## Double W-Wing/Z-Color Wing Example

Advanced methods and approaches for solving Sudoku puzzles

### Double W-Wing/Z-Color Wing Example

I wanted to use this example to illustrate that a Z-color wing could use the W cells in a color wing instead of the Z cells in an XY-wing as part of the pattern. However I noticed that in this puzzle there is actually both a 7 and a 1 W-wing in rows 8 and 9. Because of this 1 and 7 cell eliminations can be made that can not be made for a single W-wing pattern. These are indicated by a - in rows 8 and 9. What if any is the esatablished name for this pattern? You can google a name and find a pattern but not vice versa. Anyway the 7 Z-color wing is the 7 conjugate pair r3c47 ane the 7 W-cells r8c4 and r9c8. The cell eliminations for this are r1c8 and r7c7. This means 7 is in r1c2. This is Sudoku9981 extreme puzzle, Book 20 #6. I used a 1 and a 7 ER to get to this point in the puzzle.

Double W-Wing/Z-Color Wing Example
Code: Select all
` |-----------------+-----------------+-----------------| |   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  | |-----------------+-----------------+-----------------| `
Last edited by Bud on Sun Nov 16, 2008 10:02 am, edited 1 time in total.
Bud

Posts: 56
Joined: 24 August 2008

Hi Bud,

In addition to r1c8<>7, r7c7<>7, r8c5<>1,7, r9c9<>1, one could use the same arguments to show r2c2<>7. It follows from one of your eliminations, but can be made independently from it, as well.

Bud wrote:What if any is the esatablished name for this pattern?

The stock answer is that this is W-wing transport. Also, as far as I can tell, the reasoning doesn't require there to be 2 W-wings. If you only had the 1 W-wing, you could still make the 1 exclusions (and similarly for the 7's).

Incidentally, there is a transported wxyz-wing in this puzzle.
Code: Select all
` *-----------------------------------------------------------* | 5     67    3     | 9     2     68    | 4     178%  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   | 678   178& -169   | | 789   16    1689  | 17    1679  2     | 3     4     5     | | 79*   3     169   | 4     8     5     | 2     17*%  169*  | *-----------------------------------------------------------*`

The wxyz-wing creates a strong inference between the 1's in r1c9, r9c89. Transporting the 1 in r1c9 to r79c8, we can eliminate 1 from r7c9.
re'born

Posts: 551
Joined: 31 May 2007

### Double W-Wing/Z-Color Wing Example

Hi Re'born.
Thanks for your reply. I agree with you that the Z-Color Wing uses the Transport principle, but I think that the Double W-Wing is a unique pattern of itself. The key to the logic in this pattern is that 1 and 7 can not be in the same row in box 7. These Box 7 cells are already part of the pattern. With the Transport principle we are adding a conjugate pair to an existing pattern. Here we are adding nothing.
Bud

Posts: 56
Joined: 24 August 2008

Then I've missed something. Would you please explain your logic in greater detail? Also, ask keith over on dailysudoku.com for a possible name for your pattern. He and others over there have looked at w-wings extensively.
re'born

Posts: 551
Joined: 31 May 2007

### Double XY-Wing Triangle

OK, here it is.
1 and 7 obviously can not be be in the same row of box 7. If 7 is in row 9 of box 7, then r9c8 is 1 and r9c9 cannot be 1. If 7 is not in row 9 box 7, then r9c3 must be 1 and r9c9 cannot be 1. If 7 is in row 8 of box 7, then r8c4 is 1 and neither 1 or 7 can be in r8c5. If 7 is not in row 8 of box 7, then 1 must be in row 8 box 7 and r8c4 must be 7 and 1 and 7 cannot be in r8c5. QED
In other words the digits in the double W-Wing pattern can only exist in the pattern itself in rows 8 and 9. Note that these eliminations cannot be made if the two W-wings are considered separately. They are not separate because they share W cells.

Double W-Wing/Z-Color Wing Example
Code: Select all
` |-----------------+-----------------+-----------------| |   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  | |-----------------+-----------------+-----------------| `
Bud

Posts: 56
Joined: 24 August 2008

I'm a bit out of touch on the issues, but here is how I see your double w-wing situation. One gets a continuous loop by attaching the two w-wings. This, incidentally, should hold regardless of how many rows things are in and regardless of the lengths of the w-wings (in my mind, w-wings may use more than 2 intermediate cells).

[Edit: Here are the w-wings and their composite continuous loop]
(1=7)r8c4 - r8c1 = r9c1 - (7=1)r9c8 and
(7=1)r8c4 - r8c23 = r9c3 - (1=7)r9c8
combine to give:
(1=7)r8c4 - r8c1 = r9c1 - (7=1)r9c8 - r9c3 = r8c23 - (1=7)r8c4, a continuous loop.

Incidentally, did you explain how you removed 7 from r1c8 (or is just the result of removing 7 from r7c7)?
Last edited by re'born on Sun Nov 16, 2008 2:10 pm, edited 2 times in total.
re'born

Posts: 551
Joined: 31 May 2007

### Double XY-Wing Triangle

For rhe 7 Z=color wing either r3c7 or r9c8 must be 7, and since r1c8 and r7c7 are peers of both of these cells they cannot contain 7. I think we're both just looking at the double W-wing from a different viewpoint with the same result. I don't have a problem with that
Bud

Posts: 56
Joined: 24 August 2008