X-Chain Sudoku Technique

Updated 11 August 2026

An X-Chain follows one digit through a sequence of alternating strong and weak links, cell to cell, until both ends of the chain reach a cell that can see them both, which lets that candidate be erased there.

When an X-Chain applies

Every earlier technique on this list reasons about one or two units at a time. An X-Chain reasons about a single digit across an arbitrary number of units, strung together into a path. A strong link connects two cells when a unit — a row, column or box — has the digit in only those two cells: if one is not the digit, the other must be. A weak link connects two cells that merely share a unit and both carry the digit as a candidate, without either being forced.

A chain alternates strong, weak, strong, weak, and so on, starting and ending on a strong link. That alternation guarantees that the chain’s two endpoints cannot both be free of the digit at once: following the links forward from one end forces a value onto the other. The practical result is that at least one of the two endpoints holds the digit, even though it is not known which one. Any cell — other than the chain’s own cells — that sees both endpoints cannot hold the digit, because whichever endpoint turns out true, that cell is a peer of it.

Chains are also where a puzzle’s difficulty tends to climb sharply. On the widely used Sudoku-Explainer rating scale, chain techniques like this one are typically what push a puzzle’s rating past 7.0 — the point where singles, subsets, fish and wings alone are no longer enough.

Worked example

The chain on 5 runs r7c4 → r3c4 → r3c5 → r9c5 → r8c6 → r8c1, alternating strong and weak links along the way: r7c4 and r3c4 share column 4, r3c4 and r3c5 share row 3, r3c5 and r9c5 share column 5, r9c5 and r8c6 share a box, and r8c6 and r8c1 share row 8. Because the chain starts and ends on a strong link, r7c4 and r8c1 cannot both be free of 5 — at least one of them holds it.

r7c1 sits in row 7 with r7c4 and in column 1 with r8c1, so it sees both ends of the chain. It was carrying 1 and 5 before the deduction. Since 5 must land at one end of the chain or the other, r7c1 cannot hold 5 itself, and the solver removes it, leaving 1 behind.

How to spot an X-Chain

1. Pick a digit and map its strong linksFor one digit, find every unit where it has exactly two candidate cells. Each such unit is a strong link between those two cells.
2. Connect strong links with weak linksJoin strong links end to end through weak links — cells that merely share a unit and both carry the digit — building a path that alternates strong, weak, strong, weak.
3. Confirm both ends are strongA valid elimination chain must begin and end on a strong link, so at least one endpoint is guaranteed to hold the digit.
4. Erase the digit from shared peers of both endsRemove the digit from any cell, outside the chain itself, that sees both endpoints.

Common mistakes

• Ending a chain on a weak link. Without a strong link at both ends, the alternation does not guarantee either endpoint holds the digit.
• Treating any two cells that share a unit as a weak link regardless of candidate count. A weak link still requires both cells to carry the digit as a candidate.
• Eliminating the digit from a cell inside the chain. The chain’s own cells are exactly where the digit is still being debated; eliminations only apply outside it.
• Giving up after checking only short chains. Longer chains are harder to trace by hand but follow the exact same strong-weak alternation rule.