XY-Chain

Level 10 · Brown Belt

Strategy:

Multi-Digit Chains

Overview

XY-Chain links bivalue cells (cells with exactly 2 candidates) through shared candidates. Each consecutive pair of cells shares a candidate, creating alternating strong and weak inferences that force eliminations.

How It Works

  1. Find bivalue cells that share a candidate with neighboring bivalue cells.
  2. Build a chain where consecutive cells share one candidate, alternating which is the shared digit.
  3. The first and last cells must share a common candidate on their strong links.
  4. Eliminate that common candidate from cells seeing both endpoints.

Example

Starting position

Look for XY-Chain.

1

Found an XY-Chain of 3 cells

Found an XY-Chain of 3 bivalue cells from r1c7 to r2c3.

2

How the XY-Chain links bivalue cells

Each cell has exactly two candidates. Consecutive cells share one candidate, creating alternating links.

3

One endpoint must be 2

Both endpoints r1c7 and r2c3 have candidate 2 on a strong link. One of them must be 2.

4

Cells seeing both endpoints can't keep 2

Any highlighted cell that sees both endpoints cannot keep 2, because one endpoint must contain it.

5

Remove 2 from common peers

Remove 2 from cells seeing both endpoints: r1c1 and r2c9.

6

Result

Result after applying the technique.

Tips & Patterns

  • XY-Chains extend XY-Wing to longer chains.
  • Every cell in the chain must have exactly 2 candidates.
  • The key insight: the common candidate at the endpoints must be in at least one endpoint, so cells seeing both can't have it.