The New Sudoku Players' Forum

[Bud]

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  |
 |-----------------+-----------------+-----------------|

[re'born]

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.

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.

[Bud]

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.

[re'born]

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.

[Bud]

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  |
 |-----------------+-----------------+-----------------|

[re'born]

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)?

[Bud]

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