X-Chain in Sudoku: Single-Digit Chains for Beginners

An X-Chain is a chain of candidates for one digit that alternates between strong and weak links, starting and ending with a strong link. One of the two ends must hold the digit, so any cell that sees both ends can lose it. To spot one, pick a digit, find the rows, columns and boxes where it has only two candidates, and try to join those pairs end to end.

Chains sound like expert territory, but an X-Chain only needs two ideas. If you can follow "if this cell is not a 3, that one must be", you can use it.

Strong and weak links, in plain words

Every chain is built from links between two candidates of the same digit.

• Strong link: a row, column or box where the digit has exactly two candidates. One of the two must be the digit. If one is false, the other is true.
• Weak link: two candidates that see each other (same row, column or box). They cannot both be the digit. If one is true, the other is false.

A chain alternates the two. Start at one end and assume it is false. The strong link makes the next cell true. The weak link makes the cell after that false. The next strong link makes the one after that true again, and so on. If the chain ends on a strong link, the far end comes out true. So whenever the first end is false, the last end is true: at least one of the two ends holds the digit.

You have probably met the shortest X-Chains already. The skyscraper and the 2-String Kite are both strong, weak, strong. An X-Chain is the same move with more links in the middle.

A worked example on the digit 3

This position comes from a real puzzle in our library. Only the 3s are shown. The chain has three strong links joined by two weak links.

Walk it link by link, starting from R1C8 and assuming it is not a 3.

1. R1C8 to R1C1, strong (row 1). Row 1 has a 3 candidate only in R1C1 and R1C8. If R1C8 is not 3, R1C1 is.
2. R1C1 to R2C1, weak (column 1). They share column 1, so R2C1 is not 3.
3. R2C1 to R2C6, strong (row 2). Row 2 has 3 only in R2C1 and R2C6. With R2C1 out, R2C6 is 3.
4. R2C6 to R7C6, weak (column 6). They share column 6, so R7C6 is not 3. Column 6 has a third 3 candidate in R8C6, which is why this link is only weak.
5. R7C6 to R7C9, strong (row 7). Row 7 has 3 only in R7C6 and R7C9. So R7C9 is 3.

If R1C8 is a 3, fine. If it is not, R7C9 is. Either way one of the two ends is a 3.

Now find the cells that see both ends. R3C9 shares box 3 with R1C8 and column 9 with R7C9. R8C8 and R9C8 share column 8 with R1C8 and box 9 with R7C9. All three lose their 3.

The weak link from R1C1 to R2C1 happens to be strong too, since column 1 has only those two 3s. That is fine: a strong link can always stand in for a weak one, because two candidates that are the only two in a house also see each other. The reverse never works.

Puzzle 201130 from the Sudoku Royale library. Givens (row by row, dots are blanks): .....8....1..2.56..5.7..2.........92..1.653....93...7.27681....4.................. The candidates shown are what is left after singles, pointing and claiming, naked and hidden pairs, triples and quads, X-Wing and Swordfish, with nothing else applied.

How to find X-Chains

1. Pick one digit and map its strong links. Go through the rows, columns and boxes and note each one where the digit has exactly two candidates. On paper, a small line between the two cells works.
2. Join links through weak links. From the end of one strong link, look for a candidate it sees that starts another strong link. Keep going.
3. Check after every strong link. Each time the chain ends on a strong link, look for cells that see both ends. You can stop as soon as one has the digit as a candidate.
4. Keep chains short. Most useful X-Chains have three or four strong links. Longer chains work but are easy to misread.

Chains with an even number of links that loop back to the start are a different case, called a continuous loop. They clear the digit from every house along a weak link. They are rare, and you do not need them to use ordinary X-Chains.

X-Chain vs simple colouring

Simple colouring uses the same strong links on one digit but joins only strong links, giving alternate cells two colours. An X-Chain can also cross a weak link, which lets it reach cells a colouring cannot. In practice they find many of the same eliminations, and learning either one makes the other easier.

When you move from one digit to two digits per cell, you get the XY-Chain. Mixing both kinds of links on several digits gives 3D Medusa and, beyond that, alternating inference chains.

For the full order, see our list of sudoku techniques in learning order and the overview of advanced sudoku strategies.