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.
| Plain X-Cycle | Grouped X-Cycle | |
|---|---|---|
| Node | one cell | one cell or a group |
| Group location | none | box meets a row or column |
| Strong link source | two places in a unit | two places, counting a group as one |
| Rules used | 123 | 1, 2, 3, unchanged |
How to apply it, step by step
- Choose a digit and mark every position left for it.
- Mark each box where the digit is confined to one row or one column. Those cells form a group node.
- Build strong links: two nodes that are the only places for the digit in a unit.
- Fill in weak links: any two nodes sharing a unit.
- Walk a loop alternating strong, weak, strong, weak, and see which of the three rules the loop matches.
- Apply that rule's eliminations.
A worked example
Digit 4. Box 4 holds a 4 only in F1 and in the group D3, E3, F3, which all lie in column 3. Watch how the links run from F1 through the group and A3 to A5, and where the loop ends up.
The chain has three links:
- Strong, box 4: F1 and the group are the only places for 4, so one of them is 4.
- Weak, column 3: the group and A3 share column 3, so they cannot both be 4.
- Strong, row A: A3 and A5 are the only places for 4 in row A.
Read it as F1 = group, group - A3, A3 = A5. If F1 is not 4, the group is, so A3 is not, so A5 is. Either way F1 or A5 is 4. F5 sees F1 in row F and A5 in column 5, so two weak links meet at F5. That is rule 3, and it is why this loop is discontinuous there: F5 cannot be 4.
The continuous case (rule 1) needs every link to alternate all the way round. Here the loop breaks at F5, and that break is exactly what produces the elimination.
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.