Link cells that can only be two numbers, each seeing the next and sharing a number with it. Suppose the first cell isn't z: then it's its other number, which rules that number out of the next cell, forcing its other number, and so on down the chain. If the last cell ends up forced to z, then the first or the last cell is z, and no cell that sees both can be.
XY-Chain is the long form of the XY-Wing: an XY-Wing is an XY-Chain of three cells. It is the most common technique in our Extreme puzzles.
It only needs cells with two candidates, so with full pencil marks it is surprisingly easy to follow, one cell at a time.
An XY-Chain
The pattern
Candidates removed
Solved as a result
Each cell in R2C3 (2/4) → R5C3 (4/7) → R5C7 (7/5) → R3C7 (5/2) can only be its two numbers, and each sees the next. If R2C3 isn't 2, it's its other number, which rules that number out of the next cell, forcing its other number, and so on down the chain, until R3C7 has to be 2. So R2C3 or R3C7 is 2, and the 2 can be removed from every cell that sees both: R2C8 and R3C1. That leaves R2C8 with only one candidate: 4.
This position comes from a puzzle generated on this site, after every single has been filled in. Boxes are numbered 1–9 from the top left, row by row; R5C4 means row 5, column 4.
How to spot it
1Fill in the candidates and look at the cells with exactly two.
2Pick a start cell and one of its numbers, z. Assume it isn't z: write down its other number.
3Move to a two-option cell it sees that contains that number: it must be its other number. Keep going.
4If you reach a cell forced to be z, remove z from every cell that sees both the start and that cell.
Why it works
If the start cell is z, then z is at the start.
If it isn't, the chain of forced moves makes the last cell z.
So z is at one end or the other, and a cell that sees both ends can't be z.
Common mistakes
A cell in the chain with three candidates. Every cell needs exactly two.
Linking cells that don't see each other. Each step must share a row, column or box.
Removing z from cells that see only one end.
Questions
Is an XY-Wing an XY-Chain?
Yes: an XY-Wing is an XY-Chain with three cells. Longer chains work the same way.
How do I find a long chain?
Start from a cell with two candidates and follow the forced numbers. If a chain dies out, try the other number or another start.
Do I need to guess?
No. You follow a chain of if-then steps without writing anything in, then only remove what is certain.
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