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.

Great fit Classic Sudoku The natural next step after Turbot Fish, catching single-digit patterns colouring misses. → Works Killer Sudoku Occasionally needed once cage arithmetic stalls on a stubborn digit. → Works Sudoku X Diagonal strong links let cycles close that could not on a plain grid. → Limited Samurai Sudoku Deep single-grid loops rarely pay off across five grids. →

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.

Play Expert Sudoku About Expert Sudoku

Related techniques

Turbot Fish - Single-Digit Chain Sudoku Technique A four-cell chain on a single digit - two strong links joined by a weak one - that eliminates the digit where both ends can see. Simple Colouring - Sudoku Chain Technique Colour a digit's conjugate-pair chain in two alternating colours; a colour that appears twice in a unit is false, and cells seeing both colours lose the digit. Grouped X-Cycles - Sudoku Chaining Technique A single-digit loop in which a node can be a group of candidates sharing a box-line, extending the reach of an ordinary X-Cycle. 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.