Fish & Wings
XYZ-Wing
An XYZ-Wing is a pivot cell with exactly three candidates, X, Y and Z, that sees two pincers with two candidates each: one {X, Z}, the other {Y, Z}. Whatever the pivot turns out to be, one of the three cells holds Z — the pivot itself, or the pincer it forces — so Z can be removed from every cell that sees all three: the pivot and both pincers. It is the XY-Wing with Z added to the pivot, and that one extra candidate is why a target must see the pivot too — which usually leaves only a cell or two in the pivot's box.
What you need to see on the grid
Look for an XYZ-Wing right after XY-Wing: the pincers are the same, but the pivot carries Z as well. It appears once a cell is down to three candidates and sits next to two-candidate cells built from its digits. Four terms carry the method:
- Pivot
- An unsolved cell with exactly three candidates, X, Y and Z, that sees both pincers. An XY-Wing pivot has only two, X and Y; here Z is in the pivot as well. A fourth candidate breaks the pattern.
- Pincers
- Two cells with exactly two candidates that see the pivot: {X, Z} and {Y, Z}. Both digits of each come from the pivot, and the two pairs are different, sharing only Z — the same pincers as an XY-Wing. They need not see each other.
- Z, the shared digit
- The digit the two pincers share, and the only one present in all three cells. It is the only digit an XYZ-Wing removes: X and Y each sit in just two of the three cells, and nothing forces them.
- The elimination zone
- The cells that see all three — the pivot and both pincers. An XY-Wing only asks a target to see the two pincers; an XYZ-Wing also needs the pivot, because the pivot may be the Z. The zone is therefore small: usually one or two cells inside the pivot's box, sometimes none.
How to apply it, step by step
One example runs through all five steps. Every diagram shows the whole grid: the cells a step needs are filled in, the others are left blank and hatched. The pencil marks already account for digits in the blank part of the grid, and no simpler technique changes any cell that is drawn.
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Step 1Find a cell with exactly three candidates
Start from the pivot: scan for unsolved cells with exactly three candidates. That is the first difference from an XY-Wing, whose hunt starts from a cell with two. Row 5 has two such cells, R5C2 {2, 4, 7} and R5C9 {1, 2, 6}. R5C3 lists four candidates and cannot be a pivot: a fourth digit is an escape route the pivot could take, leaving both pincers free.
Every three-candidate cell is a possible pivot, so test them one at a time. We follow R5C2. Its three digits will be X, Y and Z; which of them is Z is decided by the pincers, in the next step.
Row 5. R5C2 and R5C9 have exactly three candidates; R5C3 has four, and R5C1 and R5C7 have two. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 5 9 3 8 R6 R7 R8 R9 - Digit already on the grid
- Row being scanned
- Exactly three candidates: a possible pivot
- Not needed for this step
Filled in: row 5. The other 72 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 two pincers made of the pivot's digits
A pincer is a cell with exactly two candidates that sees the pivot — it shares its row, column or box — and whose two digits both belong to the pivot. You need two of them with different pairs, one {X, Z} and the other {Y, Z}; the digit they share is Z. These are exactly the pincers of an XY-Wing: only the pivot has changed.
Search every cell R5C2 sees. R4C3 {2, 7} is in its box and R5C7 {4, 7} in its row; both pairs come from {2, 4, 7}, and they share 7. So Z = 7, X = 2 and Y = 4. The other two-candidate cells R5C2 sees do not qualify: R5C1 {6, 7} and R7C2 {2, 8} each carry a digit the pivot does not have. Notice too that the six digits missing from the pivot, 1, 3, 5, 6, 8 and 9, are all already placed among the cells it sees.
Technique: XYZ-Wing
Everything R5C2 sees: row 5, column 2 and box 4. Its pincers are R4C3 {2, 7}, in the same box, and R5C7 {4, 7}, in the same row. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 6 R3 R4 9 5 R5 5 9 3 8 R6 3 8 R7 R8 1 R9 9 - Digit already on the grid
- Pivot R5C2 and its pincers R4C3, R5C7
- Candidates 2, 4 and 7, the digits studied
- Not needed for this step
Filled in: row 5; column 2; box 4. The other 60 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 3See why Z must land in one of the three cells
Follow the pivot through its three possible values. If R5C2 is 2 (X), the pincer R4C3 {2, 7} sees it, loses 2 and must be 7. If R5C2 is 4 (Y), the pincer R5C7 {4, 7} loses 4 and must be 7. If R5C2 is 7 (Z), the pivot holds the 7 itself.
Either way, one of the three cells is 7 — but you cannot yet say which. In an XY-Wing the third case does not exist, because the pivot has no Z, so Z always ends up in a pincer. Here it may sit in the pivot, and the next step depends on exactly that.
Technique: XYZ-Wing
Only the three cells. Whichever digit the pivot R5C2 takes, one of R5C2, R4C3 and R5C7 is 7. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - Pivot R5C2 and its pincers R4C3, R5C7
- Candidate 7, the digit studied
- Not needed for this step
Filled in: cells R4C3, R5C2 and R5C7. The other 78 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 Z from the cells that see all three
One of the three cells is 7, so a cell that sees all three of them — the pivot and both pincers — cannot be 7. Here that means R5C1 and R5C3: they share row 5 with R5C2 and R5C7, and box 4 with R4C3. Both lose 7. The three pattern cells keep their 7s: one of them is the 7.
Now the trap. R4C8 sees both pincers, R4C3 along row 4 and R5C7 in box 6, and it still lists 7. An XY-Wing would clear it; an XYZ-Wing must not. R4C8 does not see the pivot, and if the pivot is 7, then R4C3 is 2, R5C7 is 4, and R4C8 is the only place left for 7 in row 4. Test every target against all three cells, never against the two pincers alone.
Technique: XYZ-Wing
Rows 4 and 5. 7 leaves R5C1 and R5C3, which see the pivot and both pincers. R4C8 sees only the two pincers and keeps its 7. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 9 5 8 1 3 R5 5 9 3 8 R6 R7 R8 R9 - Digit already on the grid
- Pivot R5C2 and its pincers R4C3, R5C7
- Candidate 7, the digit studied
- Candidate removed
- Not needed for this step
Filled in: rows 4 and 5. The other 63 cells are left blank: they are not needed for this step. Removed: 7 from R5C1 and R5C3.
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Step 5Collect what the elimination unlocks
An XYZ-Wing never places a digit by itself, so look at the cells it just cleared. R5C1 was {6, 7}; without its 7 only 6 is left: a Naked Single. R5C3 is down to {1, 4, 6}, and loses 6 as soon as R5C1 is filled.
Because the targets sit right next to the pivot, the payoff is usually close by: check each cleared cell, and its row, column and box, before you move on.
Technique: Naked Single
Row 5 after the XYZ-Wing. R5C1 has only 6 left. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 6 5 9 3 8 R6 R7 R8 R9 - Digit already on the grid
- The cell with one candidate left
- Candidate 6, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 5. The other 72 cells are left blank: they are not needed for this step. Placed: R5C1 = 6.
Common mistakes
Clearing Z from cells that see only the two pincers
This is the XY-Wing rule, and it is wrong here. In an XY-Wing, Z always lands in a pincer, so seeing both pincers is enough. In an XYZ-Wing the pivot can be the Z, and a cell that does not see the pivot is not covered: in the example, R4C8 sees both pincers, yet if the pivot is 7 it is the only place for 7 in row 4. Remove Z only where the cell sees the pivot and both pincers.
A pivot with four candidates
If the pivot can also be a fourth digit, it may end up as that digit: neither pincer is forced, Z may be nowhere in the three cells, and nothing is proved. Count the pivot's candidates: exactly three. With only two, X and Y, you are looking at an XY-Wing, which has its own, wider zone.
Removing X or Y, or picking the wrong Z
Z is the digit the two pincers share, and the only one in all three cells; X and Y each sit in just two of them, so nothing forces them anywhere. Check the pincers before you name Z: they must hold different pairs, both taken from the pivot. Two {X, Z} cells prove nothing, and a two-candidate cell with a digit the pivot lacks is not a pincer at all.
Learn the move, not just the digit
In a real game, when an XYZ-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.