Advanced Sudoku Strategy
XY-Chain Sudoku: How to Find and Use XY-Chains
Learn how to connect cells with exactly two candidates, follow a chain of implications, and eliminate a candidate that sees both endpoints.
One Chain, Two Endpoints
Each cell has two candidates. Consecutive cells see each other and share a candidate. At least one endpoint must contain the shared target number.
Find bivalue cells
Choose cells with exactly two candidates.
Follow the links
Connect cells that see each other and share a candidate.
Eliminate
Remove the endpoint candidate from cells seeing both ends.
What Is an XY-Chain in Sudoku?
An XY-Chain is an advanced Sudoku technique made from bivalue cells —unsolved cells containing exactly two candidates.
Consecutive cells must see each other (share a row, column, or 3×3 box) and share a candidate. Following the implications proves that at least one endpoint contains a particular number.
Any outside cell that sees both endpoints cannot contain that number. XY-Chains usually eliminate candidates rather than place numbers directly.
The XY-Chain Rule
Start with a cell containing X and Y. Connect it to a bivalue cell sharing Y, then continue through linked bivalue cells until the final cell also contains X. Each interior cell uses one candidate to enter the chain and the other to leave it.
If the first endpoint is not X, the chain forces the last endpoint to be X. If the first endpoint is X, the conclusion is already satisfied. Therefore at least one endpoint is X .
Worked XY-Chain Example: Remove Candidate 1
This is an illustrative candidate configuration. Each chain cell has exactly two candidates; other cells in the puzzle are omitted for clarity.
Why this works
- A and B share candidate 2 in row 1.
- B and C share candidate 3 in column 5.
- C and D share candidate 4 in row 4.
- If A is not 1, A must be 2, forcing B = 3, C = 4, and D = 1.
- If A is 1, the first endpoint already contains the target number.
Eliminate 1 from R1C8
R1C8 sees A (R1C1) through row 1 and D (R4C8) through column 8. Since at least one endpoint must contain 1, R1C8 cannot contain 1 . Remove candidate 1 there if present.
How to Find XY-Chains Step by Step
Turn On Candidate Notes
Use accurate pencil marks to identify bivalue cells.
Choose a Starting Cell
Pick a cell containing exactly two candidates, X and Y.
Find a Linked Bivalue Cell
Find another bivalue cell that sees the first and shares Y.
Follow the Other Candidate
Continue through connected cells, alternating which candidate enters and leaves each cell.
Find a Matching Endpoint
Finish at a cell that contains the original target candidate X.
Check Both Endpoints
Look for an outside cell that sees the first and last chain cells.
Eliminate and Rescan
Remove X from the outside cell and look for new singles, pairs, or other patterns.
What Does “See Both Endpoints” Mean?
A cell sees another cell if they share a row, column, or 3×3 box . The elimination target must see both ends of the chain, not merely one.
For example, R1C8 sees R1C1 through row 1 and R4C8 through column 8. This is why the example eliminates candidate 1 from R1C8.
Do not remove a candidate just because a cell lies near the chain visually. Always verify both Sudoku-unit relationships.
XY-Chain vs. XY-Wing
Already comfortable with three-cell patterns? Review our XY-Wing Sudoku guide before practising longer chains.
XY-Chain vs. Simple Coloring
Both techniques use logical chains, but they track different relationships.
XY-Chain
Uses several different candidate numbers through bivalue cells.
Simple Coloring
Follows strong links for one candidate number and alternates two colors.
Learn the single-candidate approach in our Simple Coloring Sudoku guide .
When Should You Look for XY-Chains?
Several Bivalue Cells
A grid with many two-candidate cells offers more opportunities for linked chains.
Simpler Techniques Stall
Try singles, pairs, Locked Candidates, and XY-Wings before building long chains.
Hard and Expert Puzzles
XY-Chains are useful when direct placements become difficult.
Two Ends Share a Candidate
A useful chain needs endpoints with the same target candidate and an outside cell seeing both.
Common XY-Chain Mistakes
Using a Cell With Three Candidates
Standard XY-Chains require exactly two candidates in each chain cell.
Linking Cells That Cannot See Each Other
Consecutive chain cells must share a row, column, or box.
Reusing the Wrong Candidate
At an interior cell, enter through one candidate and leave through the other.
Eliminating From a Cell Seeing Only One End
The elimination target must see both endpoints.
Mixing Separate Chains
Only use candidates connected by a valid continuous chain.
Assuming the Endpoint Value Without Proof
The chain proves at least one endpoint contains the target; it does not identify which one without more information.
XY-Chain Practice Checklist
- Every chain cell has exactly two candidates.
- Every consecutive pair of cells sees each other.
- Each link shares the candidate required by the implication.
- The chain alternates through both candidates of each interior cell.
- Both endpoints contain the same target candidate.
- The elimination target is outside the chain and sees both endpoints.
- You remove only the common endpoint candidate.
- You rescan for simpler techniques after eliminating.
Related Sudoku Learning Pages
XY-Wing Sudoku
Learn the three-cell pivot and pincer pattern.
Simple Coloring Sudoku
Track one candidate through alternating strong links.
Skyscraper Sudoku
Explore a compact strong-link elimination pattern.
Unique Rectangle Sudoku
Understand rectangle-based candidate eliminations.
Advanced Sudoku Techniques
Explore the complete advanced solving technique collection.
Locked Candidates Sudoku
Strengthen candidate elimination fundamentals.
Frequently Asked Questions
What is an XY-Chain in Sudoku?
An XY-Chain connects bivalue cells through shared candidates. Its endpoints guarantee that at least one contains the target candidate.
How do you find an XY-Chain?
Follow linked bivalue cells through shared candidates until the first and last cells contain the same target candidate.
What can an XY-Chain eliminate?
Remove the shared endpoint candidate from an outside cell that sees both endpoints.
How is XY-Chain different from XY-Wing?
XY-Wing uses a three-cell pattern; XY-Chain uses a connected sequence of bivalue cells.
Do XY-Chains require bivalue cells?
Yes. The standard technique uses cells with exactly two candidates.
Practise XY-Chains
Open an Expert Sudoku puzzle, turn on Notes, and find connected bivalue cells. Look for a chain with matching endpoint candidates and a valid elimination target.