Sudoku Written by Brian Hamilton and Karan Hamilton.
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Singles and two W-Wings, nothing else, solve puzzle 461 of our Evil sudoku bank. After 27 naked singles and 9 hidden singles, 19 squares are empty and none of them can be filled. Two W-Wings in a row break it open. The first takes a 7 out of R1C3, which leaves box 1 with only two places for a 7; the second uses exactly that to take a 9 out of R9C9, and singles do the rest.
How often will you need one? Never on our Easy to Expert levels. On Evil, all the time: 145 of the first 500 puzzles in the bank need a W-Wing, and 1,769 of all 5,000, slightly more than need an XY-Wing (1,743).
It is also easy to get subtly wrong, as our own helper did until 11 October 2026. The false version, and why it fails, is on the same puzzle below.
The Sudoku Trainer has ten W-Wing positions. Choose W-Wing, click the two matching cells and both ends of the strong link, and press Check. Hint names one of the four cells first, then the digit that comes out.
What is a W-Wing in sudoku?
A W-Wing is built from three things:
- Two cells with the same two candidates, call them x and y, that don’t see each other. If they did, they would be a naked pair.
- A strong link on x: a row, column or box where x has exactly two places left.
- One end of the link in sight of each cell. One place shares a row, column or box with the first cell, the other with the second, and neither place is one of the two cells.
Then the two cells can’t both be x, so one of them is y, and y can be removed from every cell that sees both.
The name has nothing to do with the shape, as an earlier version of this page said: SudokuWiki credits it to George Woods, who first noticed the pattern in 2007.
Two cells see each other when they share a row, a column or a 3×3 box, so they can’t hold the same digit. If a row, column or box has only two cells left for a digit, those two form a strong link, also called a conjugate pair: if one of them isn’t the digit, the other is. Both ideas only work on complete pencil marks. A forgotten candidate makes three places look like two.
How does a W-Wing work?
Take puzzle 461 after 37 moves. R2C9 and R9C1 both hold {7,9}, one near the top right of the grid and the other in its bottom-left corner, sharing no row, column or box. Box 1 can take its 7 only in R2C2 or R3C1. R2C2 is in row 2 with R2C9, and R3C1 is in column 1 with R9C1.
Suppose neither cell were 9. Then both would be 7. R2C9’s 7 would rule out R2C2, R9C1’s 7 would rule out R3C1, and box 1 would have nowhere left for a 7. That can’t happen, so R2C9 or R9C1 is 9, and no cell that sees both can be. Two cells do, the other corners of the rectangle the pair makes: R2C1, already filled, and R9C9, which loses its 9.
The links are drawn as in our XY-chain guide: a solid line where one end must be true, a dashed arrow where both can’t be.
The W-Wing only promises at least one 9. Here both cells turn out to be 9, as they did in 81 of the 190 W-Wings from our helper’s run through the first 500 Evil puzzles. Once R9C9 is down to 8, rows, columns and boxes run out of places for 9 one after another, and naked singles fill the last 19 cells.
To follow it in our Sudoku Helper, load this position. Its first Next Hint fills in pencil marks from the digits alone, which still have a 7 in R1C3, so its second names a different W-Wing, the one in the next section. Apply that hint, ask again, and the helper names this one.
Why can’t I see the W-Wing a solver found?
Usually because your marks still hold a candidate that an earlier move removed. On puzzle 461, box 1 has two places for a 7 only because the move before took the 7 out of R1C3, and that move was a W-Wing too.
Here R3C1 and R4C3 hold {3,7}. Column 2 has two 3s left, R2C2 in R3C1’s box and R4C2 in R4C3’s row, so the pair can’t both be 3 and one of them is 7. Because the two cells share a stack, more squares see both than in the first figure, three in each of their boxes; only R1C3 still had a 7 to lose. Row 7 would have done the same job, since its only 3s, R7C1 and R7C3, sit below one cell each. A W-Wing often has more than one link like that: 77 of the helper’s 190 did.
The dependency is the common case, not a curiosity. Of those 190 W-Wings, 115 would not have been on the board with pencil marks worked out from the digits alone: an earlier move had to clear a candidate first, from a wing cell or from the link’s house. If a W-Wing is supposed to be there and you can’t find it, look for marks that every move since you wrote them should have removed.
Can the strong link end on one of the two cells?
No, and it is the easiest W-Wing mistake to make, because the shape looks right. On the board of the last figure, R1C7 and R7C3 both hold {3,8}, and row 7 has only two places for an 8: R7C7, which sees R1C7 up column 7, and R7C3 itself. One end of the link sees one cell, and the other end is the other cell. Read carelessly, that is a W-Wing on 8, and it removes the 3 from R1C3, the only cell that sees both and still holds a 3.
Put it to the same test as before. Suppose neither cell is 3: both are 8, and nothing breaks, because R7C3 is itself row 7’s 8. The link says only that R7C3 or R7C7 is 8, and R7C3 can satisfy it on its own while R1C7 is 8 as well. In the solution that is exactly what happens, and R1C3 is the 3. Both ends of the link must be cells other than the two that hold the pair.
We know this mistake from the inside. Until 11 October 2026 the W-Wing in our Sudoku Helper accepted this shape, because its test for “sees” counted a cell as seeing itself. At this very position it played the false W-Wing and offered to remove the 3 from R1C3, the answer; pressing Apply Hint deleted it. The check now rejects a link with an end on either cell, and the trainer’s W-Wing lessons were rebuilt and every one checked against its answer.
We walked every puzzle in our Evil and Extreme banks with the helper, checking each step against the solution. Before the fix, a false W-Wing was the first wrong step on 1,924 of the 5,000 Evil puzzles and on 5 Extreme ones. After it, there is no wrong step on any of the 10,000. The trap is nearly always on the board: at 171 of the 190 positions where the helper now plays a real W-Wing on the first 500 Evil puzzles, there is also a false one that would remove something, and at 149 at least one would remove an answer.
Write the link digit into both cells and cross it out of every cell they see. If the link’s row, column or box is left with no place for it, the W-Wing is real. When the link ends on one of the two cells, that cell keeps the digit for its house, and the check fails, as it should.
How do you find a W-Wing?
Start from the cells, not the links. Matching pairs are rare enough to list, and each gives only two digits to try as the link.
- Update the marks, then sort the two-candidate cells by pair. On puzzle 461 before the first W-Wing, 14 of the 19 empty cells have two candidates: four hold {3,7}, and two each {3,8}, {3,9}, {7,9} and {8,9}.
- Keep the matching cells that don’t see each other. That leaves seven pairs on this board.
- Try each digit of the pair as the link. Look for a row, column or box with exactly two places for it, one in each cell’s sight and neither on the cells themselves. In 179 of the helper’s 190 W-Wings the link ran along a row or a column; only 11, like the first figure’s, sat in a box without sharing a line.
- Then look for the other digit where both cells see. If the two cells share a band or a stack, that is six squares, three in each of their boxes; if not, just the other two corners of their rectangle. The helper’s W-Wings split 92 to 98 between the two, and 128 of the 190 removed a single candidate.
On this board the seven pairs give one W-Wing that removes anything, R3C1 and R4C3, and three false shapes.
How often do you need a W-Wing?
Not on any of our classic levels. A puzzle from Easy, Medium, Hard or Expert is dealt only if singles, locked candidates and naked pairs can finish it, and across 800 of them, 200 per level, our helper never once had to reach for a W-Wing. One is sometimes there to be found, at 44 of the 277 Expert positions where the singles ran out, but something simpler always was too.
Evil is where it earns its keep. Solving with the helper’s simplest move each time, 145 of the first 500 puzzles in the bank needed a W-Wing at least once, and 1,769 of all 5,000, more than a third. Beside the XY-Wing, its nearest relative, it is needed about as often and is more often the step that finishes the job:
| Evil bank | XY-Wing | W-Wing |
|---|---|---|
| Puzzles that need it, first 500 | 157 | 145 |
| Puzzles that need it, all 5,000 | 1,743 | 1,769 |
| First step past singles, locked candidates and naked pairs | 670 | 583 |
| Hardest step, of the 3,655 puzzles the helper finishes | 543 | 689 |
| On the board where no single is left | 23.9% | 19.9% |
The helper’s eighteen techniques, simplest first, as its hints play them (XY-Wing before W-Wing). Hardest step: needed, and nothing further down the list. Last row: how often each was on the board across the 4,618 positions with no single in the first 500, which are harder than the rest because the bank is stored in score order. Measured October 2026.
Only the Almost Locked Set is the hardest step on more puzzles (1,668), and only it, a hidden pair and the XY-Wing are more often the first thing needed. With our five extra techniques added, among them finned X-Wings and finned Swordfish, which we try before the wings, 117 of the first 500 still need a W-Wing.
How is a W-Wing related to the XY-Wing and to chains?
The XY-Wing joins the same kind of outer cells through a third two-candidate cell, the pivot, where the W-Wing uses a strong link. Both clear the cells that see the two outer cells, so the six-or-two geometry above holds for both, and both hunts start from one list of two-candidate cells: look for one and you are halfway to the other.
Both are also short chains. Read from one end, the first figure says: if R2C9 isn’t 9 it is 7, so R2C2 isn’t 7, so box 1’s 7 is R3C1, so R9C1 isn’t 7, and it is 9. An XY-chain takes every step inside two-candidate cells; the W-Wing’s middle step is a strong link in a house, which makes it a short alternating inference chain, the kind an XY-chain search misses. X-cycles string several of the same strong links together on one digit.
Remote Pairs replace the link with two more cells holding the same pair. If those two see each other, exactly one of them is x, which does a strong link’s job, for both digits at once. With simpler techniques tried first, Remote Pairs were needed on none of the first 500 Evil puzzles and on 2 of the other 4,500.
What comes after a W-Wing, and what if there isn’t one?
Usually singles. Puzzle 461 needs nothing else after its second W-Wing, and 58 of the helper’s 190 were the last hard move of their puzzle; after 102 a single was ready on the very next move. Look first at the cell that lost a candidate and at the houses of the two wing cells. When the board stays shut:
- Look for another W-Wing. One removal can complete the strong link for the next, as R1C3’s 7 did on puzzle 461. The next hard move was a W-Wing again 14 times.
- Recheck locked candidates. A pointing pair came next 15 times: the removed candidate can leave a digit confined to one line of a box.
- Hunt for an Almost Locked Set. It came next 48 times, and 90 of the 145 puzzles that needed a W-Wing needed one too; 38 needed a Skyscraper and 29 Simple Colouring.
- Move to chains. 17 of the 145 outran all eighteen of the helper’s techniques. X-cycles and XY-chains come next, then a BUG check when the board is nearly full, with forcing chains kept for the very end; the order is set out in our guide to sudoku techniques.
No W-Wing anywhere is the normal state of an Evil board: at positions with no single, one was there only 19.9% of the time. The list of two-candidate cells you made for it may still hold an XY-Wing (23.9%) or an XYZ-Wing (10.0%), while Simple Colouring, at 43.2%, and the Almost Locked Set, at 95.1%, turn up far more often than any wing. Our Hard and Expert levels won’t give you a W-Wing to practise on; the Evil bank will.