X-Cycles
X-Cycles take the single-digit chain idea of the Turbot Fish and close it into a loop. Working with just one digit, you alternate strong links (the digit sits in only two cells of a unit) and weak links (two cells that share a unit) around a ring. Whether the ring closes cleanly or has a single flaw tells you what to eliminate.
The idea
Colour the loop with two alternating states, on and off. A continuous X-Cycle — where strong and weak links alternate perfectly all the way round — proves that every weak link in the ring is genuinely weak: the digit can be removed from any cell outside the loop that sees both ends of any weak link. A discontinuous loop has one spot where two links of the same type meet. That flaw is a contradiction unless the cell at the break takes a forced value: if two strong links collide there, the digit is placed; if two weak links collide, the digit is removed from that cell.
A worked example
Following the digit 4, imagine a four-cell loop r1c1 = r1c7 − r4c7 = r4c1 − r1c1, alternating strong (=) and weak (−) links and closing on itself. It is continuous because every corner joins exactly one strong and one weak link. The strong links live in rows 1 and 4 (where 4 sits in only those two cells of the row); the weak links run down columns 1 and 7. Since the loop is continuous, each weak link is confirmed: every other cell of column 1 sees both ends of that column's weak link and loses its 4, and the same holds down column 7. Even one such removal can trigger a cascade of singles.
Continuous loop on 4: strong links (solid) in rows 1 and 4, weak links (dashed) down columns 1 and 7. Every other cell in those two columns — the red ones — loses its 4.
Step by step
- Pick one digit and mark all its remaining candidates.
- Build a loop alternating strong and weak links on that digit.
- Check the loop type. Perfectly alternating means continuous; one same-type junction means discontinuous.
- Apply the rule — clear weak-link peers for a continuous loop, or force the value at the break for a discontinuous one.
Common mistakes
- Mislabelling links. A strong link needs exactly two candidates in the unit; a weak link only needs the two cells to share a unit.
- Eliminating inside the loop. For a continuous cycle the removals are on outside cells that see both ends of a weak link.
- Two flaws. A valid discontinuous loop has exactly one same-type junction — two means you have miscounted a link.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.