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From Sudopedia

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 this 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 #.

Z-Wing Example 1[code]

|-----------+-------------+-------------|
|  z  -  -  |  -   z   -  |  -   -   -  |
| #*  .  .  |  .   *   .  |  .   .   .  |
| #*  .  .  |  .   *   .  |  .   .   .  |
|-----------+-------------+-------------|
| #*  .  .  |  .   *   .  |  .   .   .  |
| xz  .  .  |  .   *   .  |  .   .   .  |
| #*  .  .  |  .   *   .  |  .   .   .  |
|-----------+-------------+-------------|
| #*  .  .  |  .   *   .  |  .   .   .  |
| #*  .  .  |  .   *   .  |  .   .   .  |
| xy  +  +  |  +  yz   +  |  +   +   +  |
|-----------+-------------+-------------| [/code]

The next example is an original puzzle. In this puzzle the xy-wing has a pivot at r7c6 and a 5 z conjugate at r3c7 and r6c4. The other 5 z conjugate in r1c36 completes the z-wing. The 5 candidates eliminated by the z-wing are in columns 3 and 6 are denoted by *. The 9 candidates that are eliminated by the z-wing in column 6 are denoted by #. Finally the 3 candidates that are eliminated by the z-wing in row 6 are denoted by +.

Z-Wing Example 2[code]

|-------------------+-------------------+----------------------|
|    3  124   2459  |   6      7   45#  | 129       8     1249 |
| 25789 248 24*6789 | 23459   349   1   | 23679  234679   2469 |
|  1270 124  24679  |  2349   349   8   |  5    1234679  12469 |
|-------------------+-------------------+----------------------|
|    6    7     29  |  159     8    59  |   4     1259     3   |
| 1289  12348 23489 | 134579 3469 34*7# | 126789 125679 123689 |
|  189    5    3489 |  13479 3469    2  |  16789   1679   1689 |
|-------------------+-------------------+----------------------|
|    4    6     35  |    8    2     39  | 1+9     1+59      7  |
|  278   238     1  |  3479   5   3467# | 23689   23469  24689 |
|  2578   9   23*78 |  3479   1   3467  | 2368    2345   24568 |
|-------------------+-------------------+----------------------| [/code]

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 3[code]

|-------------+-------------+-------------|
|  z   -   -  |  -   z   -  |  -   -   -  |
|  *   .   .  |  .   *   .  |  .   .   .  |
|  *   .   .  |  .   *   .  |  .   .   .  |
|-------------+-------------+-------------|
|  *   .   .  |  .   *   .  |  .   .   .  |
|  *   .   .  |  .   *   .  |  .   .   .  |
|  *   .   .  |  .   *   .  |  .   .   .  |
|-------------+-------------+-------------|
| xz   #   #  |  *   *   *  |  .   .   .  |
| #*   #   #  |  .   *   .  |  .   .   .  |
| +#* xy  +#* |  +  yz   +  |  +   +   +  |
|-------------+-------------+-------------| [/code]

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 4[code]

|-----------------+-----------------+-----------------|
|   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   |
|-----------------+-----------------+-----------------| [/code]

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]

|-----------+-------------+-------------|
|  #  .  .  |  .   .   -  |  .   .   .  |
|  #  .  .  |  .   .   -  |  .   .   .  |
|  #  .  .  |  .   .   -  |  .   .   .  |
|-----------+-------------+-------------|
|  #  .  .  |  .   .   -  |  .   .   .  |
| xz  *  *  |  *   *   z  |  *   *   *  |
|  #  .  .  |  .   .   -  |  .   .   .  |
|-----------+-------------+-------------|
|  #  .  .  |  *   *   z  |  .   .   .  |
|  #  .  .  |  *   *   -  |  .   .   .  |
| xy  +  +  |  *  yz   -  |  +   +   +  |
|-----------+-------------+-------------| [/code]

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

Z-Color Wing Example [code]

|-----------+-------------+-------------| 
|  z  -  -  |  -   -   z  |  -   -   -  | 
|  .  .  .  |  .   *   .  |  .   .   .  | 
|  .  .  .  |  .   *   .  |  .   .   .  | 
|-----------+-------------+-------------|
|  .  .  .  |  .   .   .  |  .   .   .  | 
| xz  .  .  |  .   .   .  |  .   .   .  | 
|  .  .  .  |  .   .   .  |  .   .   .  | 
|-----------+-------------+-------------| 
|  .  .  .  |  .   .   *  |  .   .   .  | 
|  .  .  .  |  .   .   *  |  .   .   .  | 
| xy  .  .  |  .  yz   *  |  .   .   .  | 
|-----------+-------------+-------------| [/code]

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[code]

|-------------+-------------+-------------|
|  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  |
|-------------+-------------+-------------| [/code]

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

|-----------+-------------+-------------| 
|  .  .  .  |  .   .   -  |  .   .   .  | 
|  .  .  .  |  .   .   -  |  .   .   .  | 
|  .  .  .  |  .   .   -  |  .   .   .  | 
|-----------+-------------+-------------|
|  *  *  *  |  .   .   z  |  .   .   .  |
| xz  .  .  |  *   *   *  |  .   .   .  | 
|  .  .  .  |  .   .   -  |  .   .   .  | 
|-----------+-------------+-------------| 
|  .  .  .  |  .   .   z  |  .   .   .  | 
|  .  .  .  |  .   .   -  |  .   .   .  | 
| xy  .  .  |  .  yz   -  |  .   .   .  | 
|-----------+-------------+-------------| [/code]

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

|--------------+-------------+--------------|
|  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 | 47   2   8   |
| 16 1568 5678 | 678  2   3  |  9   4  168  |
|--------------+-------------+--------------| [/code]

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 [code]

|-----------+-------------+-------------|
|  .  .  .  |  .   *   .  |  z   .   .  |
|  .  .  .  |  .   .   .  |  -   .   .  |
|  .  .  .  |  .   .   .  |  -   .   .  |
|-----------+-------------+-------------|
|  .  .  .  |  .   .   .  |  -   .   .  |
| xz  ?  ?  |  ?   -   ?  |  z   ?   ?  |
|  .  .  .  |  .   .   .  |  -   .   .  |
|-----------+-------------+-------------|
|  .  .  .  |  .   .   .  |  -   .   .  |
|  .  .  .  |  .   .   .  |  -   .   .  |
| xy  .  .  |  .  yz   .  |  -   .   .  |
|-----------+-------------+-------------| [/code]

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]

|------------------+--------------------+---------------------|
|   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  |
|------------------+--------------------+---------------------| [/code]

Crackin' Kracken X-Wing Example

Here is the only puzzle crackin' Kraken X-wing example I have found. This is a patially worked Sudoku9981 Expert puzzle Book 26 #5. I used a 1 Swordfish and 5 X-wing to get to this point.The digit is 8 and the cells in the pattern are marked with a ' and the elimination cells are marked with a *. I don't think there is a simpler pattern technique that can be used to solve this puzzle, but then I have been wrong before. XY- [code]

|-----------------+-----------------+-----------------|
| 13468 368   9   | 2468 2478  347  |  23    5  12378 |
|   7    5   1368 |  268  28    9   | 123 '1238   4   |
| '348   2   '38  |   1    5   347  |   6    9  '378  |
|-----------------+-----------------+-----------------|
|   9    4   356  |   7    1   25   |   8   236  235  |
| '68    1  '568  |   9    3   245  |   7   246   25  |
|   2   378  3578 |  458  48    6   |  135  134   9   |
|-----------------+-----------------+-----------------|
| 1368  368   4   |  25   24    1   | 2359   7   1238 |
|   5   37   1237 |  29  2479   8   | 1239  123   6   |
| *18    9  *1278 |   3    6   57   |   4   '28  1258 |
|-----------------+-----------------+-----------------| [/code]

Double XY-Wing Triangle

This is the second time I have run into this solving pattern so I thought I would pass it on. In the example below the first XY-wing has a pivot at r7c9 and 7 pincers at r7c3 and r8c8. This eliminates 7 from r8c1 which becomes the pivot for the second XY-wing with 7 pincers at r7c3 and r8c7. The resultant triangle is indicated by '. Since r8c78 are peers, at least one of these must be not 7 and its Z-conjugate r7c3 must be 7.

[code]

|-----------------+-----------------+-----------------|
|  467   8   467  |   3    5   67   |   2    9    1   |
|   9    5    1   |  468  46    2   |   3   68    7   |
|   3   67    2   | 1678   9  1678  |   4   68    5   |
|-----------------+-----------------+-----------------|
|   8   36    5   |  16    2   136  |  79   47   49   |
|   1    2    9   |  478  47   478  |   6    5    3   |
|  467  367  467  |   5   36    9   |   1    2    8   |
|-----------------+-----------------+-----------------|
|   2    9  '67   |  467   1    5   |   8    3   46   |
| *567   1    3   |   9    8   467  | '57  '47    2   |
|  567   4    8   |   2   367  367  |  579   1   69   |
|-----------------+-----------------+-----------------| [/code]

When I first made my post on the Z-Turbot Fish, I didn't have any examples of the Z-color wing. Since then I have found about a dozen puzzle-cracking examples. One of these was an alternate solution to a BUG+2 puzzle which surprized me. In this example, there is a fin x-wing in rows 6 and 9 which would eliminate 2 from r5c5 and create a BUG+2, but the alternate solution does not use this. There is an xy-wing with a pivot at r6c2 and 8 Z-conjugates at r4c3 and r6c6. By itself it does not produce any eliminations but when it is combined with the 8 conjugate pair r8c34, it becomes a Z-color wing. Consider conjugate pair r48c3. If r4c3 is not 8, its Z-conjugate r6c6 is 8. If r8c3 is not 8, its conjugate r8c4 must be 8. Consequently either r6c6 or r8c4 must be 8 and 8 can be eliminated from their peer r6c4. This cracks the puzzle and the fin x-wing and BUG+2 are redundant. Of course, if you use the fin x-wing and the BUG+2 technique, the Z-color wing is redundant. I don't want to argue the relative merits of the two solutions, since this is a personal choice. The important point here is that in this case, you do have a choice. This does not imply that this will work on all BUG+2 puzzles

Z-color wing/BUG+2 Example[code]

|-----------------+-----------------+-----------------|
|   1    7    9   |   3    5    2   |   4    8    6   |
|   2    8    6   |   7    9    4   |   5    3    1   |
|   3    4    5   |   6   18   18   |   7    2    5   |
|-----------------+-----------------+-----------------|
|  58    6   28   |   4    7    3   |   9    1   25   |
|  49    3    7   |  29   126  15   |   8   46   25   |
|  49   25    1   |  289  268  58   |   3   46    7   |
|-----------------+-----------------+-----------------|
|   6    1    3   |   5    4    9   |   2    7    8   |
|   7    9   28   |  28    3    6   |   1    5    4   |
|  58   25    4   |   1   28    7   |   6    9    3   |
|-----------------+-----------------+-----------------| [/code]

Z-color wing/BUG+1 Example[code]

|-----------------+-----------------+-----------------|
|   4    5   39   |   3    5    2   |   4    8    1   |
|   6    1    7   |   7    9    4   |   5    3    7   |
|   3    8    2   |   6   18   18   |   7    2    5   |
|-----------------+-----------------+-----------------|
|   9   36    5   |   4    7    3   |   9    1   25   |
|   1    7   46   |  29   126  15   |   8   46   25   |
|   8    2   46   |  289  268  58   |   3   46    7   |
|-----------------+-----------------+-----------------|
|   5    4    8   |   5    4    9   |   2    7    8   |
|   2   36   36   |  28    3    6   |   1    5    4   |
|   7    9    1   |   1   28    7   |   6    9    3   |
|-----------------+-----------------+-----------------| [/code]

I think this is an interesting example first because it contains a Mutant Z-Wing (which I think is a rara avis} and second because the Mutant Z-Wing reduces a BUG+3 puzzle to a BUG+1 puzzle. The puzzle is Sudoku9981 Expert Book 42 #7. I used a 7 color wing to reach this point in the puzzle. The xy-wing of interest has a pivot at r5c5 and a 5 z-conjugate at r3c5 and r5c8. The Mutant z-wing occurs when this z-conjugate is combined with the 5 conjugate pair in row1. This forces the Z-conjugate to be a true conjugate and also forces the other candidates (78) in the xy-wing to actually be in the xy-wing. Consequently, 5 can be eliminated from r2c4 and 8 can be removed from r9c5. The BUG+1 technique then reveals that 7 must be in r2c6.

BUG3/Mutant Z-Wing Example[code]

|-----------------+-----------------+-----------------|
|   8    4    6   |  57    1    9   |   2   57    3   |
|   9    1   57   | -578   2    3   |  457  48    6   |
|   2    3   57   |   6   58    6   |   9    2    5   |
|-----------------+-----------------+-----------------|
|   6   58    2   |   3    4   78   |  57    9    1   |
|   3   58    4   |   9   78    1   |   6   57    2   |
|   1    7    9   |   2    6    5   |   8    3    4   |
|-----------------+-----------------+-----------------|
|   5    6    3   |   1    9    2   |  47   48   78   |
|   7    9    8   |   6    3    4   |   1    2    5   |
|   4    2    1   |  58   57-8 78   |   3    6    9   |
|-----------------+-----------------+-----------------| [/code]

Z-Wing Example[code]

|-----------------+-----------------+-----------------|
|   5    7  12-6  |   4    8   16   |   9   26    3   |
|  26    9    4   |   3    5   17   |   8   26   17   |
|   8    3   16   | -19    2 -1679  |  15    4   157  |
|-----------------+-----------------+-----------------|
|   3    2    9   |   5    6    4   |   7    1    8   |
|   1    6    5   |   8    7    3   |   4    9    2   |
|   4    8    7   |  12    9   12   |  56    3   56   |
|-----------------+-----------------+-----------------|
|  29   15    8   |   6    4   259  |   3    7   19   |
|  69    4    3   |   7    1    8   |   2    5   69   |
|   7   15   26   |  29    3   259  |  16    8    4   |
|-----------------+-----------------+-----------------| [/code]

Double W-Wing/Z-Color Wing Example[code]

|-----------------+-----------------+-----------------|
|   5   67    3   |   9    2   68   |   4  1-78  16   |
|   4   167  168  |   5   67    3   |  678   9    2   |
|  89    2   689  |  17    4   168  |  678   5    3   |
|-----------------+-----------------+-----------------|
|   3    4    7   |   2   19   19   |   5    6    8   |
|   6    9    2   |   8    5    7   |   1    3    4   |
|   1    8    5   |   6    3    4   |   9    2    7   |
|-----------------+-----------------+-----------------|
|   2    5    4   |   3  1679  169  | 6-78  178  169  |
|  789  16  1689  |  17 -16-79  2   |   3    4    5   |
|  79    3   169  |   4    8    5   |   2   17  -169  |
|-----------------+-----------------+-----------------| [/code]

vwxyz-Wing Example[code]

|-----------------+-----------------+-----------------|
|  389   7    4   |   1  2359  29   | 3589 2389   6   |
|   1    5   389  | 2389   7    6   |  389   4   239  |
|  389   2    6   |  389  359   4   |   1    7   359  |
|-----------------+-----------------+-----------------|
|  579   6   79   |  259   8   129  | 3579  129   4   |
| 5789   4   789  |   6   129   3   |  579  129  259  |
|   2    3    1   |  59    4    7   |   59   6    8   |
|-----------------+-----------------+-----------------|
|  34   18    5   |  349   6   19   |   2   389   7   |
| 347   18'  237  | 2-349 1239' 5   |   6   389' 39'  |
|   6    9   23   |   7   23*   8   |   4    5    1   |
|-----------------+-----------------+-----------------| [/code]