XY-Chain
A chain of pairs that starts and ends on one digit forces an end.
An XY-Chain is a run of bivalue cells — cells holding exactly two candidates — linked so that each cell shares one digit with its neighbour, while the chain as a whole begins and ends on the same digit z. Because every hop forces the next, one of the two endpoints has to be z no matter which way the chain falls, so any cell that can see both ends is barred from holding z. Think of the xy-chain sudoku technique as an XY-Wing stretched out: the very same forcing logic extended from three cells to a longer chain of bivalue cells.
Worked example
The chain runs R1C1{1,2} - R1C2{2,3} - R2C2{3,4} - R2C1{1,4}, with neighbours linked on 2, then 3, then 4, and both ends carrying a 1. Suppose R1C1 is not 1: it must then be 2, which forces R1C2=3, then R2C2=4, and finally R2C1=1. So whichever way it falls, a 1 must land on one endpoint or the other. R3C1{1,9} sits in column 1 where it can see both R1C1 and R2C1, so it can never be the 1 — its 1 is eliminated and R3C1 resolves to 9.
When to use it
Reach for it in hard puzzles that are rich in bivalue cells, once XY-Wings and simpler chains have run dry. Pick a bivalue cell that contains your target digit z, then hop cell to cell along shared candidates, keeping every cell strictly bivalue, and try to loop back to another cell that also holds z. When both ends land on z, scan for any cell that sees both endpoints — that is where the elimination lives.
How to use it
- Link bivalue cells. Find a run of cells that each hold exactly two candidates, where each cell shares one candidate with the next and the two ends both contain a target digit z.
- Follow the forced alternation. Assume the first cell is not z: it must then be its other candidate, which forces the next cell, and so on until the last cell is forced to be z.
- Eliminate z from cells seeing both ends. So the first end is z or the last end is z — either way any cell that sees both ends cannot be z, and z is removed there.
Frequently asked questions
What is an XY-Chain in Sudoku?
An XY-Chain is a sequence of bivalue cells — each with exactly two candidates — linked so that consecutive cells share a digit, and the chain starts and ends on the same digit z. The links guarantee that one of the two endpoints must be z, so any cell that sees both ends can safely drop z. In the worked example, the chain R1C1-R1C2-R2C2-R2C1 forces a 1 onto one end or the other, which lets R3C1 shed its 1 and resolve to 9.
How do I spot an XY-Chain on the grid?
Start from a bivalue cell that contains your target digit z, then hop to a neighbour that shares one of its candidates, keeping every cell you pass through strictly bivalue. Keep going until you return to another bivalue cell that also holds z. Then look for a cell that sees both endpoints — like R3C1{1,9}, which sees both R1C1 and R2C1 down column 1 — and eliminate z from it.
How is an XY-Chain different from an XY-Wing?
An XY-Wing is a fixed three-cell pattern: one pivot with two pincers. An XY-Chain is the same forcing idea stretched to any odd-length run of bivalue cells, so the four-cell chain R1C1-R1C2-R2C2-R2C1 is essentially a longer wing. Both conclude that one endpoint must carry the shared digit z, which means any cell seeing both ends loses z — the wing is just the shortest possible XY-Chain.