## XY & Extended XY-Wing/Complementary cells Example

Advanced methods and approaches for solving Sudoku puzzles

### XY & Extended XY-Wing/Complementary cells Example

This post relies heavily on a post by Keith at http://dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=2143 in which he defines complementary cells, the M-Wing and the extended XY-wing both of which use complementary pairs. The example shown is Sudoku9981 Expert puzzle Book 9 #3. I used a 1 ER and a 4 W-wing to get to this point in the puzzle. At this point there are 4 important logical subpatterns that I used to solve this puzzle. First there is a 17 complementary pair in r5c2 and r9c6, i.e, both of these cells must be either 1 or 7. There is also a 47 complementary quad in r1c9, r4c7, r5c5, and r6c1. There is also an XY-wing with a pivot at r1c6 and 7 Z-xonjugates at r1c9 and r9c6 and an extended XY wing with a pivot which is the 17 complementary pair and 4 Z-conjugate at r1c6 and r6c1. Next consider the 7 Z-conjugate pair. r1c9 = 7 => r6c1 = 7 => r5c2 =1 => r9c6 = 1 & r9c6 =7 => r5c2 = 7 => r6c1 = 4 => r1c9 = 4. This proves that 7 cannot be in both r1c9 and r9c6 and therefore these are a true conjugate pair. This forces the other candidates in the XY-wing (14) to actually be in the XY-wing pattern and therefore 4 is eliminated from r1c7 and 1 is eliminated from r8c6. Next consider this 7 conjugate pair. r1c9 = 7 => r5c2 = 7 and r5c1 = not7 & r1c9 = 7 => r6c1 = 7 and r5c1 = not7. Therefore 7 is eliminated from r5c1. Next consider the 4 Z-conjugate pair. r6c1 = 4 => r1c9 = 4 and r1c6 = 1 & r1c6 = 4 => r1c9 = 7 => 46c1 = 7. As before this proves that the Z-conjugate is a true conjugate pair. Next consider this 4 conjugate pair. r6c1 = 4 => r5c5 = 4 and r6c5 = not4 & r1c6 = 4 => r6c5 = not4. Therefore 4 is eliminated from r6c5. This solves the puzzle. I think that the cells involved in the last elimination might be considered an M-wing. but then I have been wrong before.

XY and Extended XY-Wing/Complementary Cells Example
Code: Select all
` |-----------------+-----------------+-----------------| |  18    5    2   |   3   89   14   | -479   6   47   | |  18   46    7   |  16   89    2   |  49    3    5   | |   9   46    3   |   7   45   456  |   1    2    8   | |-----------------+-----------------+-----------------| |   2    8    9   |  456  457   3   |  47    1   467  | | 45-7  17  1456  |   2   47  -46   |   8    9    3   | |  47    3   46   |   8    1    9   |   2    5   467  | |-----------------+-----------------+-----------------| |   3    9   45   |  45    2    8   |   6    7    1   | |  457   2   145  | 145    6  -1457 |   3    8    9   | |   6   17    8   |   9    3   17   |   5    4    2   | |-----------------+-----------------+-----------------| `
Bud

Posts: 56
Joined: 24 August 2008

Bud

Code: Select all
`|-----------------+-----------------+-----------------| |  18    5    2   |   3   89   14   | -479   6   47   | |  18   46    7   |  16   89    2   |  49    3    5   | |   9   46    3   |   7   45   456  |   1    2    8   | |-----------------+-----------------+-----------------| |   2    8    9   |  456  457   3   |  47    1   467  | | 45-7  17  1456  |   2   47  -46   |   8    9    3   | |  47    3   46   |   8    1    9   |   2    5   467  | |-----------------+-----------------+-----------------| |   3    9   45   |  45    2    8   |   6    7    1   | |  457   2   145  | 145    6  -1457 |   3    8    9   | |   6   17    8   |   9    3   17   |   5    4    2   | |-----------------+-----------------+-----------------|`

A quicker way home (but that may not be your objective)
146r135c6=5r3c6-(5=4)r3c5-(4=1)r1c6 =>r9c6 <1> = 7
Singles from there
aran

Posts: 334
Joined: 02 March 2007

aran wrote:A quicker way home (but that may not be your objective...

Or simply:
Code: Select all
`.---------------.-------------------.-------------.| 18   5   2    |  3     89   *14   | 479  6  47  || 18   46  7    | *16    89    2    | 49   3  5   || 9    46  3    |  7     45    45-6 | 1    2  8   |:---------------+-------------------+-------------:| 2    8   9    |  45-6  457   3    | 47   1  467 || 457  17  1456 |  2     47   *46   | 8    9  3   || 47   3   46   |  8     1     9    | 2    5  467 |:---------------+-------------------+-------------:| 3    9   45   |  45    2     8    | 6    7  1   || 457  2   145  |  145   6     1457 | 3    8  9   || 6    17  8    |  9     3     17   | 5    4  2   |'---------------'-------------------'-------------'XY-Wing: 4/1/6 in r15c6,r2c4 => r3c6,r4c4<>6`
hobiwan
2012 Supporter

Posts: 321
Joined: 16 January 2008
Location: Klagenfurt

### XY & Extended XY-Wing/Complementary cells Example

Hi Aran,
First of all let me say that I recognize that I am a minor leaguer Sudoku-wise ompared to some other members of the forum. That being said, I appreciate your pointing out something that I overlooked since that helps me get better. One of my weaknesses is not being able to understand the logical notation that you used in your reply. I would appreciate it if you could make this clearer.
Bud

Posts: 56
Joined: 24 August 2008

This is Aran's hidden (almost) triple (subset) trick, which i like, so allow me to answer:
Either r3c6=5 -> r3c5=4 -> r1c6=1
or there is a triple 146 in column 6 (r135c6).

In both cases in column 6 r1c6=1.

Here a hidden pair trick would have done the same ((r3c6=5 -> r1c6=1) or (r35c6=46 -> r1c6=1)).
eleven

Posts: 2376
Joined: 10 February 2008

hobiwan wrote:
aran wrote:A quicker way home (but that may not be your objective...

Or simply:
Code: Select all
`.---------------.-------------------.-------------.| 18   5   2    |  3     89   *14   | 479  6  47  || 18   46  7    | *16    89    2    | 49   3  5   || 9    46  3    |  7     45    45-6 | 1    2  8   |:---------------+-------------------+-------------:| 2    8   9    |  45-6  457   3    | 47   1  467 || 457  17  1456 |  2     47   *46   | 8    9  3   || 47   3   46   |  8     1     9    | 2    5  467 |:---------------+-------------------+-------------:| 3    9   45   |  45    2     8    | 6    7  1   || 457  2   145  |  145   6     1457 | 3    8  9   || 6    17  8    |  9     3     17   | 5    4  2   |'---------------'-------------------'-------------'XY-Wing: 4/1/6 in r15c6,r2c4 => r3c6,r4c4<>6`

Or for more of a stroll home :
AUR(47)r14c79 : 6r4c9=9r1c7-(9=4)r2c7-4r1c79=4r1c6-(4=6)r5c6-6r4c4=6r4c9 =>r4c9=6

Bud

One of my weaknesses is not being able to understand the logical notation that you used in your reply. I would appreciate it if you could make this clearer.

I see that Eleven has answered your question.
Elegant objects, these almost hidden triples.
aran

Posts: 334
Joined: 02 March 2007

aran wrote:Elegant objects, these almost hidden triples.

And it takes a good eye to find them.
DonM
2013 Supporter

Posts: 475
Joined: 13 January 2008

### XY & Extended XY-Wing/Complementary cells Example

Thanks everyone. I like Aran's trick also now that understand it. It looks like I took a roundabout way to solve the puzzle but it was fun seeing what I could do with Keith's complementary cells.
Bud

Posts: 56
Joined: 24 August 2008

aran wrote:Bud

Code: Select all
`|-----------------+-----------------+-----------------| |  18    5    2   |   3   89   14   | -479   6   47   | |  18   46    7   |  16   89    2   |  49    3    5   | |   9   46    3   |   7   45   456  |   1    2    8   | |-----------------+-----------------+-----------------| |   2    8    9   |  456  457   3   |  47    1   467  | | 45-7  17  1456  |   2   47  -46   |   8    9    3   | |  47    3   46   |   8    1    9   |   2    5   467  | |-----------------+-----------------+-----------------| |   3    9   45   |  45    2    8   |   6    7    1   | |  457   2   145  | 145    6  -1457 |   3    8    9   | |   6   17    8   |   9    3   17   |   5    4    2   | |-----------------+-----------------+-----------------|`

A quicker way home (but that may not be your objective)
146r135c6=5r3c6-(5=4)r3c5-(4=1)r1c6 =>r9c6 <1> = 7
Singles from there

aran, you do things with chains that I didn't know were possible! Sorry for the late response but I just now stumbled back on your post.

Here's how I'm reading this: If the group isn't true, then r1c6=1 is true. Or if r1c6 isn't 1 then the group is true. Either way, r1c6=1.

Could this be called a discontinuous nice loop? Or a DNL with ALS?
Is there someone who would be so kind as to write this in NL for me?

Luke
2015 Supporter

Posts: 435
Joined: 06 August 2006
Location: Southern Northern California

Luke451 wrote:
aran wrote:Bud

Code: Select all
`|-----------------+-----------------+-----------------| |  18    5    2   |   3   89   14   | -479   6   47   | |  18   46    7   |  16   89    2   |  49    3    5   | |   9   46    3   |   7   45   456  |   1    2    8   | |-----------------+-----------------+-----------------| |   2    8    9   |  456  457   3   |  47    1   467  | | 45-7  17  1456  |   2   47  -46   |   8    9    3   | |  47    3   46   |   8    1    9   |   2    5   467  | |-----------------+-----------------+-----------------| |   3    9   45   |  45    2    8   |   6    7    1   | |  457   2   145  | 145    6  -1457 |   3    8    9   | |   6   17    8   |   9    3   17   |   5    4    2   | |-----------------+-----------------+-----------------|`

A quicker way home (but that may not be your objective)
146r135c6=5r3c6-(5=4)r3c5-(4=1)r1c6 =>r9c6 <1> = 7
Singles from there

aran, you do things with chains that I didn't know were possible! Sorry for the late response but I just now stumbled back on your post.

Here's how I'm reading this: If the group isn't true, then r1c6=1 is true. Or if r1c6 isn't 1 then the group is true. Either way, r1c6=1.

Luke451
What you write is correct in the circumstances, although I would prefer to leave it that "one end-point of the chain must be true" ie either 146r135c6 is true or r1c6 is true, and hence <1>r9c6. If for example there was an additional candidate 1 in r3c6, then the conclusion <1>r9c6 would still hold whereas r1c6=1 would not follow (at least not immediately).
aran

Posts: 334
Joined: 02 March 2007

aran wrote:What you write is correct in the circumstances, although I would prefer to leave it that "one end-point of the chain must be true" ie either 146r135c6 is true or r1c6 is true, and hence <1>r9c6. If for example there was an additional candidate 1 in r3c6, then the conclusion <1>r9c6 would still hold whereas r1c6=1 would not follow (at least not immediately).

Got it, that's an important distinction. "One end-point of the chain must be true," and any eliminations follow from there.

Luke
2015 Supporter

Posts: 435
Joined: 06 August 2006
Location: Southern Northern California