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XY-Chain example showing elimination of 9 at R9C5
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XY-Chain

Open Chains Through Bi-Value Cells

By Minimal Sudoku TeamLast updated:

XY-Chain extends Y-Wing logic to longer paths. You connect bi-value cells into a chain and use the chain to force eliminations.

Quick Summary
  • What: A chain of bi-value cells where neighbors share one candidate
  • Main Rule: If endpoints share candidate X and the chain guarantees at least one endpoint is X, remove X from cells seeing both endpoints
  • Result: Open-chain endpoint elimination
  • Difficulty: Expert — best after Y-Wing

When to Use XY-Chain

Use XY-Chain when easier techniques stop working and your grid has many cells with exactly two candidates.

Practical trigger:
"Can I connect two same-digit endpoints through a valid bi-value chain?"

The Concept

An XY-Chain is a path of bi-value cells (cells with exactly two candidates). Each link in the path must share one candidate, and linked cells must see each other (same row, column, or box).

  • Candidate: a penciled-in possible digit.
  • Endpoint: the first or last cell of a chain.

The Golden Rule

🔗 XY-Chain Rule
If an open chain has endpoints that share candidate X, and the chain guarantees at least one endpoint is X, then any other X that sees both endpoints can be removed.

Chain length alone does not make a chain valid. Correct if/then logic does.

Chain Structure

Typical XY-Chain shape:
XY-Chain structure with matching endpointsA chain of bi-value cells where neighboring cells share one candidate. The first and last cells share candidate A.A,BStartB,CC,DY,AEndEndpoints share candidate ARemove A from any cell that sees both endpoints
Neighboring cells share one candidate. When the start and end cells share the same candidate, endpoint eliminations become available.

How to Apply XY-Chains

1

Find bi-value cells

Identify cells with exactly two candidates.
2

Build links

Connect cells that see each other and share one candidate.
3

Extend the chain

Continue until endpoints share a candidate.
4

Validate endpoints

Confirm the chain guarantees at least one endpoint must hold that candidate.
5

Eliminate

Remove the candidate from any cell that sees both endpoints.

Worked Example

XY-Chain on Candidate 9

XY-Chain path from R2C5 to R9C4 eliminating 9 at R9C5

Chain:
R2C5 [9/1] → R2C1 [1/8] → R9C1 [8/7] → R9C4 [7/9]

Endpoints R2C5 and R9C4 both include 9, and the chain guarantees at least one endpoint must be 9.

Elimination
R9C5 sees both endpoints and has (2,7,9), so remove 9.

Why It Works

XY-Chain uses either/or logic on the endpoint candidate:

  • If one endpoint is not X, the chain forces the other endpoint to X.
  • If one endpoint is X directly, the rule is already satisfied.

Therefore, at least one endpoint must be X, so any third cell that sees both endpoints cannot keep X.

Detection Tips

Start from Y-Wing
If Y-Wing feels comfortable, XY-Chain is the same logic on longer paths.
Map Bi-Value Cells First
Mark all bi-value cells before chaining. It reduces scan noise and makes valid links easier to see.
Most Common Error
Do not eliminate until you verify every link is valid and endpoints share the same target candidate.

Related Techniques

XY-Chain sits between Y-Wing and broader chain frameworks:

Chain Techniques

🦋Y-Wing⭕Closed XY-Chains (Part 2)🔄X-Cycle🎨Simple Coloring
Closed XY-Chains (Part 2)Expert

Closed XY-Chains (Part 2)

Learn what changes when XY-Chains close into loop behavior.