XY-Chain

Expert9 min

Prerequisites: XY-Wing

Overview

An XY-Chain is a line of two-candidate cells, each seeing the next. If the first cell isn't Z and that forces each cell in turn until the last cell must be Z, then one of the two ends is Z, so any cell that sees both ends can't be Z.

How It Works

Every cell in the chain has exactly two candidates, and each cell sees the next one and shares a candidate with it: a different candidate from the one it shares with the cell before. Pick a digit Z that both ends hold. Suppose the first cell isn't Z: then it is its other digit, which the next cell can't be, so that cell is its other digit, and so on along the chain until the last cell is forced to Z. So either the first cell or the last cell is Z, and any cell that sees both ends can't be Z. An XY-Wing is the shortest XY-Chain, with three cells.

When To Use It

Use an XY-Chain when there are many two-candidate cells but no XY-Wing. Chains of four or five cells turn up often in Expert puzzles.

How To Detect It

Pick a two-candidate cell and one of its digits, Z. Suppose the cell isn't Z and follow the forced values: each step goes to a two-candidate cell it sees that holds the digit just forced, so that cell takes its other digit. When a cell is forced to Z, stop: remove Z from every cell that sees both the start and that cell.

Visual Example

Step 1 of 10

A Chain of Two-Candidate Cells

An XY-Chain is a line of cells with exactly two candidates each. Each cell sees the next and shares one candidate with it, a different one from the candidate it shares with the cell before. An XY-Wing is a chain of three such cells; an XY-Chain is the same idea, kept going.

Where to look

Pattern cells

Eliminations

Removed

Ruled out here

7Solved

Key candidates

This step's candidates

Supposed, not known

If this, then that

From an earlier step

Candidates already ruled out by earlier techniques aren't shown.

Practice

Your goal: fill the highlighted cell. Select a cell, then pick a number.

An XY-Chain is hiding here. Follow it from one end to the other, eliminate, and fill the Naked Single.Candidates already ruled out by earlier techniques aren't shown.

Use it in a real game

Expert puzzles are where you'll need techniques like this one.

Play an Expert puzzle