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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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 Posted: Tue Dec 11, 2007 9:54 am Post subject: Overlapping X-Wing Solutions |
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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.
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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
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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
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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
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| Posted: Mon Dec 17, 2007 9:22 am Post subject: |
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| 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
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| Posted: Mon Dec 17, 2007 12:08 pm Post subject: |
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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 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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