Grouped X-Cycles

Grouped X-Cycles are ordinary X-Cycles with one upgrade: a node no longer has to be a single cell. Two or three cells of the same digit, sitting together where a box meets a row or column, can act as one node. That single change turns a lot of dead-end chains into closed loops.

What the pattern is

An X-Cycle is a loop for one digit whose links alternate between strong and weak. A strong link means at least one end is true; a weak link means at most one end is true. Alternate them around a closed loop and the loop constrains the whole grid.

A group is the set of cells holding your digit inside one box-line intersection — the two or three cells where a box overlaps a row or column. The group behaves exactly like a single node, because "the digit is somewhere in this group" is one statement that is either true or false. It links strongly to another node when the two of them are the only places for the digit in a unit, and weakly whenever they share a unit.

The three X-Cycle rules carry over unchanged:

• Rule 1 — the loop closes with links alternating all the way round. Every weak link becomes exact, so the digit leaves every other cell that sees both ends of any weak link.
• Rule 2 — the loop closes but two strong links meet at one node. That node is true.
• Rule 3 — the loop closes but two weak links meet at one node. That node is false, so the digit leaves it.

When to look for it

Look when a digit is down to two or three positions in most units and you keep finding chains that almost close. Boxes where the digit is confined to one row or one column are the raw material — those are exactly the groups you need.

How to apply it, step by step

1. Choose a digit and mark every position left for it.
2. Mark each box where the digit is confined to one row or one column. Those cells form a group node.
3. Build strong links: two nodes that are the only places for the digit in a unit.
4. Fill in weak links: any two nodes sharing a unit.
5. Walk a loop alternating strong, weak, strong, weak, and see which of the three rules the loop matches.
6. Apply that rule's eliminations.

A worked example

Digit 5, with = for a strong link and - for a weak one:

r1c1        =  (r3c2 r3c3)     box 1 holds 5 only in these three cells
(r3c2 r3c3) -  r3c8            all three lie in row 3
r3c8        =  r1c8            column 8 holds 5 only in r3c8 and r1c8
r1c8        -  r1c1            both lie in row 1

four links, alternating, loop closed  ->  rule 1
=> 5 leaves every other cell of row 3 and of row 1

The pair r3c2 r3c3 is the group. Box 1 has only three places for 5, so either r1c1 is 5 or one of the group is — that is a genuine strong link, even though one end is two cells. Because the loop alternates all the way round, each weak link becomes an exact "one of these two", and the digit is squeezed out of the rest of both rows.

Common mistakes

• Building a group from cells that do not share both a box and a line. Scattered cells are not a node.
• Counting a group's link as strong when the digit has a fourth position elsewhere in the unit.
• Losing the alternation. One strong-strong step in the middle of a loop invalidates everything after it.
• Applying rule 1 eliminations to the loop's own cells. They stay; only outside cells lose the digit.
• Forgetting that a group node blocks eliminations across all of its cells at once — a target must see every cell of the group.

Where you meet it

Grouped X-Cycles sit in the extreme tier. They first become necessary in Master puzzles and stay useful all the way through Legendary, where single-digit logic is often the only thing left that does not require a full forcing net.