XY-chain
A chain of bivalue cells linked by shared candidates. Eliminates a digit from any cell that sees both endpoints — the workhorse intermediate-to-advanced chain technique.
An XY-chain is a sequence of bivalue cells — cells with exactly two candidates — connected by shared digits. Each adjacent pair of cells in the chain shares one of its two candidates and sits in the same unit. The chain starts and ends with cells that share the same candidate, even though they may be far apart on the grid. Any cell that sees both endpoints can have that shared digit eliminated.
How the chain runs
A worked layout. Suppose cell C1 has candidates {a, b}; cell C2 sees C1 and has candidates {b, c}; cell C3 sees C2 and has candidates {c, d}; cell C4 sees C3 and has candidates {d, a}. C1 and C4 both contain a as one of their two candidates.
Run the trial: if C1 is not a, it must be b. Then C2, which sees C1 and has {b, c}, can't be b — so it's c. C3, which sees C2 and has {c, d}, can't be c — so it's d. C4, which sees C3 and has {d, a}, can't be d — so it's a. So C1 isn't-a forces C4 is-a.
Run the other trial: if C1 is a, then C1 is a directly. So whichever way, either C1 or C4 is a. Any cell that sees both C1 and C4 must avoid being a — because one of them already is.
The chain extends to arbitrary length. As long as each step is bivalue and each adjacent pair shares the right "passing" digit, the trial-cascade carries through.
Why it's the chain technique most solvers learn first
XY-chains are the friendliest of the advanced chain techniques because every step is a single bivalue cell, with the logic running in one direction. Compare that to AIC, which alternates strong links and weak links along candidates rather than cells, and tracks two cases simultaneously. XY-chain is one case, one direction, only bivalue cells.
The closed-loop variant — XY-loop — produces additional eliminations beyond the endpoints because every step in a closed loop is constrained from both sides at once.
When you'll see it
XY-chains turn up regularly on hard and expert puzzles. The Y-wing is the simplest XY-chain, with exactly three cells. Once you've internalised Y-wing, longer XY-chains become a matter of patience and pencil-mark hygiene rather than new logic.
See also
- Y-wing (XY-wing)— Three bivalue cells where the pivot shares one candidate with each wing — eliminating the third candidate from any cell that sees both wings.
- XY-loop— A closed XY-chain — the endpoints meet rather than going off into eliminations. Every step in the loop is constrained from both sides at once, surfacing extra eliminations.
- Alternating Inference Chain (AIC)— The general-purpose chain technique. Alternates strong and weak links along a sequence of candidates, eliminating a digit from any cell that sees both endpoints' candidates.
- Candidate— A digit (1–9) a cell could still legally hold — one not yet ruled out by anything in its row, column, or 3×3 box. Every empty cell has between one and nine.
- Weak link— A relationship between two cells where at most one can hold the digit. Looser than a strong link — both might be other digits — and the steady half of every chain technique.
Read more
- Which technique is this puzzle asking for
How to read a fresh hard Sudoku and predict which intermediate technique will break it open before you've placed a single digit.