Wings

The XY-Wing

The XY-Wing is the first real chain most solvers learn — three two-candidate cells that, between them, force a number out of the grid even though not one of the three is yet solved.

The parts of the wing

An XY-Wing is built from three cells that each hold exactly two candidates. One is the pivot, carrying candidates X and Y. It connects to two pincers: one sharing a unit with the pivot and holding X and Z, the other sharing a unit with the pivot and holding Y and Z. Notice the shared Z — it is the number the wing will eliminate. The three cells form a bent chain, the pivot at the corner and a pincer reaching out along each arm.

Why it forces an elimination

Follow the only two possibilities. The pivot is either X or Y. If it is X, then the pincer sharing that unit cannot also be X, so it must be Z. If instead the pivot is Y, then the other pincer cannot be Y, so it must be Z. Either way — and there is no third way — one of the two pincers ends up as Z. So any cell that can see both pincers at once cannot hold Z, because one of those pincers is certain to claim it. You strike Z from every such cell.

Finding the pattern

Start from bi-value cells; the whole technique lives among cells with exactly two candidates, so a full candidate map is essential. Pick one as a possible pivot and read its two numbers, X and Y. Then look in the units it sees for a cell holding X and some third number Z, and another cell it sees holding Y and that same Z. When you find the pair, the wing is complete, and the cells that see both pincers are where the elimination lands.

A worked XY-Wing

Suppose the pivot holds {4,7}. In its row sits a pincer holding {4,9}, and in its column sits a pincer holding {7,9}. Here X is 4, Y is 7 and Z is 9. If the pivot is 4, the row pincer cannot be 4 and becomes 9; if the pivot is 7, the column pincer cannot be 7 and becomes 9. One of those two cells is definitely a 9. Now find any cell that lies in a unit with both pincers at once — the corner where their row and column cross is the classic spot — and remove 9 from it. That single elimination often unblocks a board that singles and pairs had left cold.

Common mistakes

Be strict, because a false wing wrecks a grid. All three cells must have exactly two candidates — no more. The pivot must genuinely see each pincer (share a row, column or box with it), and the two pincers must share the same Z. And the elimination only applies to cells that see both pincers; a cell seeing just one is untouched. When any of these conditions is loose, there is no wing and no valid elimination, so check each before you strike anything out. It also pays to confirm the payoff early: if you cannot point to at least one cell that sees both pincers, the wing may be perfectly real yet useless here, and it is not worth the effort of chasing further.

Where it fits

The XY-Wing belongs to the same family of proofs as the X-Wing: you establish a fact about cells you have not filled and act on it with total confidence. It is a hard-board technique, so meet it after singles, pairs and intersections have run out. Hunt your first wings on Medium Sudoku where they are easier to see, then wield them on Expert Sudoku. The classic grid is always ready at Sudoku AZ.