Grouped X-Cycles
Grouped X-Cycles extend the ordinary X-Cycle by letting a whole group of candidates act as a single node. Where a plain X-Cycle links one cell to one cell, a grouped version can link a cell to two or three candidates that share a box-and-line — and that extra reach closes loops the simple version leaves open.
The idea
Still working with one digit, look at the places it lives inside a box. If all of a box's copies of the digit sit in a single row or column segment, they form a group: as a unit they behave like one node in a chain. A strong link runs to the group when the digit must be somewhere in it; a weak link runs from the group when the digit being in it forbids it elsewhere in the shared line. Threading these grouped nodes into an X-Cycle lets strong and weak links alternate around a loop that no single-cell chain could form. The eliminations are the same as any X-Cycle: continuous loops clear weak-link peers, a broken loop forces the value at the flaw.
A worked example
Chasing the digit 2, box 4 holds its only 2s in r4c1 and r4c2 — a group in row 4. Treated as one node it weak-links along row 4 to a 2 further out at r4c7; that cell strong-links up column 7 to r1c7, which weak-links along row 1 to r1c1, which strong-links down column 1 back into the group. The loop alternates strong and weak all the way — a continuous grouped X-Cycle a cell-by-cell chain could not form. Its weak links then remove 2 from every cell seeing both their ends: r4c5 on the row-4 link and r1c4 on the row-1 link.
A continuous grouped X-Cycle on 2. The pair r4c1–r4c2 (amber) is box 4's only 2s, acting as one grouped node. Strong (solid) and weak (dashed) links alternate round the loop; the weak links clear 2 from r4c5 and r1c4.
Step by step
- Pick a digit and find boxes where its candidates share a single row or column.
- Treat that cluster as a group node for linking.
- Build an X-Cycle alternating strong and weak links through cells and groups.
- Apply the X-Cycle rule for the loop type you have formed.
Common mistakes
- Groups that span two lines. A group must lie in one row or column within its box, or it cannot act as a single node.
- Wrong link strength. Check whether the group forces the digit (strong) or merely forbids it (weak) at each step.
- Overlapping ends. As with any X-Cycle, eliminate on cells that see both ends of a weak link, not the ends themselves.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.