W-Wing
A W-Wing is formed by a chain of four cells:
- A: A cell with two candidates W and X.
- B: A cell with X as a candidate.
- C: A cell with X as a candidate.
- D: A cell with two candidates W and X.
It is required that A sees B, C sees D, and B and C has a strong link on the digit X. Then W can be eliminated from any cell that is seen by both A and D.
Suppose T is a cell that is seen by both A and D, and T contains W as a candidate. Then, the W-Wing can be written as follows in Nice Loop notation:
[T]-W-[A]-X-[B]=X=[C]-X-[D]-W-[T] => T<>W
Example
In this grid, note the r5c5 and r6c3 cells with the candidates 2 and 7. Also, the green 7 candidates form a chain, with r5c8 and r6c8 forming a strong link (because r5c8 and r6c8 are the only cells with candidate 7 in this box). Hence, we can eliminate the digit 2 from all cells that are seen by both r5c5 and r6c3, i. e., those marked in red.
Doubly Linked W-Wing Example
In a doubly-linked W-Wing the identical pairs are linked to 2 digit conjugate pairs rather than a single digit conjugate pair as in the previous example. This is illustrated in the following example where the 17 cells r8c4 and r9c8 are linked to the 7 conjugate pair and the 1 grouped conjugate in box 7. This means that both 1 and 7 must occur either in r8c4 or r9c8. Each of these are conjugate with the 17 conjugates in box 7. Therefore 1 and 7 can be eliminated from other cells in rows 8 or 9. Candidate eliminations are indicated with a -.
Doubly Linked W-Wing Example
|-----------------+-----------------+-----------------| | 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 -16-79 2 | 3 4 5 | | 79 3 169 | 4 8 5 | 2 17 -169 | |-----------------+-----------------+-----------------|
