Peak Sudoku
Fish, wings, chains & uniquenessPeak Score 70–85

XY-Chain

A chain of bivalue cells that forces a digit at both endpoints.

Also known as: Bivalue Cell Chain, Y-Chain

Quick Summary

An XY-Chain is a sequence of two-candidate cells where adjacent cells share a candidate. If the start cell does not take a digit, a domino effect forces the end cell to take it, eliminating that digit from any cell seeing both ends.

Understanding the XY-Chain Technique

An XY-Chain generalizes the XY-Wing to chains of arbitrary length. Every link in the chain is a bivalue cell (a cell with exactly two candidates). If the starting cell does not take the starting digit, its second candidate forces the next cell in the chain, which forces the next, all the way to the final cell, which is forced to take that original digit. Since either the starting cell or the ending cell must take that number, any cell in the grid that sees both endpoints can never contain that candidate.

Visual Examples

Green cells highlight the XY-Chain pattern, amber cells denote where candidates are affected, and struck-through pencil marks show the eliminations. Use the buttons or arrow keys to browse examples.

Example 1 of 5
A chain of bivalue cells links R1C1 to R9C4: R1C1 → R5C1 → R8C1 → R8C9 → R8C8 → R9C8 → R9C4. Each adjacent pair shares one candidate, so if R1C1 isn't 2 the links force R9C4 to be 2. Either way one end holds 2, so any cell that sees both ends can drop 2.
Removes: 2 from R1C4

How to Spot & Solve a XY-Chain (Step-by-Step Guide)

The complete visual scanning and logical deduction process to execute this technique:

  • 1Start at a two-candidate cell (e.g. with candidates 2 and 6) and follow a chain of bivalue cells that share candidates.
  • 2Notice the domino effect: if the starting cell is not 2, it is forced to be 6, which forces the next cell's value, and so on along the chain.
  • 3Trace the chain until it reaches an ending cell that finishes on the original digit 2.
  • 4Either the starting cell is 2, or the chain reaction forces the ending cell to be 2.
  • 5At least one of the two endpoint cells must be 2.
  • 6Eliminate candidate 2 from any cell in the grid that sees both endpoints of the chain.

Common Traps & Mistakes to Avoid

  • Accidentally using a cell with three candidates in an XY-chain. Every cell in the chain MUST be strictly bivalue.
  • Linking two cells that do not share a house (row, column, or box).
  • Ending the chain on a different digit than the starting digit.

Frequently Asked Questions

How long can an XY-Chain be?

An XY-Chain can be as short as 3 cells (which is just an XY-Wing!) or can span 8, 10, or more cells across the entire grid.

Expand your repertoire by learning prerequisites and advanced counterparts:

XY-WingPeak Score 55–70

Three bivalue cells – a hinge and two pincers.

Read guide →
Nice Loop (AIC)Peak Score 75–90

An alternating chain of strong and weak links – open or closed.

Read guide →
Forcing ChainPeak Score 90–100

Both branches from a bivalue cell reach the same conclusion.

Read guide →

Ready to Practice XY-Chain?

Try a puzzle in the Expert tier (Peak Score 71–80), or browse all 25 techniques.

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