Peak Sudoku

XY-Chain

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

Tier 4
Fish, wings, chains & uniqueness
Typical difficulty
Peak Score 70–85
Also known as
Bivalue Cell Chain, Y-Chain

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.

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.

Worked examples

1 / 5
Reading the exampleCells forming the patternCells losing candidatesEliminated candidates

Suppose R1C1 is not 2.

  1. Then R1C1 is 9, since only 2 and 9 are left there.
  2. So R5C1 is 3, since it sees R1C1 and only 3 and 9 are left there.
  3. So R8C1 is 2, since it sees R5C1 and only 2 and 3 are left there.
  4. So R8C9 is 8, since it sees R8C1 and only 2 and 8 are left there.
  5. So R8C8 is 3, since it sees R8C9 and only 3 and 8 are left there.
  6. So R9C8 is 1, since it sees R8C8 and only 1 and 3 are left there.
  7. So R2C8 is 8, since it sees R9C8 and only 1 and 8 are left there.
  8. So R2C5 is 2, since it sees R2C8 and only 2 and 8 are left there.

Either R1C1 or R2C5 is 2, so 2 can be removed from any cell that sees both.

Removes: 2 from R2C2

How to apply it

Spotting the pattern and making the deduction, in order.

  1. Start at a two-candidate cell (e.g. with candidates 2 and 6) and follow a chain of bivalue cells that share candidates.
  2. Notice 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.
  3. Trace the chain until it reaches an ending cell that finishes on the original digit 2.
  4. Either the starting cell is 2, or the chain reaction forces the ending cell to be 2.
  5. At least one of the two endpoint cells must be 2.
  6. Eliminate candidate 2 from any cell in the grid that sees both endpoints of the chain.

Common mistakes

  • 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.
Peak Score 55–70XY-WingThree bivalue cells – a hinge and two pincers.Peak Score 75–90Nice Loop (AIC)An alternating chain of strong and weak links – open or closed.Peak Score 90–100Forcing ChainBoth branches from a bivalue cell reach the same conclusion.

Put XY-Chain into practice

Try a puzzle from the Expert tier (Peak Score 71–80).

Play a Expert puzzleAll 25 techniques