Alternating chains of strong and weak links for a single digit that reveal contradictions or force eliminations.
X-Cycles extend Simple Coloring by adding weak links to the chain. In Simple Coloring, every link is a strong link — a conjugate pair where the digit has exactly two positions in a unit. X-Cycles allow weak links too: connections where a digit appears in more than two cells in the unit, so the link says "if this cell has the digit, that cell doesn't" but NOT the reverse.
By mixing strong and weak links in alternating fashion, X-Cycles can reach cells that pure coloring cannot. The chain forms a cycle (a closed loop) with an odd or even number of links. When the chain creates a contradiction, like a cell being both "on" and "off", you know that cell's assignment is impossible, and you can eliminate the digit.
X-Cycles are a powerful generalization that bridges the gap between simple coloring techniques and full Alternating Inference Chains (AICs). They work on a single digit at a time.
Choose a digit. Build a chain of cells where the digit is a candidate, alternating between strong links (s) and weak links (w). A strong link means the digit appears in exactly two cells in a shared unit, if one is false, the other must be true. A weak link means both cells share a unit but there may be more than two positions.
Try to form a closed loop (cycle). There are three cases. Continuous Nice Loop: if the loop has an even number of links with perfect strong-weak alternation, eliminate the digit from any cell outside the loop that sees both endpoints of a weak link in the loop.
Discontinuous Nice Loop (elimination): if the loop has a cell where two weak links meet (both of the cell's links in the loop are weak), assuming that cell holds the digit forces a contradiction around the loop — eliminate the digit from it. Discontinuous Nice Loop (placement): if the loop has a cell where two strong links meet, that cell is forced true from both directions — place the digit there.
Example 1: X-Cycle (Discontinuous Nice Loop)
Follow digit 5 around this six-cell loop: R2C9 → R4C9 → R4C3 → R9C3 → R9C4 → R2C4, then back to R2C9. The links alternate: a solid arrow is a strong link (digit 5 has exactly two spots in that unit), a dashed arrow is a weak link.
Colour the loop by alternating green and blue. Assume R2C9 = 5 and follow the alternation around the loop — it forces R2C9 to NOT be 5 as well, a contradiction. So R2C9 cannot be 5.
Eliminate digit 5 from R2C9, which reduces to {9}.
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