XY-Chains in Sudoku: A Step-by-Step Guide

The structure of an XY-Chain

Every cell in an XY-Chain is bivalue, meaning it has exactly two candidates. Consecutive cells must see each other and share one candidate, but the chain passes through a different candidate at each step. A simple sequence might be AB, BC, CD, and DA. The first and last cells share candidate A, while the internal links pass through B, C, and D. The endpoints do not need to see each other. Instead, the deduction targets any outside A candidate that sees both endpoints. In Shapedoku terms, the first cell might contain heart or circle, the next circle or crescent, the next crescent or diamond, and the last diamond or heart. Each link works because if the incoming candidate is false, the other candidate in that bivalue cell must be true. The candidate choices therefore alternate through the chain. Accurate Notes are essential, since one hidden third candidate would break the forced relationship inside that cell.

Why the endpoint elimination is valid

Consider the first endpoint, which contains A and B. If A is true there, any outside A candidate seeing that endpoint is false. If A is false, B must be true. That forces B false in the next linked cell, so C becomes true. The logic continues through every bivalue cell until the final endpoint is forced to A. In other words, either the first endpoint is A or the last endpoint is A. At least one endpoint must contain A, even though you do not know which one. Any outside A candidate that sees both endpoints is therefore impossible. It would conflict with the first endpoint in one branch and with the last endpoint in the other. This is not guessing because both possible states of the starting cell produce the same elimination. The chain proves only what both branches share. It does not allow you to choose an endpoint or place any internal candidate without additional evidence.

How to search without losing the chain

Begin with bivalue cells whose candidates repeat in nearby bivalue cells. Pick one candidate as a possible endpoint value, then follow the other candidate to a cell that sees the first and contains that same symbol. Continue by leaving each new cell through its second candidate. Keep the chain short while learning. Four or five cells are easier to verify than a long route crossing the whole board. Stop when you reach another cell containing the original endpoint candidate. Then look for an outside candidate of that shape that sees both ends. A chain with no common target is valid but useless, so do not force an elimination. On an Extreme Shapedoku board, use Notes to trace the candidate pairs and recheck every shared unit before deleting anything. After one successful elimination, return to ordinary scanning. XY-Chains often matter because a single removal creates a hidden single or breaks a stubborn pair.

1. Find a bivalue cell and choose one candidate as the endpoint value.
2. Follow the other candidate to a seeing bivalue cell that contains it.
3. Continue through linked bivalue cells by alternating candidates.
4. End at a cell containing the original endpoint value.
5. Find an outside candidate of that value that sees both endpoints.
6. Remove only that shared target, then verify every link again.