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Overlapping X-Wing Solutions

 
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Asellus



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

PostPosted: Tue Dec 11, 2007 9:54 am    Post subject: Overlapping X-Wing Solutions Reply with quote

In solving yesterday's (10-Dec-2007) One-Trick 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   ab-23-56 A13-9 | 24    59    8    |
+------------------+--------------------+------------------+

There is an X-Wing on <3> marked "aA" and an X-Wing 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 X-Wings, 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 X-Wing Solutions Involving a Third Digit Strong Link

First, the case of a single-cell overlap. Then, the much less interesting case of a two-cell overlap.

ONE-CELL OVERLAP:

Two X-Wings, 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" X-Wing and one in the "z" X-Wing. (These "x" cells can be remote and the link can be strongly inferential, as in the pincer ends of a wing or chain.) Each X-Wing has two possible solutions, which I will denote with "Y" and "y" and with "Z" and "z", respectively. In each X-Wing, 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 y-y and Z-Z.
                    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 X-Wing.)


TWO-CELL 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 non-overlapping cells of the X-Wings. If the linked x's do not share a row or column, then the diagonals of each X-Wing 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

PostPosted: Mon Dec 17, 2007 9:22 am    Post subject: Reply with quote

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 136-hidden 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

PostPosted: Mon Dec 17, 2007 12:08 pm    Post subject: Reply with quote

Well, I could say, "Who wants to find a Hidden Triple in the same old way all the time, anyway?" Wink

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 y-y and Z-Z.
                    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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