X-Chain
A single-digit chain of alternating links forces one end to be true.
An X-chain is a single-digit solving method that follows one number as it weaves through the grid along a trail of linked cells. The x-chain sudoku technique alternates strong links — units where the digit has only two possible squares, so one of them must be true — with weak links, where the digit is simply shared along a row, column, or box. When the trail both starts and ends on a strong link, one of its two endpoints is guaranteed to hold the digit, which lets you erase that digit from any cell that can see both ends of the chain.
Worked example
Here the chain is built on the digit 1. Row 1 holds only two 1s, at R1C1 and R1C5, so they form a strong link — one of them must be the 1. Following column 5 downward, R1C5 connects by a weak link to R3C5, and row 3 also has only two 1s, making R3C5 and R3C3 another strong link. The chain reads R1C1 = R1C5 - R3C5 = R3C3, so because it opens and closes on strong links, either R1C1 or R3C3 must be 1. R2C2 sees both of those endpoints inside box 1, so it can never be the 1 — the candidate is eliminated from R2C2.
When to use it
Reach for an X-chain once the basic scans stall and one digit stays stubborn across the grid. Scan that single number for conjugate pairs — units where it has exactly two candidate squares — since each pair is a ready-made strong link. String those pairs together with weak links along shared rows, columns, or boxes, then check whether some common cell can see both ends of the chain. If it can, that digit comes out of the common cell.
How to use it
- Chain strong and weak links. Pick a digit and build a chain that alternates strong links (a unit where the digit has only two spots) with weak links (two spots sharing a unit), starting and ending on a strong link.
- Read the endpoints. Because the chain starts and ends with a strong link, at least one of its two endpoints must hold the digit — you do not know which, but you know one of them does.
- Eliminate from cells seeing both ends. Any cell that sees both endpoints sees the digit no matter which end is true, so the digit can be removed from it.
Frequently asked questions
What is an X-chain in Sudoku?
An X-chain is an alternating inference chain built on a single digit. It links candidate cells through strong links — a unit where the digit has only two spots, so one must be true — and weak links, where the digit is shared along a unit. When the chain starts and ends on strong links, one of its two endpoints must hold the digit, so any cell that sees both endpoints loses that candidate. In our example R2C2 sees both ends, R1C1 and R3C3, so the 1 is removed from R2C2.
How do I spot an X-chain on the grid?
Pick one stubborn digit and mark every unit where it has just two candidate cells; those are your strong links. Try to walk from one conjugate pair to the next, alternating with weak links wherever the digit is shared along a row, column, or box. When the two ends of such a chain are both visible to a single cell — as R1C1 and R3C3 both are to R2C2 above — that cell cannot hold the digit.
What is the difference between an X-chain and an X-Wing?
An X-Wing is a rigid 2x2 fish: the digit sits at the corners of a fixed rectangle spanning two rows and two columns. An X-chain is a flexible chain of any length on one digit, so it can bend through the grid however the available conjugate pairs allow, like the R1C1 = R1C5 - R3C5 = R3C3 path in the example. In fact an X-Wing, and the skyscraper, are just short, special-case X-chains.