X-Chain
X-Chain is a single-digit chain of three or more nodes connected by alternating strong and weak links. Unlike X-Cycles, an X-Chain doesn't close into a loop — it has two endpoints, and the eliminations live on cells that see both endpoints.
Note: This technique uses chain colouring to display candidates. Square (▢) and Diamond (◇) shapes represent the two colour partitions.
The Pattern
An X-Chain on digit d is a sequence of cells where:
- Every cell has d as a candidate
- Adjacent cells alternate strong → weak → strong → weak → … links
- Both endpoints sit on strong links (this is what makes the chain "discontinuous" rather than a continuous loop)
A ══ B ── C ══ D (endpoints A and D, both connected by strong links)
The minimum length is three nodes (one strong + one weak + one strong); two-node single-digit chains are classified as Skyscraper, Two-String Kite, or Turbot Fish depending on the unit types involved.
The Elimination Rule
Both endpoints A and D carry the same digit d. The chain proves that one of A or D must be d. Any cell that sees both A and D therefore cannot be d — it would conflict with whichever endpoint takes the digit.
Eliminate d from every cell that shares a unit (row, column, or box) with both endpoints.
How to Spot an X-Chain
- Pick a single digit that has several conjugate pairs across the grid.
- Build the strong-link skeleton — every conjugate pair for that digit is a candidate strong link.
- Look for weak hops — two strong links can be bridged by a weak link if they share a unit (the linking cells share a row, column, or box).
- Extend the alternating chain until both ends sit on strong links.
- Check the endpoints' shared peers — any cell those endpoints both see is an elimination target.
Example
This puzzle reaches an X-Chain on digit 8 after a few early placements. The chain is:
R6C9 ══ R3C9 ── R3C2 ══ R9C2 ── R7C1 ══ R7C4
Five links, six cells. Endpoints R6C9 and R7C4 both carry digit 8 on strong links. R6C4 sees R6C9 (row) and R7C4 (column + box), so 8 is eliminated from R6C4.
puzzle: .3.2....62..19..5.5....83.......1.4...768..........5....5..6731..3........2.7.6.4
mode: free
technique: X-Chain
settings:
showCandidates: true
Open the puzzle, enable hints, and run the v2 engine — the X-Chain step appears after the initial Hidden Singles and Naked Singles clear out.
Relationship to Other Techniques
| Technique | Links | Pattern |
|---|---|---|
| Skyscraper | 2 strong + 1 weak (parallel lines) | Two-node X-Chain |
| Two-String Kite | 2 strong + 1 weak (perpendicular) | Two-node X-Chain |
| Turbot Fish | 2 strong + 1 weak (one in a box) | Two-node X-Chain |
| X-Chain | 3+ alternating, both ends strong | Open chain |
| X-Cycles | 3+ alternating, closes into a loop | Closed chain |
X-Chain and X-Cycles share the same single-digit alternating-link mechanism; the difference is whether the chain forms a loop.
Tips
- Trust the conjugate pairs — strong links on a single digit are the backbone.
- Hop through cells with two candidates of the digit in some unit — those are your weak links.
- Watch for short chains first — three-node X-Chains are the easiest to see and equivalent to "Skyscraper / Two-String Kite / Turbot Fish" patterns under a different name.
- Both endpoints must be on strong links — if a chain ends on a weak link, the elimination doesn't follow.
Related Techniques
- X-Cycles — Same pattern, but closed into a loop
- Skyscraper — Shortest X-Chain (parallel lines)
- Two-String Kite — Shortest X-Chain (perpendicular lines)
- Turbot Fish — Shortest X-Chain (through a box)
- Simple Colouring — Strong links only, no weak links