Fish & Wings
XY-Wing
An XY-Wing is three cells with exactly two candidates each: a pivot {X, Y} and two pincers it sees, {X, Z} and {Y, Z}. Whichever digit the pivot takes, the pincer holding that digit loses it and becomes Z, so at least one pincer is Z. Every cell that sees both pincers can therefore drop Z; the pivot itself never holds Z, and X and Y are never removed.
What you need to see on the grid
Look for an XY-Wing once singles, locked candidates, subsets and the fish have run dry but the grid still holds a fair number of cells with just two candidates. It is the first wing in the course, and four terms carry the method:
- Pivot
- An unsolved cell with exactly two candidates, X and Y, that sees both pincers. It is where the argument starts: the pivot must be X or Y, and each choice has a consequence. A third candidate in the pivot breaks the pattern (with Z as the third, it is an XYZ-Wing).
- Pincers
- Two cells with exactly two candidates that see the pivot: one holds {X, Z}, the other {Y, Z}. Each shares exactly one digit with the pivot, and not the same one. They need not see each other, and they may sit in three different boxes with the pivot.
- Z, the shared digit
- The second digit of both pincers — the same in each — and a digit the pivot does not hold. It is the only digit an XY-Wing removes: the pivot may end up X or Y, so neither of those is forced anywhere.
- The elimination zone
- The cells that see both pincers. When the pivot sees one pincer along its row and the other along its column, the zone is a single cell, the fourth corner of the rectangle. When one pincer shares the pivot's box, as in the example below, the zone grows to two strips of three cells.
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. A digit missing from a drawn cell is one its row, column or box already holds — even where that part of the grid is blank — or one an earlier step removed, and no simpler technique changes any cell that is drawn.
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Step 1Find a cell with exactly two candidates
Start from the pivot: scan for unsolved cells with exactly two candidates, often called bivalue cells. Row 2 has two of them, R2C2 {3, 7} and R2C7 {5, 7}. R2C3 and R2C5 have three candidates each and cannot be a pivot: a third digit is an escape route the pivot could take, leaving both pincers free.
Every bivalue cell is a possible pivot, so test them one at a time. We follow R2C2. Its two digits are X = 3 and Y = 7; Z will come from the pincers, in the next step.
Row 2. R2C2 and R2C7 have exactly two candidates; R2C3 and R2C5 have three. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 4 9 1 8 2 R3 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Row being scanned
- Exactly two candidates: a possible pivot
- Not needed for this step
Filled in: row 2. 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 that share the same Z
A pincer is a bivalue cell that sees the pivot — it shares its row, column or box — and holds exactly one of the pivot's two digits. Its other digit is a candidate Z. You need two pincers, one holding X and the other Y, whose other digit is the same.
R2C2 sees row 2, column 2 and box 1. Three bivalue cells there touch its digits. R3C1 {3, 5}, in box 1, holds X = 3 and 5. R2C7 {5, 7}, in row 2, holds Y = 7 and 5. Both second digits are 5: these are the pincers, and Z = 5. R8C2 {3, 8} is a decoy: it shares 3 with the pivot, but its partner would have to be {7, 8}, and no cell in sight is. The other cells have three candidates.
Technique: XY-Wing
Row 2, column 2 and box 1 — everything R2C2 sees. Its pincers are R3C1 {3, 5} and R2C7 {5, 7}; R8C2 {3, 8} does not fit. Every digit missing from the pivot is placed in sight. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 8 1 R2 4 9 1 8 2 R3 9 2 R4 5 R5 2 R6 6 R7 R8 R9 4 - Digit already on the grid
- Everything the pivot sees
- Pivot R2C2 and its pincers R3C1, R2C7
- Candidates 3, 5 and 7, the digits studied
- Not needed for this step
Filled in: row 2; column 2; box 1. 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 3Follow both cases: one pincer is always Z
The pivot is 3 or 7; take each case in turn.
If R2C2 is 3, then R3C1, in the same box, cannot be 3, so it is 5. If R2C2 is 7, then R2C7, in the same row, cannot be 7, so it is 5.
There is no third case. Whatever the pivot turns out to be, at least one pincer is 5 — perhaps both. You do not know which one, and you do not need to.
Technique: XY-Wing
Only the three cells. Pivot 3 makes R3C1 a 5; pivot 7 makes R2C7 a 5. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - Pivot R2C2 and its pincers R3C1, R2C7
- Candidates 3, 5 and 7, the digits studied
- Not needed for this step
Filled in: cells R2C2, R2C7 and R3C1. 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 every cell that sees both pincers
A cell that sees both pincers sees the 5, wherever it lands, so it cannot be 5. List those cells. R3C1 sees row 3, column 1 and box 1; R2C7 sees row 2, column 7 and box 3. They overlap in row 2 inside box 1 — R2C1, R2C3 and the pivot — and in row 3 inside box 3 — R3C7, R3C8 and R3C9.
Because R3C1 shares a box with the pivot, the zone is these two strips, not the single rectangle corner you get when all three cells are in different boxes. Of its cells, R2C3, R3C8 and R3C9 list 5: remove it from all three. R2C1 and R3C7 already hold digits, and the pivot never lists Z. The pincers keep their 5.
Technique: XY-Wing
Rows 2 and 3 and boxes 1 and 3: everything the pincers see there. 5 leaves R2C3, R3C8 and R3C9, the cells that see both pincers and list it. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 8 1 9 4 R2 4 9 1 8 2 R3 9 2 8 4 1 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- Pivot R2C2 and its pincers R3C1, R2C7
- Candidates 3, 5 and 7, the digits studied
- Candidate removed
- Not needed for this step
Filled in: rows 2 and 3; boxes 1 and 3. The other 57 cells are left blank: they are not needed for this step. Removed: 5 from R2C3, R3C8 and R3C9.
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Step 5Collect what the elimination unlocks
An XY-Wing places nothing by itself, so look at the cells it cleared. R3C9 had only 5 and 6. With 5 gone, 6 is all that is left: a Naked Single.
Wings often pay out this way, on a bivalue cell in the zone, so check every cell that lost a candidate before you move on.
Technique: Naked Single
Row 3 and box 3 after the XY-Wing. R3C9 has one candidate left, 6. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 9 4 R2 8 2 R3 9 2 8 4 1 6 R4 R5 R6 R7 R8 R9 - Digit already on the grid
- The cell that must be 6
- Candidate 6, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 3; box 3. The other 66 cells are left blank: they are not needed for this step. Placed: R3C9 = 6.
Common mistakes
Removing Z from a cell that sees only one pincer
The guarantee is that one of the two pincers is Z, not which one. R3C6 lists 5 and sees R3C1 along row 3, but not R2C7: if R2C7 is the 5, R3C6 may still be 5. R2C5 is the same case the other way round. Seeing the pivot adds nothing either — test every target against both pincers.
A pivot with three candidates
If the pivot can also be some other digit, it may take that one and leave both pincers free to avoid Z, and the argument collapses. When the third digit is Z itself the pattern is an XYZ-Wing, whose zone is narrower: a target must see the pivot as well. Count the pivot's pencil marks: exactly two.
Removing X or Y along with Z
Only Z is pinned down. The pivot may be 3 or 7, so neither digit is forced into a pincer, and neither can be cleared anywhere. In the example, R2C3 keeps its 7 and R3C8 keeps its 3 and 7: only the 5s go.
Recent daily puzzles that needed this technique
Keep going
- XYZ-WingA three-candidate pivot and two pincers trap Z in three cells — only cells that see all three lose it.
- W-WingTwo cells with the same pair, joined by a strong link on one digit — cells that see both lose the other digit.
- XY-ChainA run of bivalue cells pins one digit to one end of the chain or the other — every cell that sees both ends loses it.
Learn the move, not just the digit
In a real game, when an XY-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.