X-Cycles
X-Cycles are single-digit loops built from alternating strong and weak links; an X cycle in Sudoku works on one digit at a time. They pick up where Simple Colouring stops, because a cycle is allowed to travel through units where the digit has three or more places.
What the pattern is
Everything happens on one digit. Two kinds of link connect its candidate cells inside a shared unit:
- A strong link joins the only two places the digit has in that unit. If one is false the other is true.
- A weak link joins any two of its places in a unit that has three or more. If one is true the other is false.
A cycle is a closed path through those cells in which links alternate: strong, weak, strong, weak. Follow it around and the digit alternates true, false, true, false. Where that alternation holds cleanly the loop is continuous; where it breaks at exactly one cell the loop is discontinuous, and the break is what you cash in.
| Loop type | What you see at the joint | What it gives you |
|---|---|---|
| Continuous | Links alternate the whole way round | For every weak link, the digit leaves all other cells of that link's unit |
| Break with two strong links | Both links meeting one cell are strong | That cell holds the digit |
| Break with two weak links | Both links meeting one cell are weak | The digit comes off that cell |
A continuous four-cell cycle is exactly an X-Wing written in link notation, which is a useful sanity check when you are learning. In practice a continuous loop is usually caught earlier as an X-Wing or another fish, and a break with two strong links is rare, so the break with two weak links is the shape you will meet most.
When to look for it
- One digit is placed six or more times, so its remaining candidates form a thin, connected web.
- You already coloured that digit, found nothing, and can see a unit with three places that would bridge two coloured groups.
- Wings and fish are exhausted but you do not want to open a full multi-digit chain yet.
How to apply it, step by step
- Pick a digit and mark every unit where it has exactly two places. Those are your strong links.
- Start at any candidate of that digit and walk out along a strong link.
- Leave the next cell along a weak link: any other cell of the digit in a unit it belongs to.
- Keep alternating strong, weak, strong, weak until you return to a cell already on the path.
- Inspect the joint where the path closes and read off the matching row of the table.
A worked example
Only the digit 8 matters. Three units have exactly two places for it, which gives three strong links: row E (E3 and E8), column 7 (C7 and F7) and column 2 (C2 and G2). Two weak links join them: E8 and F7 share box 6, and C7 and C2 share row C.
The chain runs E3, E8, F7, C7, C2, G2, and it alternates strong and weak links all the way. H3 closes the loop: it sees E3 through column 3 and G2 through box 7. Both links meeting H3 are weak, so this is the third row of the table.
Test it directly. If E3 is not 8, row E forces E8 to be 8, which rules out F7, so column 7 forces C7. That rules out C2, so column 2 forces G2. Either E3 or G2 is always 8, and H3 sees both, so 8 comes off H3.
Common mistakes
- Treating every shared unit as a strong link. A strong link needs exactly two places for the digit.
- Breaking the alternation twice. One break is a result; two breaks prove nothing at all.
- Reading a continuous loop as a placement. Continuous loops only clear the units of their weak links.
- Eliminating from cells on the loop itself. Continuous-loop eliminations hit the other cells of the unit.
- Mixing digits. A single X-Cycle never leaves the digit it started with.
Where you meet it
X-Cycles are the first properly chain-based technique on the ladder and you will need them for expert classic Sudoku. Once you can read strong and weak links comfortably, the same notation carries straight over to multi-cell and multi-digit chains, so the effort pays for itself twice.