Chains

X-Chain

An X-Chain follows a single digit through the grid. A strong link joins two cells that are the only homes for the digit in some row, column or box, so one of them must hold it; a weak link joins two cells that see each other, so at most one of them does. Chain them in turn, strong, weak, strong, starting and ending on a strong link, and one of the two ends must hold the digit. Any cell that sees both ends can't.

Written by Georgi Nedyalkov

Level
Advanced

The pattern

X-Chain: what it does.

Read it from one end. If the first cell isn't the digit, its strong link puts the digit in the next cell; that cell sees the one after, so that one isn't the digit, and its strong link puts the digit in the next. The chain carries the conclusion to the far end.

So either the first end holds the digit, or the reasoning arrives at the last end and that one does. A cell that sees both ends loses the digit either way.

Some techniques on this site are short X-Chains: the skyscraper and the two-string kite are two strong links joined by one weak link. A longer chain reaches removals those shapes can't.

Example

X-Chain: a worked example.

  1. Choose a digit

    Follow one digit and mark its strong links: every row, column and box where it has exactly two homes.

  2. Start on a strong link

    Begin at one cell of a strong link and step to its partner. That partner is where the chain stands now.

  3. Alternate weak and strong

    From there, step to any cell that sees it and holds the digit, then along that cell's strong link to its partner. Repeat, never visiting a cell twice.

  4. Clear what sees both ends

    Whenever the chain stands at the end of a strong link, check the cells that see both that end and the first cell. Remove the digit from them.

Spotting it

X-Chain: how to spot it.

  • Draw the digit's strong links first. A chain is a path through that drawing, and with the links in front of you it's much easier to see.
  • A chain always has an even number of cells, since it starts and ends with a strong link. Two strong links make the shortest chains, such as the skyscraper and the kite; longer ones are where X-Chains earn their name.
  • A weak link only needs the two cells to see each other. It can even be a strong link used as a weak one.
  • Look at the target before the path. A cell holding the digit that sees two distant candidates for it is a hint that a chain might join them.

Where it stops working

An X-Chain follows one digit only, so it can't use what the other digits say, and long chains are easy to misread. Every strong link has to be real: a unit with a third, forgotten home for the digit breaks the link, and the removal stops being safe.

Keep going

Related topics.

New to sudoku? The rules take two minutes to read, and the first-puzzle walkthrough covers the basics that come before any of these.

Beyond the lessons.

The game's lessons stop at X-Wing, so there's no X-Chain lesson. The eight it does teach will get you through almost any puzzle; this page is here for when you run into the pattern anyway.

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