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.

Great fit Classic Sudoku Group nodes close loops that a plain X-Cycle leaves open, on hard Classic grids. → Limited Killer Sudoku Cage logic usually resolves Killer before grouped loops are needed. → Works Sudoku X The diagonals add group links to draw the loop through. → Limited Samurai Sudoku Seldom worth the effort on Samurai. →

Where to practise this

This technique starts paying off at Expert, which is where puzzles first stop yielding to anything simpler. Reading about a pattern and spotting it on a live grid are different skills - the second one only comes from playing.

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Related techniques

X-Cycles - Single-Digit Loop Sudoku Technique A loop of alternating strong and weak links on one digit; continuous loops eliminate around the ring, a broken loop forces the digit at the discontinuity. X-Wing Sudoku Technique Explained One digit confined to the same two columns across two rows (or vice versa) - eliminate it from the rest of those lines. XY-Wing Sudoku Technique (Y-Wing) Explained A pivot cell with two candidates links two pincer cells; the digit they share is removed from any cell both pincers see. Swordfish Sudoku Technique Explained One digit confined to the same three columns across three rows (or vice versa) - eliminate it from the rest of those columns.