X-Cycles Sudoku: Single-Digit Loops, Weak-Link Eliminations, and Forced Digits

X-Cycles are AICs restricted to one candidate digit: every node is a “digit d in cell” position, and links follow that digit’s conjugate structure through rows, columns, and boxes. Because the object is one digit, the diagram is easier to sanity-check than a full multi-digit net—until the loop type is wrong by one strong/weak placement.

Use this page when your search intent is specifically x-cycles sudoku or “one digit, loop, eliminate everywhere at weak edges”—not the whole extreme syllabus.

Pattern

Build the graph for digit d:

• Strong edges: conjugate pairs (exactly two d-candidates in a unit).
• Weak edges: two d-candidates share a unit but are not conjugate (more than two d in that unit).

Walk alternating strong / weak around the graph. When the path closes, classify the loop:

1. Continuous (even length): strong–weak–strong–weak… around the loop. Weak-link endpoints sit in shared units; eliminate d from other cells in those units at each weak link.
2. Discontinuous: two strong links meet at a break cell → that candidate is often forced true; two weak links at the break → the candidate is often forced false.

If your classification does not match the parity you drew, the loop is not yet valid—redraw only d before mixing other digits back in.

Logic

Prefer X-Cycles when:

• One digit feels “almost trapped” but fish do not close.
• Simple coloring on d already shows a contradiction or a trap—you are one step from naming the cycle.
• Multi-digit AIC felt like noise; collapsing to d only clarifies the spine.

For the general alternating framework (multi-digit), see AIC in Sudoku. For two-branch case splits, see forcing chains.

Example (schematic)

Imagine candidate 6 cycling through rows and columns so that weak links land in columns where other 6s still live. At each weak link, off-loop cells in that column lose 6. The exact geometry is puzzle-specific; the reusable skill is tagging S/W on every edge before claiming an elimination.

Pitfalls

• Treating bilocal as automatic strong while a third d still hides in the same unit.
• Confusing continuous vs discontinuous break rules—draw the loop once on paper.
• Ignoring box conjugate edges when the loop lives mostly in rows/columns.

Where to go next

• Broader chain hub: how to solve extreme Sudoku.
• Set logic that often shortens a digit hunt: ALS-XZ.

For single-digit highlighting and extreme-rated boards that reward conjugate discipline, use Sudoku Face Off. Prefer a quieter session without ad breaks while you loop-hunt: Sudoku app no ads (2026).