## Hidden Power of the XY-Wing. The Z-Wing & Z-Turbot Fish

Advanced methods and approaches for solving Sudoku puzzles

### Hidden Power of the XY-Wing. The Z-Wing & Z-Turbot Fish

The hidden power of the XY-wing is due to the fact that the ends of the chain are a diagonal Z conjugate pair whose cells are not in the same house. A Z-conjugate pair differs from a true conjugate pair since if A and B are Z-conjugates then A inclusive or B must be z. For a conjugate pair this relationship would be A exclusive or B. If a Z-conjugate replaces a conjugate pair in an X-wing pattern then the Z-conjugate is forced to be a true conjugate. If a Z-conjugate replaces a conjugate pair in one of the turbot fish patterns, the resultant pattern will still work. The z-conjugate can also be used with caution in chain techniques. To illustrate these here are several examples..

The first example is a Z-wing, which closely resembles an X-wing and has the same cell z-eliminations as an X-wing. These cells are denoted by *. The logic for this is essentially the same as an X-wing. Next consider the z conjugate pair in row 1. If r1c1 is z then r5c1 is x and r9c9 is y. If r1c5 is z then r9c5 is y and r9c1 is x. This means that y must be either in r9c1 or r9c5 and x must be in either r5c1 or r9c1. This means that no other y can be in row 9 which is denoted by + and no other z can be in column 1 which is denoted by #. I want to thank Ronk for his help on the Z-wing portion of this post.

Z-Wing Example 1
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` |-----------+-------------+-------------| |  z  -  -  |  -   z   -  |  -   -   -  | | #*  .  .  |  .   *   .  |  .   .   .  | | #*  .  .  |  .   *   .  |  .   .   .  | |-----------+-------------+-------------| | #*  .  .  |  .   *   .  |  .   .   .  | | xz  .  .  |  .   *   .  |  .   .   .  | | #*  .  .  |  .   *   .  |  .   .   .  | |-----------+-------------+-------------| | #*  .  .  |  .   *   .  |  .   .   .  | | #*  .  .  |  .   *   .  |  .   .   .  | | xy  +  +  |  +  yz   +  |  +   +   +  | |-----------+-------------+-------------| `

The next z-wing example has a 2 box xy-wing with pivot at r9c2. For this case additional z eliminations dnoted by * in r9b7 and r7b8 are due to the xy-wing. Also the x cell eliminations denoted by # occur in box 7. The y eliminations denoted by + occur in row 9.

Z-Wing Example 2
Code: Select all
` |-------------+-------------+-------------| |  z   -   -  |  -   z   -  |  -   -   -  | |  *   .   .  |  .   *   .  |  .   .   .  | |  *   .   .  |  .   *   .  |  .   .   .  | |-------------+-------------+-------------| |  *   .   .  |  .   *   .  |  .   .   .  | |  *   .   .  |  .   *   .  |  .   .   .  | |  *   .   .  |  .   *   .  |  .   .   .  | |-------------+-------------+-------------| | xz   #   #  |  *   *   *  |  .   .   .  | | #*   #   #  |  .   *   .  |  .   .   .  | | +#* xy  +#* |  +  yz   +  |  +   +   +  | |-------------+-------------+-------------| `

The next example is a partially worked Sudoku 9981 Expert puzzle, Book 47 #7. I used a 6 2-string kite and a 1 xy-wing to get to this point. An xy-wing has a pivot at r3c1 and a 6 z-conjugate at r6c1 and r2c2. The 6 conjugate pair in column 5 and the z-conjugate form the z-wing. The z=wing cell eliminations are marked with a -. These are a pair of 9's in column 1 and a 4 in box 1. This cracks the puzzle.

Z-Wing Example 3
Code: Select all
` |-----------------+-----------------+-----------------| |   2    8    3   |   1    9    7   |   6    5    4   | |   7   46    1   |   5   46    8   |   2    9    3   | |  49 -469    5   |   3    2   46   |   8    1    7   | |-----------------+-----------------+-----------------| |   8    2   69   |  49    7   16   |  14    3    5   | |   3    1    4   |   2    8    5   |   9    7    6   | |  69    5    7   |  49   16    3   |  14    8    2   | |-----------------+-----------------+-----------------| | 146-9 469  69   |   7    5   19   |   3    2    8   | |   5    3    2   |   8   14  149   |   7    6   19   | | 1-9    7    8   |   6    3    2   |   5    4   19   | |-----------------+-----------------+-----------------| `

The next example is a mutant Z-wing since 2 of the cells in the Z-loop are in the same box. The z cell eliminations denoted by * occur in row 5 and box 8. The x cell eliminations denoted by # are in column 1 and the y cell eliminations denoted by + are in row 9.

Mutant Z-Wing Example
Code: Select all
` |-----------+-------------+-------------| |  #  .  .  |  .   .   -  |  .   .   .  | |  #  .  .  |  .   .   -  |  .   .   .  | |  #  .  .  |  .   .   -  |  .   .   .  | |-----------+-------------+-------------| |  #  .  .  |  .   .   -  |  .   .   .  | | xz  *  *  |  *   *   z  |  *   *   *  | |  #  .  .  |  .   .   -  |  .   .   .  | |-----------+-------------+-------------| |  #  .  .  |  *   *   z  |  .   .   .  | |  #  .  .  |  *   *   -  |  .   .   .  | | xy  +  +  |  *  yz   -  |  +   +   +  | |-----------+-------------+-------------| `

The next example is a Z color wing. The logic for this and the cell eliminations are the same as a clor wing.

Z-Color Wing Example 1
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` |-----------+-------------+-------------|  |  z  -  -  |  -   -   z  |  -   -   -  |  |  .  .  .  |  .   *   .  |  .   .   .  |  |  .  .  .  |  .   *   .  |  .   .   .  |  |-----------+-------------+-------------| |  .  .  .  |  .   .   .  |  .   .   .  |  | xz  .  .  |  .   .   .  |  .   .   .  |  |  .  .  .  |  .   .   .  |  .   .   .  |  |-----------+-------------+-------------|  |  .  .  .  |  .   .   *  |  .   .   .  |  |  .  .  .  |  .   .   *  |  .   .   .  |  | xy  .  .  |  .  yz   *  |  .   .   .  |  |-----------+-------------+-------------| `

The next example is a partially worked Sudoku9981 expert puzzle, book 28 #2. I used a 4 2-string kite and a 9 x-wing to get to this point. The xy-wing 3 conjugate pair is at r7c3 amd r9c8 with the pivot at r9c1. The other conjugate pair in the z-color wing is at r6c37. The cell eliminations for 3 are at r4c8 and r9c7. This cracked the puzzle. Note that there were no cell eliminations for the xy-wing by itself.

Z-Color Wing Example 2
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` |------------+-------------+-------------| |  6   5 49  |  3  49   2  |  8   7   1  |  | 479 47  8  |  1   6  49  |  2   5   3  |  |  3   1  2  |  7   5   8  | 49  49   6  |  |------------+-------------+-------------| |  1 2347  5 |  8  379 79  |  6  349 247 |  |  8  347 49 |  2  379  6  |  5   1   47 | | 279  6  37 |  4   1   5  | 379  8   27 | |------------+-------------+-------------|  | 247 2347 37 |  9   8   47 |  1   6   5 | |  5   9   1  |  6   47  3  | 47   2  8  | | 47   8   6  |  5   2   1  | 347  34  9 | |------------+-------------+-------------| `

The next example is a z-2-string kite. The 2 strings are the conjugate pair is in column 6 and the xy-wing z conjugate is in b4 and b8.

Z-2-String Kite Example 1
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` |-----------+-------------+-------------|  |  .  .  .  |  .   .   -  |  .   .   .  |  |  .  .  .  |  .   .   -  |  .   .   .  |  |  .  .  .  |  .   .   -  |  .   .   .  |  |-----------+-------------+-------------| |  *  *  *  |  .   .   z  |  .   .   .  | | xz  .  .  |  *   *   *  |  .   .   .  |  |  .  .  .  |  .   .   -  |  .   .   .  |  |-----------+-------------+-------------|  |  .  .  .  |  .   .   z  |  .   .   .  |  |  .  .  .  |  .   .   -  |  .   .   .  |  | xy  .  .  |  .  yz   -  |  .   .   .  |  |-----------+-------------+-------------| `

The next example is a partially worked Sudoku9981 expert puzzle, book 17 #1. I used an xy-wing with pivot at r2c5 and 3 z conjugate at r2c3 and r6c5 to get to this point. There is another xy-wing with pivot at r5c8 and 3 z conjugate pairs at r5c4 and r1c8. These two conjugate pairs both have a cell in box 5 and together they form a z-2-string kite for z. The 3 candidates in r1b1 and r2b3 can be removed and the puzzle is cracked.

2-String-Kite Example 2
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` |--------------+-------------+--------------| |  2  56   356 |  1  49  57  |  4  38  78   |  |  8   9   35  |  4  57   6  | 23  123 127  |  |  4   7    1  | 38  38   2  |  6   9   5   |  |--------------+-------------+--------------| | 16   2   68  |  5   4   9  | 18   7   3   |  |  7  38    9  | 23   6   1  |  5  28   4   | |  5  13    4  | 237 37   8  | 12   6   9   | |--------------+-------------+--------------|  |  3 14568  2  |  9  58  45  |  7  158 1686 | |  9 4568 5678 | 678  1  457 | 238 2358 268 | | 16 1568 5678 | 678  2   3  |  9   4  168  | |--------------+-------------+--------------| `

The next example can be either a Z color trap or an AIC trap. If z is a candidate in any of the cells marked ?, then it is the latter. Otherwise it is the former. The rule for using a Z-conjugate in an AIC are this. The z-conjugate can replace any of the strong links in an AIC, but it cannot replace any of the weak links. If a color chain is considered to be a special case of an AIC in which all of the weak links are replaced by strong links, then the rule is this. The Z-conjugate can replace any of odd number link (starting at the end) in the color chain, but it cannot replace an even number link..

Z Color Trap/AIC Trap Example
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` |-----------+-------------+-------------|  |  .  .  .  |  .   *   .  |  z   .   .  |  |  .  .  .  |  .   .   .  |  -   .   .  |  |  .  .  .  |  .   .   .  |  -   .   .  |  |-----------+-------------+-------------| |  .  .  .  |  .   .   .  |  -   .   .  |  | xz  ?  ?  |  ?   -   ?  |  z   ?   ?  |  |  .  .  .  |  .   .   .  |  -   .   .  |  |-----------+-------------+-------------|  |  .  .  .  |  .   .   .  |  -   .   .  |  |  .  .  .  |  .   .   .  |  -   .   .  |  | xy  .  .  |  .  yz   .  |  -   .   .  |  |-----------+-------------+-------------| `

The next example is a partially worked Sudoku9981 expert puzzle, book 14 #7. The xy- wing has a pivot at r7c7 and a 1 z-conjugate pair at r3c7 and r7c1. This eliminates 1 from r3c1. The 1 ER occurs in r23c23. Using the 1 z-conjugate pair as part of the z-ER gives a 1 cell elimination in r1c8. This sets up a 1 z-color trap. The colors are denoted by ' and ^. all of the 1's in c6b58 can be eliminated. This cracks the puzzle.

Z-ER & Z-Color Trap Example 2
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` |------------------+--------------------+---------------------| |   4   ^15     9  |   6     3     '17  |    2    -157     8  | | '125    8     7  |   12   149   1249  |  156    1456     3  | | -12     3     6  |   5     8   ^1247  |  '17      9     47  | |------------------+--------------------+---------------------| | 567     9     1  |  378   4567  34578 |   78      2     47  | |  57    45     8  |  127  14579 124579 |    3    147      6  | |   3    46     2  |  178   1467   1478 |    9   1478      5  | |------------------+--------------------+---------------------| | ^16     2     4  |  1378  157   13578 |   67    367      9  | |   8    16     5  |    9    17    137  |    4    367      2  | |   9     7     3  |    4    2      6   |   58     58      1  | |------------------+--------------------+---------------------| `
Last edited by Bud on Mon Nov 10, 2008 1:41 am, edited 18 times in total.
Bud

Posts: 56
Joined: 24 August 2008

### Re: The Hidden Power of the XY-Wing. The Z-Wing,etc

Bud wrote:For a 2 box XY-wing, the pattern is both a Z-wing and a Z color wing as shown in the example below. I call this a Z blockbuster wing. The vertical cell eliminations are due to the Z-wing and the horizontal cell eliminations are due to the Z color wing.

Z-Blockbuster Wing Example
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` |-----------+-------------+-------------|  |  z  -  -  |  -   z   -  |  -   -   -  |  |  *  .  .  |  .   *   .  |  .   .   .  |  |  *  .  .  |  .   *   .  |  .   .   .  |  |-----------+-------------+-------------| |  *  .  .  |  .   *   .  |  .   .   .  |       |  *  .  .  |  .   *   .  |  .   .   .  |  |  *  .  .  |  .   *   .  |  .   .   .  |  |-----------+-------------+-------------|  | xz  .  .  |  *   *   *  |  .   .   .  |  |  *  .  .  |  .   *   .  |  .   .   .  |  | xy  *  *  |  .  yz   .  |  .   .   .  |  |-----------+-------------+-------------| `

You've illustrated a continuous loop, for which there are 20 more potential eliminations -- 7 for 'y' and 13 for 'x'. Similarly, your "z-wing" has 14 more potential eliminations.

Continuous loops occur infrequently -- relative to discontinuous loops, that is -- so an actual example would be nice.
ronk
2012 Supporter

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Joined: 02 November 2005
Location: Southeastern USA

### Hidden Power of the XY-Wing. The Z-Wing & Z-Turbot Fish

I've done a lot of editing since I first made this post. At Ron's suggestion I have added actual puzzle examples, one for Z-wing, one for the Z-color-wing, one for the Z-2-string kite, and one for the Z-ER. I've also added a diagram of a mutant Z-wing, and added the x and y cell eliminations for the Z-wing diagrams as Ron pointed out. I also added a diagram of a Z-2-string kite and rules for chains.
Last edited by Bud on Tue Oct 28, 2008 4:40 am, edited 1 time in total.
Bud

Posts: 56
Joined: 24 August 2008

Well, maybe you should have taken a little longer editing. The PM for your Z-Wing Example 2 is a nightmare. There is a zero (0) among the candidates in [r3c1]. Removing it and clearing out all of the special characters results in a puzzle with 949 solutions according to Simple Sudoku.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### Remote XY-wing ?!

Hello Bud,

You seem to study only classical zx-xy-yz wing with an external strong link z'=z", which sometime helps.

I often look for a XY-wing in the grid and I find only two bivalue cells in the same house, say zx-xy, but the third cell yz does not share a house with either zx or xy!
Here is an example (Luke451-62920):
Code: Select all
` +--------------------------------------------------------------------+  | 3      14     7      | 1458   1248   9      | 2458   2458   6      |  | 5      1469   1469   | 1468   12348  7      | 2489   23489  3489   |  | 8      2      469    | 456    34     36     | 1      7      3459   |  |----------------------+----------------------+----------------------|  | ≠29    5      23489  | 7      6     *28     | 2489   23489  1      |  | 1269   1689   12689  | 3      5      4      | 7      289    89     |  | 7      348    2348   |*18     9      1'28   | 2458   6      3458   |  |----------------------+----------------------+----------------------|  | 69     378    5      | 468    3478   368    | 4689   1      2      |  |*12     1378   1238   | 9      13478  1"3568 | 4568   458    458    |  | 4      1689   1689   | 2      18     56     | 3      59     7      |  +--------------------------------------------------------------------+  (28)r4c6 = zx (18)r6c4 = yx (12)r8c1 = yz target cell 2r4c1`

Problem is, 1r6c4 and 1r8c1 does not "see" each other..
To make it work, we should couple the two cell Y-wing style! And indeed we find in c6 a strong link (1)r68c6

Since XY-Wing is a subset of AIC we may write the whole chain:
(2=8)r4c6 - (8=1)r6c4 - (1'=1")r68c6 - (1=2)r8c1 => r4c1≠2
DanG

Posts: 20
Joined: 28 March 2007

### Re: Remote XY-wing ?!

DanG wrote:Problem is, 1r6c4 and 1r8c1 does not "see" each other..
To make it work, we should couple the two cell Y-wing style! And indeed we find in c6 a strong link (1)r68c6

Since XY-Wing is a subset of AIC we may write the whole chain:
(2=8)r4c6 - (8=1)r6c4 - (1'=1")r68c6 - (1=2)r8c1 => r4c1≠2

Whether XY-Wing, XYZ-Wing, ALS Chains, finned X-Wings etc., the way I see it, if the same digit value in 2 different cells are connected by a conjugate link, they 'see' each other just the same as if they were in the same house. By always keeping that in mind, I am more likely to see situations where a pattern (no matter what it is) can cause an elimination where otherwise I might have missed it. It also means that sometimes life is simplified- patterns that have different names turn out to be essentially the same pattern except that one of them uses a conjugate link (eg. Kraken X-Wing vs. Finned X-Wing).
DonM
2013 Supporter

Posts: 475
Joined: 13 January 2008

A new approach needs some serious backing. Your approach is only serious in what it's lacking.

Z-Wing Ex 3: Solves w/ a simple XY-Wing. No need for a new approach.

Z-Color Wing Ex 2: Solves w/ Singles. No need for a new approach.

2-String Kite Ex 2: Invalid grid w/ [b9]~3.

After three PM examples, your approach is fruitless.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### Hidden Power of the XY-Wing. The Z-Wing & Z-Turbot Fish

Hi Daj,
I think I fixed the grid errors on the Z-2-string kite and Z-color-wing examples. I apologize for the careless errors. Since I added the Z-color wing example, I have found several more examples. I agree that having a simpler technique makes an example a bad one. I'm think I can find other examples from the others if there is still a problem. As far the Z-wing, this must be a rara avis as Ron indicated because I've found very few examples of this. But I'll keep looking.
Hi DanG. I have seen something similar to your example except that it was a simple xy-change rather than a W-wing, It was called an xy-wing chain. Is there another name for this type of xy-wing?
Bud

Posts: 56
Joined: 24 August 2008

### Re: Hidden Power of the XY-Wing. The Z-Wing & Z-Turbot

Bud wrote:The next example is a partially worked Sudoku9981 expert puzzle, book 14 #7. The xy- wing has a pivot at r7c7 and a 1 z-conjugate pair at r3c7 and r7c1. This eliminates 1 from r3c1. The 1 ER occurs in r23c23. Using the 1 z-conjugate pair as part of the z-ER gives a 1 cell elimination in r1c8. This sets up a 1 z-color trap. The colors are denoted by ' and ^. all of the 1's in c6b58 can be eliminated. This cracks the puzzle.

Z-ER & Z-Color Trap Example 2
Code: Select all
` |------------------+--------------------+---------------------| |   4   ^15     9  |   6     3     '17  |    2    -157     8  | | '125    8     7  |   12   149   1249  |  156    1456     3  | | -12     3     6  |   5     8   ^1247  |  '17      9     47  | |------------------+--------------------+---------------------| | 567     9     1  |  378   4567  34578 |   78      2     47  | |  57    45     8  |  127  14579 124579 |    3    145      6  | |   3    46     2  |  178   1467   1478 |    9   1478      5  | |------------------+--------------------+---------------------| | ^16     2     4  |  1378  157   13578 |   67    367      9  | |   8    16     5  |    9    17    137  |    4    367      2  | |   9     7     3  |    4    2      6   |   58     58      1  | |------------------+--------------------+---------------------| `

Your eliminations in c6 are invalid. The xy-wing permits both r3c7=1 and r7c2=1 to be true. This would make both r1c6<>1 and r3c6<>1. Hence, no eliminations of digit 1 in c6.
ronk
2012 Supporter

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