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 Post subject: X-Wing
PostPosted: Fri Apr 16, 2010 8:49 pm 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
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Location: Lethbridge, Alberta, Canada
Many "plain vanilla" sudoku techniques are also useful when solving Killers; some of these have already been included in the Killer Techniques forum and I expect that some others will be added. It should also be noted that at least one jigsaw sudoku technique, Law of Leftovers, can be used for Killers which have a 45(9) cage covering more than one row/column/nonet.

One of the simplest "plain vanilla" sudoku techniques is X-Wing.

The first example is a very simple one, taken from Killer 1476 Moderate on the first sudoku site I used, http://www.sudoku.org.uk.

Image

The 3(2) cage at R7C6 and the 4(2) cage at R9C6 contain naked pairs {12} and {13} respectively, locking 1 for R7 and R9. They form a X-Wing for 1 in C67 so there can be no other 1s in C67.

In this particular case there is also the interesting point that there can be no other 1s in N89.


There is an extension from simple X-Wings, which only involve two 2-cell cages, for Killers. Grouped X-Wings are formed when the combinations of two cages lock one, or occasionally two, values within those cages.

An example of this comes from A188V2.

Image

In this puzzle the 7(3) cage at R6C6 is a naked triple {124} while all combinations for the 25(6) cage at R6C4 must contain 1,2 leading to

Grouped X-Wing for 1,2 in 25(6) cage at R6C4 and 7(3) cage at R6C6 for R67, no other 1,2 in R67

[See later post for comment about terminology.]


Last edited by Andrew on Fri Feb 17, 2012 10:31 pm, edited 1 time in total.

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 Post subject: Re: X-Wing
PostPosted: Sat May 01, 2010 5:52 am 
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Grand Master
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Thanks Andrew for starting this thread.

Generalised X-Wing
Image

Here is another example that comes from vanilla sudoku, but this time works because of an extra constaint, the diagonals (1-9 cannot repeat on the diagonals) from a sudoku X. It works the same way as normal X-Wing, but involves one diagonal. This comes from my walk-through for Cage-Pairs 6 Overlap X here

Note: the 9s are highlighted in blue.
Quote:
17. Generalized X-wing on 9 in n9 and D/ -> 9 must be in r29c8, locked for c8; and in r8c29, locked for r8
ie, 9 is only in two places in n9 and only two places in D/ and each of these sees one 9 in nonet 9 or diagonal / -> 9 locked for all common peers.

Afmob's walk-through for A130 here (step 4a) introduced me to the term "generalised X-Wing" for this move involving the diagonal.

Some thoughts on Andrew's Grouped X-Wing example
Andrew post gives the really nice example of a killer version of X-Wing for A188V2. He calls it "Grouped X-Wing". Just to note that Afmob calls it "Caged X-Wing" in his walk-through here. I quite like Afmob's usage and will use it when two cages make the move work. I'll use "Grouped X-wing" for situations that lock a candidate(s) in two rows or columns using more than just two cages.

Another observation is that Caged X-Wing are only needed to make eliminations when the two cages require the same digit and share two rows or columns and both cross a nonet boundary. If they don't cross a nonet boundary then the simpler way is using locked candidates for 3 different nonets. Don't worry if that doesn't make much sense. It's just By The Way (BTW) really.

BTW2. SudokuSolver finds "X-Wing", "Generalized X-Wing" and "Caged X-wing" in the "X-cycle simple" routine.

BTW3. Caged X-Wing is a type of killer sub-set. Larger killer subsets involving more than two cages in 2 rows or columns are possible. They are really fun to find.

Cheers
Ed


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 Post subject: Re: X-Wing
PostPosted: Fri Feb 17, 2012 10:59 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
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Location: Lethbridge, Alberta, Canada
Thanks Ed for commenting on the terminology of my A188 V2 example. Yes, I agree that Caged X-Wing is a better description when it results from two cages in the puzzle diagram. At the time I did that post I hadn't managed to finish the puzzle. It was only recently that I came back to it and finished it; then I looked at Afmob's walkthrough.

Grouped X-Wing is more appropriate when cages are grouped to form one of the components of the X-Wing. I'm also inclined to use that term when one of the components is a hidden cage.

Thanks Ed for posting a neat example of a Generalised X-Wing, where the components are connected by a diagonal and a nonet, rather than by cages.


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