Chains

XY-Chain

An XY-Chain is a line of cells that each hold exactly two candidates, where each cell sees the next and shares a digit with it. Suppose the first cell isn't the digit Z: then it's its other digit, which forces the next cell to its other digit, and so on down the chain until the last cell is forced to Z. So one of the two ends is always Z, and Z can be removed from any cell that sees both ends.

Written by Georgi Nedyalkov

Level
Advanced

The pattern

XY-Chain: what it does.

Each step is forced. A two-candidate cell that loses one digit must be the other, and a cell that sees it then loses that digit. The chain is that argument, repeated.

Only the ends matter for the removal. The chain proves that the first cell or the last cell is Z, so a cell that sees both can't be Z, however long the chain between them is.

An XY-Wing is the shortest XY-Chain: three cells, with the pivot in the middle. A longer chain finds the same kind of removal when the two ends are too far apart for one pivot to join them.

Example

XY-Chain: a worked example.

  1. Start from a pair

    Pick a cell with two candidates and choose one of them as Z, the digit the chain will try to prove. Leave the cell through its other digit.

  2. Pass the digit on

    Find a two-candidate cell that sees the current one and holds the digit you left through. Enter it on that digit and leave it through its other one.

  3. Stop when you reach Z

    When the digit you'd leave a cell through is Z, the chain can end there. It needs at least four cells to be more than an XY-Wing.

  4. Clear Z from what sees both ends

    Remove Z from every cell that shares a unit with both the first cell and the last. If there are none, carry the chain on or try another start.

Spotting it

XY-Chain: how to spot it.

  • Mark the two-candidate cells first. A chain is made of nothing else, and a grid with plenty of pairs is where chains turn up.
  • Write the chain down as you go: each cell and the digit you leave it through. Long chains are easy to follow forwards and easy to lose.
  • Look at where the ends could meet before you build. A chain whose ends share no cell they both see gives nothing, however neat it is.
  • A chain can't visit the same cell twice. If it loops back, start again from a different cell.

Where it stops working

An XY-Chain uses only two-candidate cells, so a grid with few of them has few chains, and it removes one digit from the cells both ends see. It's also among the easiest techniques to get wrong: one stale candidate anywhere along the chain breaks the argument and makes the removal false.

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 XY-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.

Pre-order on the App Store

Out 1 December

Thanks. You'll get one email on launch day.

One email on launch day, then your address is deleted.Privacy