What is an XY-Chain?
An XY-Chain is a logical pattern that uses a chain of bivalue cells (cells with exactly two candidates) to eliminate one candidate from another cell. Each link in the chain connects two bivalue cells that share a candidate. If both ends of the chain contain the same candidate X but do not see each other, any other cell that sees both ends cannot be X and so X can be removed from that cell.
This technique is a parity-based deduction: along the chain each bivalue cell forces one of its two candidates to be true, and that choice propagates through the chain. If one end is X, the chain logic forces the other end to be not‑X, and vice versa. Because the ends are linked to the same candidate X, whichever end is actually X prevents X from appearing in any cell that sees both ends, allowing a safe elimination.
When to look for it
Look for XY-Chains when you have several bivalue cells left and eliminations by simpler techniques (singles, pairs, pointing, box/line reductions, and standard chains like XY-wings) are no longer producing progress. It’s most useful in medium-hard to hard puzzles where many cells have been reduced to two candidates. You should try it when:
- You can spot a sequence of bivalue cells that share candidates in overlapping houses (rows, columns or boxes).
- Two bivalue cells both contain the same candidate X but are not peers (they don’t share a row, column or box).
- There’s a third cell that sees both of those endpoints and still contains candidate X.
How to build an XY-Chain (step-by-step)
1. List the bivalue cells. Find cells with exactly two candidates — these are the only cells that can form a valid XY-Chain.
2. Pick a starting bivalue cell. One of its two candidates will be the first link.
3. Find a neighboring bivalue cell that shares one of those two candidates and sits in the same house (row, column, or box). That shared candidate becomes the link between the two cells.
4. Continue building the chain by moving to another bivalue cell that shares a candidate with the last cell in the chain. Alternate shared candidates as you go.
5. Stop when you reach a bivalue cell that contains the same candidate X as the first cell but is not in the same house (it does not see the first cell). Those two cells are the chain endpoints.
6. Any cell that sees both endpoints cannot be X. Remove X from such cells.
Worked example (follow the candidate-grid below)
In the diagram that follows, the chain runs through four bivalue cells. The cells are:
- r1c2 = {5,7}
- r1c5 = {7,2}
- r3c5 = {2,9}
- r3c1 = {9,5}
The endpoints are r1c2 and r3c1; both contain candidate 5 and do not see each other (they are not in the same row, column, or box). The chain links are: 5–7 between r1c2 and r1c5, 7–2 between r1c5 and r3c5, and 2–9 between r3c5 and r3c1. Because of the alternating shared candidates, if r1c2 were 5 then r1c5 cannot be 7 and must be 2; that forces r3c5 to be 9 and then r3c1 cannot be 9 and must be 5. The opposite assumption (r1c2 not 5) leads to r3c1 not 5. Either way, r3c1 is forced to be the opposite truth value to r1c2 with respect to candidate 5. Since both endpoints include 5, any cell that sees both endpoints cannot be 5.
In the example, r2c1 sees both endpoints (it shares the left box with r1c2 and the column with r3c1). If r2c1 currently has candidate 5, that candidate can be eliminated safely.
Practical tips and common pitfalls
- Only bivalue cells are allowed in the chain. A cell with three or more candidates breaks the alternating logic.
- Links must be peer-to-peer: two cells form a valid link when they share a candidate and lie in the same row, column, or box (so the truth of a candidate in one forces the other candidate in the neighbor).
- The endpoints must not see each other. If they do see each other, the chain becomes a closed loop and you might be back to other techniques (or nothing new).
- Remember to check every cell that sees both endpoints; sometimes multiple eliminations are possible.
- Keep your candidate list tidy. XY-Chains are easier to spot when candidates are written in cells so you can trace links visually.
- Start with short chains. Long chains are harder to verify and more prone to mis-tracing. If you can find a 3–5 link chain that does the job, that is usually better than hunting for a long chain.
Summary
XY-Chains are a powerful, purely logical technique based on bivalue propagation. They are most useful late in a solve when many cells are down to two candidates. Build a chain of bivalue cells linked by shared candidates, find endpoints that contain the same candidate but don’t see each other, and eliminate that candidate from any cell that sees both endpoints. The included candidate-grid shows a clear, short XY-Chain and the single safe elimination it produces — study it and try spotting similar chains in your next medium to hard puzzle to add this technique to your solving toolbox.