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Asellus
Joined: 05 Jun 2007 Posts: 865 Location: Sonoma County, CA, USA

Posted: Tue Dec 11, 2007 9:54 am Post subject: Overlapping XWing Solutions 


In solving yesterday's (10Dec2007) OneTrick Pony from sudocue.net, I came across the following:
Code:  ++++
 28 28 9  3 7 6  5 1 4 
 6 5 7  9 1 4  8 2 3 
 1 3 4  5 8 2  6 79 79 
++++
 3 9 28  6 4 5  1 78 27 
 7 4 6  1 a23 a38  9 58 25 
 5 28 1  28 9 7  3 4 6 
++++
 9 7 3  248 25 18  24 6 15 
 248 b16 258  24 b256 19  7 3 159 
 24 B16 25  7 ab2356 A139  24 59 8 
++++ 
There is an XWing on <3> marked "aA" and an XWing on <6> marked "bB" and they overlap in a single cell (r9c5). There is also a strongly linked pair of <1>s in r9, one <1> in each of the two XWings, the cells marked "A" and "B". I realized that this led to the elimination of <2> and <5> in r9c5 and <9> in r9c6, as I will explain below.
I hadn't encountered this pattern before and it made me think about the general case, which I describe below. No doubt all this has been described elsewhere. Yet, since I hadn't encountered it, I thought I'd post it. If I've made any errors, I'm certain someone will post corrections.
Overlapping XWing Solutions Involving a Third Digit Strong Link
First, the case of a singlecell overlap. Then, the much less interesting case of a twocell overlap.
ONECELL OVERLAP:
Two XWings, one on "y" and one on "z", overlap on a single cell. A third digit, "x", is strongly linked between two of these cells, one in the "y" XWing and one in the "z" XWing. (These "x" cells can be remote and the link can be strongly inferential, as in the pincer ends of a wing or chain.) Each XWing has two possible solutions, which I will denote with "Y" and "y" and with "Z" and "z", respectively. In each XWing, there is a diagonal solution that includes the overlap cell, and a diagonal solution that excludes the overlap cell. There are four possible configurations, one of which is trivial, based on the locations of the strongly linked x's:
POSSIBILITY 1: Both linked x's occur in cells on the diagonals that exclude the overlap cell. Result: The overlap cell and and "x" cells become bivalues ({yx}, {zx}, and {yz}) with all other digits in these three cells eliminated.
Code:  Example:
Xy Y The diagonals that exclude the overlap are yy and ZZ.
Here, one x is in an y cell and the other in a Z cell.
z xZ Polarity ("color") is induced as shown by the capitalization.
The Xy, xZ and Yz cells become bivalues; all other candidates
Z Yz y are eliminated from these three cells.

POSSIBILITY 2: One linked "x" is in a diagonal excluding the overlap cell and the other "x" in a diagonal including the overlap cell. Result: The diagonal that contains neither the overlap cell nor one of the linked x's is True and those digits can be placed.
Code:  Examples:
xy Y The y diagonal contains one of the linked x's.
The Y diagonal contains the overlap cell.
xz Z The z diagonal contains one of the linked x's and the overlap cell.
The Z diagonal contains no linked x and no overlap cell.
Z Yz y The two Z values are True and can be placed in those two cells.
y Y Only the y diagonal contains neither a linked x
nor the overlap cell.
z xZ The two y values are True and can be placed in those two cells.
Z xYz y

POSSIBILITY 3: The linked x's are each diagonally opposite the overlap cell. No eliminations or placements result from this configuration.
Code:  Example:
a xA The "x" cells are both diagonally opposite the overlap cell.
No eliminations or placements result from this configuration.
xb B
B Ab a

POSSIBILITY 4: The linked x's are in the overlap cell and a cell diagonally opposite it. The diagonal opposite that of the "x" cells is True and can be placed. (This is the trivial possibility since it really involves only a single XWing.)
TWOCELL OVERLAP:
This is not so interesting. The two overlapping cells are necessarily a locked pair. So, any linked "x" pair can only occur in the nonoverlapping cells of the XWings. If the linked x's do not share a row or column, then the diagonals of each XWing without one of these x's are true and can be placed. (Since these cells are never peers, the x's would have to be linked by a wing or coloring or some other implication chain.) If the linked x's share a row or column, no eliminations or placements result. 

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Myth Jellies
Joined: 27 Jun 2006 Posts: 64

Posted: Mon Dec 17, 2007 9:22 am Post subject: 


I don't wish to stifle your creativity, but I note that your example (as well as theoretical possibility 1) works out to be an elaborate way to find a 136hidden triple in row 9. Some of your other theoretical setups might be more interesting though. 

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Asellus
Joined: 05 Jun 2007 Posts: 865 Location: Sonoma County, CA, USA

Posted: Mon Dec 17, 2007 12:08 pm Post subject: 


Well, I could say, "Who wants to find a Hidden Triple in the same old way all the time, anyway?"
Yes, it is (now) obvious that "Possibility 1" is necessarily a Locked Triple when the the "x" cells and the overlap cell are colinear. However, I believe Possibility 1 (might?) still have value when the cells are not colinear:
Code:  Example:
Xy Y The diagonals that exclude the overlap are yy and ZZ.
Here, one x is in an y cell and the other in a Z cell.
z Z Polarity ("color") is induced as shown by the capitalization.
The Xy, xZ and Yz cells become bivalues; all other candidates
xZ Yz y are eliminated from these three cells. 
I don't believe that these cells are inherently a Hidden Triple provided they don't share a box. Since the "x" pair would be remote in that case, the strong link would need to be induced externally, by a wing or chain for instance. 

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