X-Cycles - Sudoku69

X-Cycles are single-digit loops built from alternating strong and weak links. 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.

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 6 matters. Column 1 has just two places for 6, at r5c1 and r9c1. Column 8 also has just two, at r5c8 and r9c8. Row 5 has three: c1, c5 and c8.

Example

r5c5  - row 5 -  r5c1  = col 1 =  r9c1
                                    |
                                 row 9
                                    |
r5c5  - row 5 -  r5c8  = col 8 =  r9c8

  =  strong link      -  weak link

Walk the loop from r5c5: weak to r5c1, strong to r9c1, weak to r9c8, strong to r5c8, weak back to r5c5. Both links meeting r5c5 are weak, so this is the third row of the table.

Test it directly. If r5c5 were 6, then r5c1 and r5c8 could not be, so column 1 would force 6 into r9c1 and column 8 would force 6 into r9c8 — two 6s in row 9. Impossible, so 6 comes off r5c5.

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.