X-Chain
An X-Chain follows a single digit through the grid and joins its candidates with two kinds of links. A strong link joins the only two places for the digit in a row, column or box: if one of them is not the digit, the other is. A weak link joins two candidates of the digit that see each other: if one of them is the digit, the other is not. An X-Chain alternates the two kinds and starts and ends with a strong link, so whichever way the digit is placed, one of its two ends holds it. Every cell that sees both ends can lose the digit. The Skyscraper and the 2-String Kite are the shortest X-Chains, with three links. The X-Chain is a Master technique in Bare Sudoku.
When to use it
When the single-digit patterns, such as the Skyscraper, the 2-String Kite and the Empty Rectangle, find nothing, but some digit has many rows, columns or boxes where it can go in only two places. Those pairs are the strong links that chains are built from.
How to spot it
Choose a digit and mark every unit where it has exactly two candidates; those are strong links. Start at one candidate, take a strong link to its partner, then a weak link to any candidate of the digit that the partner sees, then a strong link again, and so on. Each time you finish a strong link, look at the cells that see both the starting cell and the current one: if any of them holds the digit, it can be removed. Short chains are easier to check, and Bare Sudoku always looks for the shortest one.
Example
| c1 | c2 | c3 | c4 | c5 | c6 | c7 | c8 | c9 | |
|---|---|---|---|---|---|---|---|---|---|
| r1 | 189 | 159 | 6 | 3 | 89 | 7 | 589 | 2 | 4 |
| r2 | 7 | 4 | 358 | 59 | 2 | 1 | 3589 | 6 | 358 |
| r3 | 2 | 59 | 358 | 569 | 689 | 4 | 1 | 39 | 7 |
| r4 | 49 | 7 | 45 | 8 | 1 | 56 | 23569 | 39 | 2356 |
| r5 | 3 | 569 | 2 | 4 | 69 | 569 | 7 | 8 | 1 |
| r6 | 189 | 1569 | 158 | 7 | 3 | 2 | 569 | 4 | 56 |
| r7 | 5 | 2 | 9 | 1 | 4 | 68 | 368 | 7 | 368 |
| r8 | 14 | 3 | 14 | 69 | 7 | 689 | 268 | 5 | 268 |
| r9 | 6 | 8 | 7 | 2 | 5 | 3 | 4 | 1 | 9 |
Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.
- Follow digit 9 along the chain r4c8 = r3c8 - r2c7 = r2c4 - r8c4 = r8c6. Each = is a strong link: the two cells are the only places for 9 in a row, column or box. Each - is a weak link: the two cells see each other.
- Suppose r4c8 is not 9. The first strong link then makes the next cell 9, the weak link after it rules 9 out of the cell after that, and so on, until the last cell r8c6 is 9.
- So 9 is in r4c8 or in r8c6, possibly in both.
- A cell that sees both r4c8 and r8c6 cannot be 9. Remove 9 from r4c6.
Common mistakes
- Treating two cells as a strong link when the digit has a third place in their unit. A strong link needs the two cells to be the only places for the digit in that row, column or box.
- Ending the chain on a weak link. Both ends must be reached through strong links, or it does not follow that one end holds the digit.
- Removing the digit from a cell that sees only one end of the chain. The eliminated cell must see both ends.
In Bare Sudoku
Needed from level: Master
In the game, the hint says: X-Chain
Play this puzzle · Play Sudoku · Sudoku solver
Practice puzzles
In these puzzles, X-Chain is the hardest technique you need. Each link opens the puzzle in the game; the solver link shows every step.
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.5 · Sudoku solver
- Master 6.6 · Sudoku solver
Frequently asked questions
What is an X-Chain in Sudoku?
A chain of candidates of one digit that alternates strong and weak links and starts and ends with a strong link. One of the two ends must hold the digit, so cells that see both ends cannot.
What is the difference between a strong link and a weak link?
A strong link says that at least one of two candidates is true: they are the only two places for the digit in a unit. A weak link says that at most one of them is true: they are the same digit in cells that see each other. A strong link inside a unit also works as a weak link, but not the other way round.
How is an X-Chain related to the Skyscraper and the 2-String Kite?
Both are X-Chains with three links: two strong links joined by one weak link. The [Skyscraper](skyscraper) joins them through a shared row or column, the [2-String Kite](two-string-kite) through a box. Longer X-Chains follow the same logic over more links.
How long can an X-Chain be?
There is no fixed limit, but every link must be checked. Bare Sudoku looks for the shortest chain that removes something, and longer chains raise a puzzle's rating.
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