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The generalized m-wing (gm-wing): simple recipe
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Sun Nov 23, 2008 5:15 pm    Post subject: The generalized m-wing (gm-wing): simple recipe Reply with quote

This is a distilled (and graphically enhanced) version of the one-step solution in a recent thread in the "other puzzles"
_____________________________________________________________
Nataraj's gm-wing recipe:

two strong links (in different candidates a,b), sharing one common cell
a bi-value cell with those same candidates {a,b}, which sees either free end of the strong links.
_____________________________________________________________

As an example, I'd like to take this Menneske M2534023 super hard (transposed):
Code:

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


After basics, this is what we have:
Code:
+--------------------------+--------------------------+--------------------------+
| 9       6       2        | 378     5       37       | 48      1       34       |
| 8       5       4        | 123     6       13       | 9       7       23       |
| 7       1       3        | 28      9       4        | 5       68      26       |
+--------------------------+--------------------------+--------------------------+
| 3       49      78       | 5       1       67       | 2       4689    469      |
| 1       2       6        | 9       4       8        | 3       5       7        |
| 5       49      78       | 67      3       2        | 48      4689    1        |
+--------------------------+--------------------------+--------------------------+
| 2       7       1        | 34      8       35       | 6       49      459      |
| 46      3       5        | 46      7       9        | 1       2       8        |
| 46      8       9        | 146     2       156      | 7       3       45       |
+--------------------------+--------------------------+--------------------------+


The one-step winner in this puzzle is the generalized m-wing

(8)r6c3=r1c7;r6c7<>8

In previous posts, I have compared the generalized m-wing to a flail. It seems that the reference was a bit outdated, so I try something else.

A gun.

There are two strong links in the (g)m-wing:

one strong link (7) between r6c4 and r1c4 --- the "grip"


one strong link ( 8 ), connected to the first one, between r1c4 and r1c7 --- the "barrel"


one bi-value cell {7,8} --- the "hand", reaching (7 to 7) for the grip


Together --- bad luck for the "8" in cell r6c7 that sees both the hand (r6c3) and the business end of the barrel (r1c7):



For the more mathematically inclined among us:

Start with the "hand" (r6c3).
If it is "8", the victim is toast ( r6c3=8 => r6c7<>8 )

If r6c3 is not 8, it must be 7 (remember, the hand is always bi-value)

Let's see what follows from this assumption (notice how we elegantly switched over to "7" as our workhorse ...)

Since r6c3 "sees" one end of the grip, that end (r6c4) can then not be 7
But since we have a strong link on 7 in col 4, the other end of the grip (r1c4) must be 7.
... now it is time to switch back again to "8" ....
If r1c4=7 then it cannot be "8". (regardless of how many other candidates there are in that cell Smile )
The strong link (on "8") in row 1 assures us that if r1c4<>8 => r1c7=8.

But with r1c7=8, the poor victim 8 in r6c7 is dead once more ...

Boooohahaaaaaaaa !!!!
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Mon Nov 24, 2008 1:20 am    Post subject: Reply with quote

OK, nice work and thanks for taking the time. But back to my fragment:

Code:
+--------------+-------+-------+
| .   .    .   | . . . | . . . |
| .   .    .   | . . . | . . . |
| .   .    .   | . . . | . . . |
+--------------+-------+-------+
| 4   1    7   | . . . | . . . |
| 28  28   3   | . . . | . . . |
| 6   9    5   | . . . | . . . |
+--------------+-------+-------+
| 3   267  126 | . . . | . . . |
| 9   2567 126 | . . . | . . . |
| 258 4    268 | . . . | . . . |
+--------------+-------+-------+


Quote:
two strong links (in different candidates a,b), sharing one common cell
a bi-value cell with those same candidates {a,b}, which sees either free end of the strong links.


I'm obviously missing something (what else is new?) since it seems that r5c1 meets those criteria.
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re'born



Joined: 28 Oct 2007
Posts: 80

PostPosted: Mon Nov 24, 2008 2:08 am    Post subject: Reply with quote

Marty R. wrote:
OK, nice work and thanks for taking the time. But back to my fragment:

Code:
+--------------+-------+-------+
| .   .    .   | . . . | . . . |
| .   .    .   | . . . | . . . |
| .   .    .   | . . . | . . . |
+--------------+-------+-------+
| 4   1    7   | . . . | . . . |
| 28  28   3   | . . . | . . . |
| 6   9    5   | . . . | . . . |
+--------------+-------+-------+
| 3   267  126 | . . . | . . . |
| 9   2567 126 | . . . | . . . |
| 258 4    268 | . . . | . . . |
+--------------+-------+-------+


Quote:
two strong links (in different candidates a,b), sharing one common cell
a bi-value cell with those same candidates {a,b}, which sees either free end of the strong links.


I'm obviously missing something (what else is new?) since it seems that r5c1 meets those criteria.

r5c1<28> sees r9c1<258> which is strongly linked via 8 to r9c3 which is strongly linked via 2 to ???. If ???=r78c3, then you can eliminate 2 from r9c1 (it's cannibalistic Exclamation ). If ???=the rest of block 7, then you can't eliminate anything.
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Mon Nov 24, 2008 9:01 pm    Post subject: Reply with quote

Marty R. wrote:

I'm obviously missing something


Marty, try putting together the complete set of ingredients (not just one cell)

You need:
One bi-value cell: the hand
A strong link in one of the two candidates (seen by the hand): the grip
Another strong link in the other of the two candidates (connected to the first one): the barrel
And the victim

In your fragment, which is which (hand, grip, barrel, victim) ?

I think what you are missing is the barrel (and the victim) but we'll see ...
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Mon Nov 24, 2008 10:27 pm    Post subject: Reply with quote

Nataraj,

Let's temporarily forget grips and barrels. We are connecting two remote pairs (not literally since one's a triple) XY and aXY. And I know the first part: those two cells are connected by two strong links on either X or Y.

Coincidentally or not, in both your example and my fragment, it's the triple that needed to be the common cell that shares both strong links.

So is it true that it's the aXY cell that must always be the common cell?
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Tue Nov 25, 2008 7:14 am    Post subject: Reply with quote

Marty,

I guess we're getting closer to the essence of the matter.

First, we should decide whether to use "notation" or not. Initially I wrote this little guide here to get away from notation and present a more intuitive, graphical picture (there was someone over in the "others" forum said something like "I might be the only one here who doesn't understand notation, ..." lol)

One excellent example how notation for a gm-wing looks like, is Danny's explanation there in the other thread ("Another good Menneske"):
Code:

      a         b           c         d
 (8=7)r3c6 - (7)r4c6 = (7-8)r4c1 = (8)r8c1  =>  [r8c6]<>8


In a less formal way, this is Keith's notation in the "remote pairs" thread

Quote:
XY ... (X) ... bXY = aY


Less formal in the sense that it has the "...(X)..." part, which is explained earlier in the text (when there was no b in bXY): "They are connected in some way so that you know they have the same value in the solved puzzle"

In my presentation of the gm-wing I do not use the notion of remote pairs at all (because there are no longer any remote pairs) but made that a weak link plus strong link in X (which makes sure that if the first cell is X than the third cell is, too - which is enough to perfom the elimination and not quite the same as saying the two cells always have the same value. This is mainly where "generalized" comes in)

The ingredients in the above chain:
hand: XY
grip: part of ...(X)...
barrel: bXY=aY
victim: any cell that sees the first (XY) and last (aY) cell.

To get back to your post:
XY and aXY are not a remote pair
they are NOT connected by two strong links (but a weak plus strong in X)

The bXY is the connecting cell all right.
It could be anything from a simple pair XY to a cell packed with candidates, not necessarily a triple.

I'm still missing the barrel (the "other" strong link, the "aY" cell) in your fragment. (Please. It would be helpful indeed if you could relate the cells/links in your fragment with either the hand,grip,barrel or the XY, ...(X)..., bXY, aX elements of Keith's notation.)
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Tue Nov 25, 2008 9:20 am    Post subject: Reply with quote

Marty: The typical M-Wing is four cells joined by strong links in two candidates -- the ones in the initial bivalue cell. Your fragment is just that! It only supports three cells of the necessary four. You lack a final strong link to join your third cell to a fourth cell Exclamation

The reason keith's general notation includes ...(X)... is because M-Wings with 6/8/... cells are also possible.
(Somewhere, I posted a 6-cell M-Wing.)
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Tue Nov 25, 2008 5:59 pm    Post subject: Reply with quote

Thank you both but I'm still not sure. Maybe it means that each of the four cells must be part of a strong link?

I didn't realize that this was as complicated as it looks to me. And as much as I appreciate the help, I think we're splitting hairs as to whether XY and aXY are remote or not (they're in different houses) and whether they're connected by two strong links or a strong and weak (to me a strong link is when there are two possibilities in a house).

Neither of you were around when I joined here and started pestering people with my questions. My spatial skills are zero and the only way I've been able to learn techniques is to somehow get a rule and then apply that rule to situations that I subsequently recognize. I'm strictly a mechanic and don't have an understanding of the underlying theory and have a hard time visualizing all this stuff.

Once again, I always appreciate the fact that people are willing to help.
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nataraj



Joined: 03 Aug 2007
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PostPosted: Tue Nov 25, 2008 7:30 pm    Post subject: Reply with quote

Fair enough. The "generalized" part is indeed a very abstract / mathematical concept. Last thing I would want to do is get into splitting hairs ... Smile

(and I will learn to tell better stories Very Happy)
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Tue Nov 25, 2008 10:16 pm    Post subject: Reply with quote

Quote:
(and I will learn to tell better stories)

Instead of that, how about learning to help me get connected to your format tool instead of it giving me the message:

"Connection Interrupted

The connection to the server was reset while the page was loading.

The network link was interrupted while negotiating a connection. Please try again."


Just kidding, I don't think you have control over the server, but I've encountered this a few times in the last couple of days. Smile
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Wed Nov 26, 2008 2:01 pm    Post subject: Reply with quote

I noticed the problems, too.

I'd only added the backup link in the "site help" forum, here goes:
nataraj wrote:
Free webhosting being, well, free (and sometimes unreliable) I've opened a backup site. Two possibilities:

- http://nataraj.eluhost.de/fmtsudoku.htm (currently giving lots of errors) and
- http://nataraj.0catch.com/fmtsudoku.htm (more pop-up ads)
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Marty R.



Joined: 12 Feb 2006
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PostPosted: Wed Nov 26, 2008 4:35 pm    Post subject: Reply with quote

Yes, thanks, I used that second one yesterday when the first wouldn't connect.
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sun Feb 07, 2010 8:07 pm    Post subject: Reply with quote

[Withdrawn: An extraneous rehash of information already present.]

Last edited by daj95376 on Fri Mar 12, 2010 10:40 pm; edited 1 time in total
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Marty R.



Joined: 12 Feb 2006
Posts: 5162
Location: Rochester, NY, USA

PostPosted: Sun Feb 07, 2010 10:54 pm    Post subject: Reply with quote

Quote:

My apologies if this leaves you even more confused about gM-Wings!

If that was directed at me, no, it does not further confuse me. I understand the one here; it's that one in the other puzzle that I could never see.
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Mogulmeister



Joined: 03 May 2007
Posts: 695

PostPosted: Fri Mar 12, 2010 1:54 pm    Post subject: Reply with quote

Just to say to Nataraj that I am more graphically inclined and I like the spatial references! I like "flail" - it is not outdated, just more literary. Smile
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nataraj



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PostPosted: Fri Mar 12, 2010 7:17 pm    Post subject: Reply with quote

Mogulmeister wrote:
Just to say to Nataraj that I am more graphically inclined and I like the spatial references! I like "flail" - it is not outdated, just more literary. Smile


Thanks !
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keith



Joined: 19 Sep 2005
Posts: 3175
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PostPosted: Sat Mar 13, 2010 8:10 am    Post subject: Reply with quote

Another recipe is posted here:

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=4366&start=21
Quote:

The original recipe for M-wings involved finding two bivalue cells and looking for how they might be connected. That is a variant on how to find W-wings, so I still use it.

Here is a different recipe:

Choose any cell, a, that has only two candidates, XY. See if you can find another cell, b, such that
1) a = X forces b = X.
2) b contains Y as a candidate.

If you can find a third cell, c, such that bc is a strong link in Y, then ac are the pincers of an M-wing, and eliminate Y.
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Bud



Joined: 06 May 2010
Posts: 47
Location: Tampa, Florida

PostPosted: Mon May 10, 2010 2:07 pm    Post subject: Reply with quote

I am a new member and I think I understand Keith's M-Wing and I wanted to find out what a gM-wing is. However after looking at the example the only thing I can see is XY-Chain forcing or if you start at R6C7 an XY-Chain forcing wrap. This doesn't seem to be a pattern technique at all. Of course I have been wrong before.
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Marty R.



Joined: 12 Feb 2006
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PostPosted: Mon May 10, 2010 3:24 pm    Post subject: Reply with quote

Hi Bud,

Welcome to the forum. I'd also like to find out what a gM-Wing is.
You can see Nataraj's definition at the start of this thread. You can see Keith's definition here:

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=4366&highlight=humpty+dumpty&sid=5d128acf34316e16d11802c519e4a338

If you do find out, please let me know the answer. Wink
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Bud



Joined: 06 May 2010
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PostPosted: Mon May 10, 2010 4:40 pm    Post subject: Reply with quote

Thanks Marty,
The example in the link looked like a pattern and I was able to follow Keith's logic. But the example here is completely different and still looks like an example of XY-Chain forcing.

Bud
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