to look very far to find an example of a W-Wing "excited" by a Color Wing. The following is from the "DB Saturday Puzzle: August 18, 2007" thread in the "Other puzzles" forum:
| Code: | +-----------+------------+----------+
| 179 8 3 | 4 2 79 | 59 6 15 |
| 19 6 4 | 159 8 59 | 2 3 7 |
| 2 5 79 | 169 67 3 | 89 4 18 |
+-----------+------------+----------+
| 8 2 6 | 59 3 579 | 1 57 4 |
| 79 3 579 | 2 1 4 | 57 8 6 |
| 4 1 57 | 56 67 8 | 3 2 9 |
+-----------+------------+----------+
| 5 7 8 | 3 9 6 | 4 1 2 |
| 6 4 1 | 78 5 2 | 78 9 3 |
| 3 9 2 | 78 4 1 | 6 57 58 |
+-----------+------------+----------+ |
The focus in that thread was on the 79 W-Wing in Boxes 1 and 2 that is excited by the strong link on <9> in C7.
However, there is also a 79 W-Wing in Boxes 2 and 4 that is excited by a Color Wing on 9. The weak link "Bridge" is R3C37. There are single strong links to the Pincers at R1C7 and R5C3.
This second W-Wing results (in part indirectly) in the same eliminations as those of the first W-Wing. And, it clearly shows that it is possible to excite a W-Wing with a Color Wing. |
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strmckr
Joined: 18 Aug 2009
Posts: 64
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| Posted: Sat Aug 22, 2009 8:58 pm Post subject: |
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an exhaustive list of basic
W-wings
posted by ronk Here
| Code: | . . . | . . . | . / . . . a | . . . | . . .
. ab . | . -b . | . a . . ab a | . -b . | . . .
. . . | . . . | . / . . . a | . . . | . . .
---------+----------+---------- ---------+----------+----------
. . . | . . . | . / . . . / | . . . | . . .
. -b . | . ab . | . a . . -b a | . ab . | . . .
. . . | . . . | . / . . . / | . . . | . . .
---------+----------+---------- ---------+----------+----------
. . . | . . . | . / . . . / | . . . | . . .
. . . | . . . | . / . . . / | . . . | . . .
. . . | . . . | . / . . . / | . . . | . . .
Type A: Type B:
. . -b | . . . | . / .
. ab -b | . . . | . a .
. . -b | . . . | . / .
---------+----------+----------
. -b . | . . . | . / .
. -b ab | . . . | . a .
. -b . | . . . | . / .
---------+----------+----------
. . . | . . . | . / .
. . . | . . . | . / .
. . . | . . . | . / .
Type C:
Based on empirical tests, Types D1 and D2 below are equivalent:
-b . a | . . . | . . . -b . . | . . . | . . .
-b ab a | . . . | . . . -b ab . | . . . | . . .
-b . a | . . . | . . . -b . . | . . . | . . .
---------+----------+---------- ---------+----------+----------
. -b a | . . . | . . . . -b . | . . . | . . .
ab -b a | . . . | . . . ab -b . | . . . | . . .
. -b a | . . . | . . . . -b . | . . . | . . .
---------+----------+---------- ---------+----------+----------
. . / | . . . | . . . a a / | . . . | . . .
. . / | . . . | . . . a a / | . . . | . . .
. . / | . . . | . . . a a / | . . . | . . .
Type D1: Type D2:
. . . | / a / | . . .
. ab . | a a-b a | . . .
. . . | / a / | . . .
---------+----------+----------
. . . | . . . | . . .
. -b . | . ab . | . . .
. . . | . . . | . . .
---------+----------+----------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
Type E:
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Non-exhaustive list of
Extended W-wings
purposed by strmckr Here
| Code: |
. . . | / a / | . . .
. ab . | a a a | . -b .
. . . | / a / | . . .
---------+----------+----------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
---------+----------+----------
. . . | / a / | . . .
. -b . | a a a | . ab .
. . . | / a / | . . .
Group type E:
. . . | / a / | . . .
. ab . | a a-b a | . -b .
. . . | / a / | . . .
---------+----------+----------
. . . | . . . | . . .
. -b . | . ab . | . -b .
. . . | . . . | . . .
---------+----------+----------
. . . | / a / | . . .
. -b . | a a-b a | . ab .
. . . | / a / | . . .
Extended group type E:
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KEY: '/' <=> cells void of candidate 'a'
'a' <=> cells with candidate 'a'; not all are required
"-b" <=> potential eliminations of candidate 'b'[/code]
Last edited by strmckr on Fri Aug 28, 2009 6:48 pm; edited 1 time in total |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sat Aug 22, 2009 10:42 pm Post subject: |
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strmckr,
I do not see the need to enumerate every permutation, unless you are a computer programmer. Surely. defining the general pattern is sufficient?
Keith |
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wapati
Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.
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| Posted: Sat Aug 22, 2009 11:45 pm Post subject: Dunderhead |
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| keith wrote: | strmckr,
I do not see the need to enumerate every permutation, unless you are a computer programmer. Surely. defining the general pattern is sufficient?
Keith |
Keith, skycraper you expanded to cover all sorts of not related stuff.
You ignored that pattern and tried to plaster it on many other distinct solving methods.
Sky pattern is sufficient. |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sun Aug 23, 2009 1:14 am Post subject: Re: Dunderhead |
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| wapati wrote: | | keith wrote: | strmckr,
I do not see the need to enumerate every permutation, unless you are a computer programmer. Surely. defining the general pattern is sufficient?
Keith |
Keith, skycraper you expanded to cover all sorts of not related stuff.
You ignored that pattern and tried to plaster it on many other distinct solving methods.
Sky pattern is sufficient. |
Wapati,
Really?
I would appreciate a quote of where I said that.
Keith
And yes, I take offense at your post title of "Dunderhead". You think you are superior to others on this forum? |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sun Aug 23, 2009 1:21 am Post subject: |
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| Well, after reviewing what's been written in this thread, I now have to allow for a W-Wing with more that one strong link between cells. |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sun Aug 23, 2009 1:34 am Post subject: |
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| daj95376 wrote: | | Well, after reviewing what's been written in this thread, I now have to allow for a W-Wing with more that one strong link between cells. |
Danny,
Yes, but I don't know quite what you mean.
A W-wing with a strong link (or an equivalent connection) in one candidate is, well, a W-wing.
Sometimes, and we have a number of examples, there is another connection in the other candidate. So, we have a double W-wing, or a remote pair. The pincer cells make eliminations in both candidates.
Keith |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sun Aug 23, 2009 3:16 am Post subject: |
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| keith wrote: | | daj95376 wrote: | | Well, after reviewing what's been written in this thread, I now have to allow for a W-Wing with more that one strong link between cells. |
Danny,
Yes, but I don't know quite what you mean.
A W-wing with a strong link (or an equivalent connection) in one candidate is, well, a W-wing.
Sometimes, and we have a number of examples, there is another connection in the other candidate. So, we have a double W-wing, or a remote pair. The pincer cells make eliminations in both candidates.
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Keith,
The Group type E and the Extended group type E use two ER patterns (strong links in <a>) in [b2] and [b8] to form the W-Wing.
In this example by Asellus, there are two strong links -- [c3] and [c7] -- needed to form his second W-Wing.
| Code: | +-----------+------------+-----------+
| 179 8 3 | 4 2 *79 | 59@ 6 15 |
| 19 6 4 | 159 8 59 | 2 3 7 |
| 2 5 79@ | 169 67 3 | 89@ 4 18 |
+-----------+------------+-----------+
| 8 2 6 | 59 3 579 | 1 57 4 |
|*79 3 579@| 2 1 4 | 57 8 6 |
| 4 1 57 | 56 67 8 | 3 2 9 |
+-----------+------------+-----------+
| 5 7 8 | 3 9 6 | 4 1 2 |
| 6 4 1 | 78 5 2 | 78 9 3 |
| 3 9 2 | 78 4 1 | 6 57 58 |
+-----------+------------+-----------+
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| Asellus wrote: | Yes, the external strong link can involve multiple cells. For instance, the "Pincers" of an XY-Chain can do the trick. Or, the ends of some other sort of Alternate Implication Chain (AIC). More important than an even number of cells (which holds for basic coloring or simple chains such as XY-Chains, but not necessarily for more involved structures) is to say that they must have opposite polarity ("+/-" or "red/green" or the like).
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| Keith wrote: | As Asellus says, any chain with opposite polarity on the ends will theoretically do. If the chain is not simple coloring, I find it hard to conceive how one would recognize it, or why the fact that the ends of the (extended) chain have the same two candidates is remarkable.
|
What I find disturbing is that most of this thread's discussion preceeds your Remote Pairs thread where you say:
| Keith wrote: | A W-wing consists of two cells that have the same two (and only two) candidates. They are supplemented by two more cells that are a strong link on either of the candidates:
YX-X=X-XY
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I based my W-Wing detection module on this simplified statement! |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sun Aug 23, 2009 4:28 am Post subject: |
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Danny,
Let's go back to the history.
Back in the day, there was a flurry of messages related to a pattern that George Woods and TexCat had (sort of) recognized.
I tried to document the pattern, in the "W-wing" thread that you quote.
At about the same time, I noticed the connection between pairs that I called an M-wing. I also noticed that a remote pair did not need to be formed by a chain of identical candidate pairs.
Hence, my "Remote Pair" thread.
My current contention is that any of these wings starts with a pair of cells that have the same two candidates. The precise details of how they are connected are varied. |
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wapati
Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.
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| Posted: Sun Aug 23, 2009 6:03 am Post subject: |
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| keith wrote: | Danny,
Let's go back to the history.
Back in the day, there was a flurry of messages related to a pattern that George Woods and TexCat had (sort of) recognized.
I tried to document the pattern, in the "W-wing" thread that you quote.
At about the same time, I noticed the connection between pairs that I called an M-wing. I also noticed that a remote pair did not need to be formed by a chain of identical candidate pairs.
Hence, my "Remote Pair" thread.
My current contention is that any of these wings starts with a pair of cells that have the same two candidates. The precise details of how they are connected are varied. |
I wish you would solve the x-chain thing. even better the double same.
Triple, go for gold?
It has been done.
Keith, you need a rest. |
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strmckr
Joined: 18 Aug 2009
Posts: 64
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| Posted: Sun Aug 23, 2009 6:55 am Post subject: |
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| Quote: | strmckr,
I do not see the need to enumerate every permutation, unless you are a computer programmer. Surely. defining the general pattern is sufficient?
Keith |
I enumerated the differences in patterns as they are not all 4 cells specifically as covered by the patterns which you have addressed previously.
this is a more generalized version of them.
there is inferences of links in mini row/column ie grouped strong links that allows the pattern to function.
I'm a more kin to pattern based manually solving and like classification of move sets into groups of similar patterns or logic that works off the same premises .
i posted these examples for a few reason:
simply knowing how they function is some times not enough to fully comprehend how they actually function and diagrams reinforced there ideas and perhaps enlighten some on how different variations can be formed out of the idea.
i posted the diagram of possible occurrences of these with the exception of
w-Rings
which can be seen postedby Ronk.
{besides there is some variation perhaps not thought of before by some users as Danny noticed which are not covered by the earlier idea.}
interesting example Danny never thought of those before...
| Code: |
. . . | . / . | . / .
. -b . | AB / . | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | . / . | . / .
. . . | . a . | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | . / . | . / .
. AB . | -b a . | . / .
. . . | . / . | . / . |
| Code: |
. . . | . / -b | . / .
. . . | AB / -b | . a .
. . . | . / -b | . / .
---------+----------+----------
. . . | . / . | . / .
. . . | . a . | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | -b / . | . / .
. . . | -b a AB | . / .
. . . | -b / . | . / . |
...easy enough to see how they work but its a question where to draw the line for ease of use...
like add a fin a/b cell #: true\false *'s <> b
| Code: |
. . . | . / -b | . / .
. * . | AB / -b | . a .
. . . | . / -b | . / .
---------+----------+----------
. . . | . / . | . / .
. # . | * a * | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | -b / . | . / .
. * . | -b a AB | . / .
. . . | -b / . | . / . |
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Laimonas123
Joined: 17 Sep 2010
Posts: 1
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| Posted: Fri Sep 17, 2010 9:41 am Post subject: |
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| strmckr wrote: | | Quote: | strmckr,
I do not see the need to enumerate every permutation, unless you are a computer programmer. Surely. defining the general pattern is sufficient?
Keith |
I enumerated the differences in patterns as they are not all 4 cells specifically as covered by the patterns which you have addressed previously.
this is a more generalized version of them.
there is inferences of links in mini row/column ie grouped strong links that allows the pattern to function.
I'm a more kin to pattern based manually solving and like classification of move sets into groups of similar patterns or logic that works off the same premises .
i posted these examples for a few reason:
simply knowing how they function is some times not enough to fully comprehend how they actually function acomplia online and diagrams acomplia dosage reinforced there ideas and perhaps enlighten some on how different variations can be formed out of the idea.
i posted the diagram of possible occurrences of these with the exception of
w-Rings
which can be seen postedby Ronk.
{besides there is some variation perhaps not thought of before by some users as Danny noticed which are not covered by the earlier idea.}
interesting example Danny never thought of those before...
| Code: |
. . . | . / . | . / .
. -b . | AB / . | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | . / . | . / .
. . . | . a . | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | . / . | . / .
. AB . | -b a . | . / .
. . . | . / . | . / . |
| Code: |
. . . | . / -b | . / .
. . . | AB / -b | . a .
. . . | . / -b | . / .
---------+----------+----------
. . . | . / . | . / .
. . . | . a . | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | -b / . | . / .
. . . | -b a AB | . / .
. . . | -b / . | . / . |
...easy enough to see how they work but its a question where to draw the line for ease of use...
like add a fin a/b cell #: true\false *'s <> b
| Code: |
. . . | . / -b | . / .
. * . | AB / -b | . a .
. . . | . / -b | . / .
---------+----------+----------
. . . | . / . | . / .
. # . | * a * | . a .
. . . | . / . | . / .
---------+----------+----------
. . . | -b / . | . / .
. * . | -b a AB | . / .
. . . | -b / . | . / . |
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Thanks, but is not easy. And about Danny example. Danny example is very intresting, I have saw similar exaples. Danny nice job. Post same new examples 
Last edited by Laimonas123 on Fri Oct 15, 2010 12:01 pm; edited 1 time in total |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Fri Sep 17, 2010 6:56 pm Post subject: |
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| Laimonas123 wrote: | Danny nice job. Post some new examples.
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I don't specifically search for extended W-Wings anymore -- internal or external extended. Their eliminations are often duplicates from shorter W-Wings -- see groups 2 & 4 below. Sometimes, the eliminations aren't duplicates -- see group 3 below. In this case, even though the group 3 eliminations ar different than the group 1 eliminations, the final effect is the same because either group's eliminations generate a Locked Candidate that would eliminate the other group's eliminations.
| Code: | Puzzle 10/09/05: C -- after basics
+--------------------------------------------------------------+
| 124 18-24 6 | 248 124 5 | 3 7 9 |
| *24 9 5 | 7 6 3 | 1 *24 8 |
| 3 18-24 7 | 2489 1249 124 | 456 56 256 |
|--------------------+--------------------+--------------------|
| 6 7 *24 | 1 5 9 | 8 *24 3 |
| 1-24 5 8 | 3 24 246 | 7 9 126 |
| 9 124 3 | 246 7 8 | 456 56 1256 |
|--------------------+--------------------+--------------------|
| 7 3 @24+9 | 249 8 24 | 56 1 56 |
| 5 @24 @24+1 | 246 3 1246 | 9 8 7 |
| 8 6 19 | 5 19 7 | 2 3 4 |
+--------------------------------------------------------------+
# 63 eliminations remain
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The Remote Pair as 2x W-Wing: -plus- internally extended W-Wings
| Code: | * W-Wing (2=4)r2c1 - r2c8 = r4c8 - (4=2)r4c3 => r5c1<>2
* W-Wing (4=2)r2c1 - r2c8 = r4c8 - (2=4)r4c3 => r5c1<>4
W-Wing (2=4)r2c1 - r2c8 = r3c7 - r6c7 = r4c8 - (4=2)r4c3 => r5c1<>2
W-Wing (4=2)r2c1 - r2c8 = r3c9 - r56c9 = r4c8 - (2=4)r4c3 => r5c1<>4
W-Wing (2=4)r2c1 - r2c8 = r4c8 - r4c3 = r78c3 - (4=2)r4c3 => r5c1<>2
*@ W-Wing (2=4)r2c1 - r2c8 = r4c8 - r4c3 = r78c3 - (4=2)r8c2 => r13c2<>2
*@ W-Wing (4=2)r2c1 - r2c8 = r4c8 - r4c3 = r78c3 - (2=4)r8c2 => r13c2<>4
W-Wing (2=4)r2c1 - r2c8 = r3c7 - r6c7 = r4c8 - r4c3 = r78c3 - (4=2)r4c3 => r5c1<>2
W-Wing (2=4)r2c1 - r2c8 = r3c7 - r6c7 = r4c8 - r4c3 = r78c3 - (4=2)r8c2 => r13c2<>2
W-Wing (4=2)r2c1 - r2c8 = r3c9 - r56c9 = r4c8 - r4c3 = r78c3 - (2=4)r8c2 => r13c2<>4
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The Remote Pair as 2x M-Wing: (for completeness)
| Code: | M-Wing 3A (2=4)r2c1 - r2c8 = (4-2)r4c8 = (2)r4c3 => r5c1<>2
M-Wing 3A (4=2)r2c1 - r2c8 = (2-4)r4c8 = (4)r4c3 => r5c1<>4
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