XY-Chain Sudoku Technique

Updated 11 August 2026

An XY-Chain links bivalue cells end to end, each one sharing a candidate with the next, so that the chain’s two endpoints agree on one value between them and that value can be erased from every cell both endpoints see.

When an XY-Chain applies

An XY-Wing is a three-cell XY-Chain. The same idea extends to any length: string together cells that each carry exactly two candidates, where consecutive cells share one value in common. At each interior cell, one of its two candidates connects back to the previous cell and the other connects forward to the next one — the two values alternate down the chain like a line of dominoes.

The proof follows the chain from one end. Assume the first cell does not hold the value that links it to the second cell. Then it must hold its other candidate. That forces the second cell away from the shared value, so the second cell must take its own other candidate, which forces the third cell, and so on to the last cell. That reasoning shows that whenever the first cell is not its own “other” candidate, the last cell ends up holding that same candidate. Combined with the case where the first cell is that candidate directly, one of the two endpoints always holds it.

Worked example

r4c5 holds 1 and 9. r9c5 holds 1 and 5, sharing 1 with r4c5. r9c7 also holds 1 and 5, sharing 5 with r9c5. r6c7 holds 1 and 4, sharing 1 with r9c7. r6c2 holds 4 and 9, sharing 4 with r6c7. If r4c5 is not 1, it must be 9 — the chain ends immediately. If r4c5 is 1, then r9c5 cannot be 1 and must be 5; then r9c7 cannot be 5 and must be 1; then r6c7 cannot be 1 and must be 4; then r6c2 cannot be 4 and must be 9. Either way, one of r4c5 or r6c2 ends up holding 9.

r4c2 sits in row 4 with r4c5 and in column 2 with r6c2, so it sees both endpoints. r6c6 sits in the same box as r4c5 and in row 6 with r6c2, so it sees both endpoints too. r4c2 was carrying 4 and 9, and r6c6 was carrying 1 and 9, before the deduction. Since 9 must land at one endpoint or the other, neither r4c2 nor r6c6 can hold it, and the solver removes both.

XY-Chains, alongside X-Chains and simple coloring, are the kind of technique that typically pushes a Sudoku-Explainer rating past 7.0 — the point where the grid needs reasoning across a whole path of cells rather than a fixed small pattern.

How to spot an XY-Chain

1. Map every bivalue cellList every cell with exactly two remaining candidates, along with what those two candidates are.
2. Link cells that share a candidateConnect two bivalue cells if they see each other and share exactly one candidate. That shared candidate is the link between them.
3. Follow the chain to a consistent endpoint valueTrace a path where each cell’s incoming and outgoing links use its two different candidates, until the chain’s two endpoints share their remaining, unlinked candidate.
4. Erase that candidate from shared peersRemove the endpoints’ shared candidate from any cell, outside the chain, that sees both endpoints.

Common mistakes

• Allowing a chain cell with three or more candidates. Every cell in an XY-Chain must be strictly bivalue for the alternating argument to hold.
• Reusing the same candidate for both the incoming and outgoing link at one cell. Each interior cell must alternate between its two different values.
• Eliminating the wrong candidate. Only the value shared by both endpoints — not any value seen along the way — can be erased from their common peers.
• Stopping the search at short chains. XY-Chains can run through many cells; a pattern that fails at three or four cells might still close a few links later.