X-Chain

An X-Chain follows a single digit along a chain of links that alternate between strong and weak. It generalises simple colouring and the X-Wing into one flexible tool, and it can reach eliminations that neither of those can.

Strong and weak links

Two kinds of link connect cells for a single digit:

• A strong link is an either/or: a unit where the digit has exactly two homes. If one is off, the other must be on.
• A weak link is a not-both: two cells sharing a unit where the digit appears (possibly among others). They can't both be on, but they could both be off.

An X-Chain alternates them — strong, weak, strong, weak — starting and ending on a strong link. Follow the alternation and the two ends of the chain have a fixed relationship: at least one of them must be the digit.

The elimination

Because at least one end of the chain holds the digit, any cell that can see both ends cannot hold it — whichever end is true, this cell is attacked. Delete the digit from every such cell. It's the same conclusion as an X-Wing, but reached along a chain that can bend across the whole grid rather than sitting in a neat rectangle.

Step by step

1. Choose a digit and identify its strong links (two-home units).
2. Build an alternating chain — strong, weak, strong… beginning and ending strong.
3. Take the two ends. One of them must be the digit.
4. Eliminate the digit from any cell seeing both ends.

Where it fits

X-Chains are single-digit chains; the XY-Chain is their bi-value cousin that moves between digits. Together with colouring they form the backbone of advanced manual solving, short of the full forcing chains.

Common mistakes

• Not alternating. The links must strictly alternate strong/weak, and both ends must be strong.
• Only one end seen. The elimination needs cells that see both ends of the chain.

Which Sudoku types is this best for?

The same logic applies across variants, but it pays off more in some than others.