Fish & Wings
W-Wing
A W-Wing removes one digit from the cells that see two matching cells. It needs two pincers — unsolved cells holding exactly the same two candidates {X,Y} that do not see each other — and a strong link on X: a row, column or box where X has only two places, one seeing each pincer. The link stops both pincers from being X, so at least one of them is Y, and Y can be removed from every cell that sees both pincers, while X stays where it is.
What you need to see on the grid
Look for a W-Wing once singles, subsets and fish have stopped producing moves and the grid still holds a few cells with only two candidates. The trigger is a repeat: two such cells, far apart, with exactly the same pair. Four terms carry the whole method:
- Pincers
- Two unsolved cells with exactly the same two candidates: {X,Y} and {X,Y}, not {X,Y} and {X,Z}. They must not see each other. Two cells with the same pair in one row, column or box are a Naked Pair, a different and simpler move.
- Strong link
- A row, column or box where a digit has exactly two places left. One of the two must hold the digit, so if one is not X, the other is. Count before you trust it: three places is not a strong link.
- Link ends
- The two places of X in the link's unit. Each must see a different pincer, and neither may be a pincer itself. If both ends see the same pincer, or one sees neither, the other pincer is free to be X and nothing is proved.
- What gets removed
- Only Y, the pincers' digit that is not on the link, and only from cells that see both pincers. A W-Wing never removes X, and the pincers keep both their candidates.
How to apply it, step by step
One example runs through all five steps: the pair {4,7}, part-way through a hard puzzle. Every diagram shows the whole grid. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them. A digit missing from a drawn cell was ruled out earlier in the solve, by a digit in its column or box (drawn or not) or by an earlier deduction.
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Step 1Find two cells with the same pair that do not see each other
Scan for cells with only two candidates and compare their pairs. Rows 1 and 6 hold six of them: {4,7} at R1C2, {4,9} at R1C6, {8,9} at R1C9, {2,7} at R6C2, {4,5} at R6C6 and {4,7} at R6C8. Only one pair repeats: R1C2 and R6C8 both read exactly {4,7}. {2,7} shares a digit with them, but a W-Wing needs both digits to match.
Now check that the two cells do not see each other. R1C2 is in row 1, column 2 and box 1; R6C8 is in row 6, column 8 and box 6, so they have no unit in common. These are the pincers. Had they shared a unit, they would be a Naked Pair, and you would play that instead.
Rows 1 and 6 in full. Six cells have two candidates; only R1C2 and R6C8 share the same pair, {4,7}, and they have no row, column or box in common. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 5 2 6 1 R2 R3 R4 R5 R6 3 9 1 6 R7 R8 R9 - Digit already on the grid
- Rows being read
- A pincer: exactly {4,7}
- Candidates 4 and 7, the digits studied
- Not needed for this step
Filled in: rows 1 and 6. The other 63 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Find a strong link on one of the two digits
Take each digit of the pair in turn and look for a unit where it has exactly two places, one seeing each pincer. Row 9 works for 4: it fits only R9C2 and R9C8. The 4 already placed at R7C4 shares box 8 with R9C4 and R9C5, which is why they cannot hold it. R9C2 shares column 2 with the pincer R1C2, and R9C8 shares column 8 with the pincer R6C8: one end for each pincer.
The same row is no link for 7, which has three places there: R9C4, R9C5 and R9C8. Row 6 also leaves 4 just two places, R6C6 and R6C8, but R6C8 is a pincer itself, and a link end has to be another cell. The link can be in a row, a column or a box.
Row 9 in full, the two pincers and the 4 at R7C4. In row 9, 4 fits only R9C2 and R9C8, in the columns of R1C2 and R6C8; 7 fits three cells. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 4 R8 R9 6 1 8 5 3 - Digit already on the grid
- Row 9: two places for 4
- A link end: a place for 4
- Candidate 4, the digit studied
- Not needed for this step
Filled in: row 9; cells R1C2, R6C8 and R7C4. The other 69 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Follow the link: one pincer must be 7
Suppose R1C2 is not 7. Then it is 4, so R9C2, in the same column, is not 4. Row 9 still needs its 4, and the only place left is R9C8, which shares column 8 with R6C8. So R6C8 is not 4: it is 7. Either R1C2 is 7, or R6C8 is.
The argument runs just as well starting from R6C8, and you never need to know which pincer holds the 7. Both may even be 7. What the link rules out is both being 4, because row 9 would then have no place left for its 4.
The four cells of the W-Wing. Column 2 joins R1C2 to R9C2, column 8 joins R9C8 to R6C8, and row 9 must put its 4 at one end — which keeps 4 out of one pincer. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 4 R8 R9 6 1 8 5 3 - Digit already on the grid
- Row 9: the strong link on 4
- Column joining a pincer to a link end
- The W-Wing: two pincers and two link ends
- Candidate 4, the digit studied
- Not needed for this step
Filled in: row 9; cells R1C2, R6C8 and R7C4. The other 69 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 4Remove 7 from every cell that sees both pincers
One of the pincers is 7, so a cell that sees both of them cannot be 7: whichever pincer holds the 7 shares a unit with it. Two cells see both here: R1C8 (row 1 with R1C2, column 8 with R6C8) and R6C2 (row 6 with R6C8, column 2 with R1C2). Both lose 7.
The digit that goes is 7, the one not on the link, never 4. R1C8 keeps its 4: the pattern only proves that the pincers are not both 4, which says nothing against a 4 in R1C8. Leave alone the cells that see just one pincer, such as R1C5 or R6C9, too: the 7 may be in the other pincer.
Technique: W-Wing
Rows 1, 6 and 9 in full. R1C8 and R6C2 are the only cells that see both pincers: they lose 7 and keep everything else. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 5 2 6 1 R2 R3 R4 R5 R6 3 9 1 6 R7 4 R8 R9 6 1 8 5 3 - Digit already on the grid
- Row 9: the strong link on 4
- The W-Wing: two pincers and two link ends
- Candidates 4 and 7, the digits studied
- Candidate removed
- Not needed for this step
Filled in: rows 1, 6 and 9; cell R7C4. The other 53 cells are left blank: they are not needed for this step. Removed: 7 from R1C8 and R6C2.
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Step 5Collect what the elimination unlocks
A W-Wing never places a digit by itself, so look at the cells it just cleared. R6C2 had two candidates, 2 and 7. With 7 gone, only 2 is left: a Naked Single, R6C2 = 2.
R1C8 is down to {3,4,9} and gives nothing yet. Check every row, column and box that lost a candidate before you move on — that is where a W-Wing pays you back.
Technique: Naked Single
Row 6 after the W-Wing. R6C2 has one candidate left: 2. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 3 2 9 1 6 R7 R8 R9 - Digit already on the grid
- The cell that must be 2
- Candidate 2, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 6. The other 72 cells are left blank: they are not needed for this step. Placed: R6C2 = 2.
Common mistakes
Counting three places as a strong link
A strong link needs exactly two places. In the example, 7 has three places in row 9, so the row links nothing on 7. Had 4 kept a third place there too, both pincers could be 4 while row 9 put its 4 in that third cell, and nothing would be proved. Recount the places in the unit right before you use it.
Removing the link digit instead of the other one
The link is about 4, so erasing 4 feels natural. But the pattern proves only that the pincers are not both 4: it guarantees a 7, not a 4. In the example R1C8 can still be 4, and erasing it would be a real error. The digit that leaves is always the one that is not on the link.
Clearing a cell that sees only one pincer
R1C5 shares row 1 with R1C2 and lists 7, but it does not see R6C8. If R6C8 is the 7 and R1C2 is 4, R1C5 is still free to be 7. Only a cell that sees both pincers is covered whichever one turns out to hold the 7.
Recent daily puzzles that needed this technique
Keep going
- XY-WingThree two-candidate cells force a digit into one of two pincers — cells seeing both lose it.
- Simple ColoringChain one digit's strong links and colour them in turn — one colour is true, so a cell seeing both loses the digit.
Learn the move, not just the digit
In a real game, when a W-Wing is what opens the next cell, the hint in the app names it and shows the cell it unlocks — the same move as step 5 above.