XY-Chain
An XY-Chain is a string of bivalue cells — cells with exactly two candidates — each seeing the next. The chain passes a carry digit along like a baton: every cell contains the digit handed to it and, if forced, hands over its other digit to the next cell. The chain targets one digit: the start holds the target plus its first carry, and the far end's second digit is the target again. The conclusion is that at least one of the two end cells must be the target, so the target can be removed from every cell that sees both ends.
Trace one. Target 5, starting at r1c1 holding 5/9. Suppose r1c1 is not 5 — then it is 9, and the chain fires: r1c5 (9/2, same row) loses its 9 and becomes 2; r5c5 (2/7, same column) loses its 2 and becomes 7; r5c8 (7/5, same row) loses its 7 and becomes 5. So either r1c1 is 5, or r5c8 is. The cells r1c8 and r5c1 each see both ends — one by row, one by column — and neither can be 5.
The Y-Wing is the shortest XY-Chain, three cells long, with the pivot as the middle link. Longer chains snake through four, five or more bivalue cells, and each hand-off must be exact: the next cell must actually contain the carry digit, and it surrenders its other digit onwards. Both ends must contain the target — that's what makes the elimination two-sided rather than a guess.
XY-Chains reward a well-marked grid rich in bivalue cells, typically late in a hard puzzle. Pick a target digit that appears in several bivalue cells and try to link a pair of them through carries.
- Every link cell must be bivalue and must see the previous cell — no exceptions, or the forcing breaks.
- Check the hand-offs: each cell receives the carry and passes its other digit; the end's other digit must equal the target.
- Eliminations apply only to cells seeing both ends — the chain's interior cells are never resolved by the technique.