Sudoku Written by Brian Hamilton and Karan Hamilton.
Brian and Karan are the puzzle lovers and game makers behind Sudoku Online Puzzles. You can learn more on the About page.
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An XYZ-Wing is an XY-Wing whose middle cell keeps a third candidate, and that third candidate is the very digit the pattern removes. One extra pencil mark changes a lot. The wing’s reach shrinks from as many as five cells to exactly two, and the XY-Wing habit of clearing every cell that sees both pincers starts removing correct answers.
We show it on puzzle 299 of the bank our Evil sudoku page draws on, where a single XYZ-Wing is the only step past naked and hidden singles that the puzzle needs. We also counted. On Evil, 79 of the first 500 puzzles need an XYZ-Wing, about half as many as need an XY-Wing; our Easy to Expert levels never call for one.
The Sudoku Trainer has ten XYZ-Wing positions. Choose XYZ-Wing, select the pivot and both pincers and press Check; Hint names one of the three cells first, then the digit. All ten have the pincers apart, one in the pivot’s box and one along its row or column, the shape this guide is about. Each was the simplest move our helper had on its board, and each clears a single candidate.
What is an XYZ-Wing in sudoku?
Three cells and three digits, usually labelled X, Y and Z. The pivot holds all three. It sees two cells with two candidates each, the pincers: one holds X and Z, the other Y and Z. Some guides call the pincers wings, and SudokuWiki calls the pivot the apex or hinge. Whatever the pivot turns out to be, one of the three cells is Z, so Z can go from any cell that sees all three.
The pincers must not see each other. If they did, all three cells would sit in one row, column or box with three digits between them: a naked triple, which our helper looks for long before any wing. At the 87 positions on the first 500 Evil puzzles where it played an XYZ-Wing, the board held 412 XYZ-Wing shapes; 177 of them sat inside one house, and those 177 removed nothing between them.
How does an XYZ-Wing remove a candidate?
Take puzzle 299 after 38 moves, every one of them a single. Seventeen squares are empty and there is no single left anywhere. The pivot is R3C6, which can be 2, 4 or 8. It sees R2C6 {2,8}, the square above it, and R3C9 {2,4}, at the end of row 3, and those two don’t see each other. Go through the pivot’s digits:
- If R3C6 is 8, R2C6 can’t be 8, so it is 2.
- If R3C6 is 4, R3C9 can’t be 4, so it is 2.
- If R3C6 is 2, the 2 is in the pivot.
Every branch puts a 2 in one of the three cells. R3C5 sees all three, R3C6 and R3C9 along row 3 and R2C6 in box 2, so it can’t be 2. It held {2,3}, and now it is 3.
That one elimination is enough: naked singles fill every remaining square from here. In the solution it is the third branch that comes true: R3C6 is the 2, R2C6 is 8 and R3C9 is 4. An XY-Wing has no such branch, and it is the reason the XYZ-Wing reaches so few cells.
To watch the helper find it, open the position in our Sudoku Helper, where Next Hint’s first press only pencils in the candidates; they match the figure exactly, since no earlier move took any away. The second press names this XYZ-Wing and reports two techniques available; besides it there is an Almost Locked Set, which turns out to be a close relative.
Why can an XYZ-Wing only clear two cells?
A target has to see the pivot as well as both pincers, and the pincers can’t share a house. Those two rules fix the shape. We went through all 8,748 ways to place a pivot and two pincers it sees with the pincers apart, and every one is one of two kinds:
- One pincer in the pivot’s box, the other along the pivot’s row or column, outside the box (5,832 placements). Exactly two cells see all three: the two other cells of the pivot’s box on that row or column. On puzzle 299 the outside pincer, R3C9, is on row 3, so the two cells are R3C4 and R3C5.
- Both pincers outside the pivot’s box, one along its row and one along its column (2,916). No cell sees all three, so the wing can never remove anything, however good it looks.
Apply the XY-Wing’s rule to the same three cells and six cells qualify, in two strips of three: one through the pivot, the other in the outside pincer’s box. The XYZ-Wing keeps only the strip through the pivot.
Two cells is a ceiling, and the Zs rarely fill it: 84 of the helper’s 87 XYZ-Wings on the first 500 Evil puzzles removed one candidate, and 3 removed two.
What is the difference between an XY-Wing and an XYZ-Wing?
The pivot’s third digit. An XY-Wing’s pivot holds only X and Y, so when the pattern’s Z is anywhere among its cells it is in a pincer. An XYZ-Wing’s pivot holds Z too, so the pivot itself may be the Z. Everything else in the table follows from that.
| XY-Wing | XYZ-Wing | |
|---|---|---|
| Pivot’s candidates | X, Y | X, Y, Z |
| Where the Z can be | One pincer or both | One pincer, both, or the pivot |
| A target must see | Both pincers | Both pincers and the pivot |
| Cells within reach | Up to five, or one corner cell | Two, in the pivot’s box (none if neither pincer is in it) |
| On the board where singles ran out | 23.9% | 10.0% |
| Needed, first 500 Evil puzzles | 157 | 79 |
| Needed, all 5,000 | 1,743 | 753 |
| Hardest step, of the 3,655 our helper finishes | 543 | 78 |
A puzzle counts as needing a technique if, at least once, it was the simplest move available to our helper’s eighteen techniques; the hardest step means nothing further down the helper’s list was used. “On the board” is measured at all 4,618 positions without a single in the first 500 Evil puzzles. Score order makes those 500 tougher than the rest of the bank. Measured October 2026.
The third row is where solvers go wrong. Clear every cell that sees both pincers, as you would for an XY-Wing, and you can remove an answer: on puzzle 299 that would take the 2 from R2C9, which is the 2 of row 2 in the solution.
In 25 of the helper’s 87 XYZ-Wings, the pivot itself turned out to be the Z. In the other 62 a pincer was, and in 33 of those both pincers were. In 63 of the 87, some cell that sees both pincers but not the pivot still held a Z, and in 18 that Z was the answer. Read as XY-Wings, those 18 would have thrown away a correct digit.
Puzzle 199, the one our XY-Wing guide opens with, has both kinds on the board at once. Its XYZ-Wing, pivot R9C9 {1,8,9} with R7C9 {8,9} in its box and R9C3 {1,8} along row 9, reaches R9C7 and R9C8. R9C8 is already filled, so the wing takes the 8 from R9C7, which is the XY-Wing’s own pivot.
Is an XYZ-Wing an Almost Locked Set move?
Yes, the smallest kind. On puzzle 299, take the pivot R3C6 {2,4,8} together with R2C6 {2,8}, both in box 2, as set A, and R3C9 {2,4} on its own as set B. The sets share two digits, 4 and 2, and the 4 can’t be in both: set A’s only 4 is in the pivot, which sees R3C9. If set A loses the 4, its two cells hold just 2 and 8, so one of them is the 2. If set B loses it, R3C9 is the 2. Either way a 2 lands in one of the three cells, which is the XYZ-Wing’s argument told another way, and it clears the same cell, R3C5.
A group of N cells in one row, column or box holding N+1 digits between them. Take away any one digit and the rest is locked, like a naked pair or triple. A single two-candidate cell counts (one cell, two digits), and so does the pivot with the pincer in its box (two cells, three digits). Two sets joined by a digit they can’t both keep make the move called ALS-XZ, which our helper reports as an Almost Locked Set.
Our helper’s Almost Locked Set search counts single cells as sets, so every XYZ-Wing lies inside it, and at all 87 positions where the helper played an XYZ-Wing the ALS search had a move too. The helper simply tries the XYZ-Wing first. On puzzle 199 the ALS move it would report is that very XYZ-Wing, as R9C3 on its own and R7C9 with R9C9; on puzzle 299 it reports a different pair of sets first, one that takes the 5 from R2C4 and R3C2. Other guides go one size up to a WXYZ-Wing, with a fourth cell and digit; our helper has no detector by that name, so we have no figures for it.
Where should you look for an XYZ-Wing?
Because the shape is fixed, you can search for it in a fixed order:
- Keep every pencil mark current. In 47 of the 87 positions, an earlier move had taken a candidate out of at least one of the three cells. Marks worked out from the digits alone would not show those wings.
- Go box by box. In each box, look for a three-candidate cell with a two-candidate cell in the same box made of two of its digits. That is a pivot and its box pincer.
- Look along the pivot’s row and column, outside the box, for a two-candidate cell holding the pivot’s third digit, the one the box pincer lacks, and one of the box pincer’s two digits; that shared digit is Z. Skip it if it sees the box pincer.
- Check two cells. Only the rest of the pivot’s box on that row or column can lose Z. If neither holds it, move on.
Starting from the pivots saves time, because three-candidate cells are the scarcer kind by then: across those 87 positions the median board had 13 of them against 18 two-candidate cells, and the two-candidate cells were the more numerous at 66.
Or start from the candidate you want gone. A candidate can only fall to an XYZ-Wing whose pivot shares its box and its row or column, which leaves four cells to check. For the 2 in R3C5 they are R1C5, R2C5, R3C4 and R3C6, and R3C6 is the only one with three candidates including a 2.
How rare is the XYZ-Wing on real puzzles?
We let our helper play 200 boards from each classic level, then the first 500 and all 5,000 puzzles of the Evil bank, taking the simplest move it could find every time, and counted the puzzles where that move was an XYZ-Wing at least once.
- Easy, Medium, Hard and Expert: never. Every board at those levels is one that singles, locked candidates and naked pairs can finish, or our generator wouldn’t deal it. At Expert, singles run out at 277 positions, and an XYZ-Wing is present at 44 of them, though never as the simplest move.
- Evil: about 15% of puzzles. 79 of the first 500 need one, and 753 of all 5,000. It is the first step beyond singles, locked candidates and naked pairs on 244 of the 5,000.
It is seldom the whole difficulty. Our helper finishes 3,655 Evil puzzles; just 78 of them needed an XYZ-Wing and nothing further down its list, and puzzle 299 is one of the 7 such puzzles in the first 500. Of the 79 puzzles in the first 500 that needed one, 61 also needed an Almost Locked Set, 27 a W-Wing and 23 a Skyscraper, and 23 stalled with none of the helper’s eighteen left. With five extra detectors run ahead of the wings, as our full measurement does, the first-500 count falls from 79 to 66, because finned fish get to some of those positions first.
What comes after an XYZ-Wing, and what if there isn’t one?
Don’t expect a single straight away. An XYZ-Wing usually takes out one candidate, and after only 15 of the helper’s 87 was a single ready at once (9 naked, 6 hidden). After an XY-Wing it is more than half. Puzzle 299 is the lucky kind. More often the next move beyond singles was an Almost Locked Set (27 times), a W-Wing (14), a Skyscraper (8), or a pointing pair or hidden pair (6 each); after 9, singles alone finished the puzzle.
And often there is no XYZ-Wing at all. Across the first 500 Evil puzzles, singles ran out at 4,618 positions, and an XYZ-Wing was on the board at only 10.0% of them, less often than an XY-Wing (23.9%) or a W-Wing (19.9%), which start from the same two-candidate cells, so check those too. Simple Colouring and Almost Locked Sets are on the board far more often. Past the helper’s eighteen, the XY-chain is the natural next step, the same either/or reasoning carried through any number of two-candidate cells; X-cycles, a BUG check in the endgame and forcing chains follow, in the order our guide to sudoku techniques sets out. Real practice means Evil puzzles, since our Easy to Expert levels never ask for one.
Evil Sudoku
Every board in this guide comes from this bank, and about 15% of its 5,000 puzzles need an XYZ-Wing.
▶ Play Evil Sudoku