XY-Chains
Learn the XY-Chain: a chain of bi-value cells linked by shared digits whose two ends carry a common digit Z, eliminating Z from any cell seeing both ends.
An XY-Chain is a chain of bi-value cells, each linked to the next by a shared digit. Walk the chain and each cell "flips" its value to the other candidate, passing the link along: {Z,a} to {a,b} to {b,c} and so on, until the final cell reads {…,Z} again. Both ends carry the same digit Z.
Whatever the first cell turns out to be, the chain forces one of the two endpoints to equal Z, so any cell that can see both ends cannot be Z, and you remove it. The XY-Wing is simply the shortest XY-Chain, three cells long.
XY-Chains reach deep eliminations on the hardest boards without guessing, which is what makes them so versatile. Walk the chain in the example cell by cell and you'll see why one end has to be Z.
Practise the XY-Chain
The best way to learn a technique is to use it. Play a puzzle at the level where it first appears, or drop a tricky board into the solver to watch it in action.
Frequently asked questions
What is an XY-Chain in Sudoku?
A chain of bi-value cells linked by shared candidates, with both endpoints holding a common digit Z. One endpoint must be Z, so any cell seeing both ends loses Z.
How long can an XY-Chain be?
Any length. Three cells is an XY-Wing; longer chains of four, five or more bi-value cells reach eliminations that shorter wings cannot.
Is an XY-Chain the same as an X-Chain?
No. An X-Chain follows a single digit through strong and weak links; an XY-Chain hops between digits through bi-value cells. Both prove one of two endpoints holds a digit.
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