Alternating chains of strong and weak links for a single digit that reveal contradictions or force eliminations.
Chain cells that hold one digit, alternating strong links (exactly two positions in a unit) with weak links (any shared unit), then close the chain into a loop. A perfectly alternating loop eliminates the digit outside itself; a loop with two weak links meeting at one cell eliminates there, and two strong links meeting places it.
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}.
A strong link means the digit has exactly two possible cells in a shared unit, so if one is false the other must be true. A weak link only means two cells share a unit and the digit may sit in other cells there too, so if one is true the other is false — but not the reverse.
Simple Coloring uses strong links only, so the chain stops wherever a digit has three or more positions in a unit. X-Cycles add weak links between those links, which lets the chain keep going and reach cells that coloring cannot. Both work on one digit at a time.
No. Where the elimination lands depends on the loop type. A continuous loop with perfect alternation eliminates the digit from cells outside the loop that see both ends of one of its weak links. A discontinuous loop acts on the cell where the alternation breaks, either removing the digit or placing it.
When two strong links meet at the same cell in the loop. Following the alternation from either direction forces that cell to hold the digit, so you can place it. The mirror case — two weak links meeting at one cell — forces a contradiction instead, and the digit is eliminated from that cell.
Assume the digit sits in the cell you suspect and follow the loop. A weak link turns a true cell false, and a strong link turns a false cell true. If the alternation carries you back to the starting cell in the opposite state, the assumption is impossible and the digit comes out.
Not quite. An X-Cycle works on a single digit, whereas Alternating Inference Chains generalize the same strong-and-weak-link machinery across several digits at once. X-Cycles are the single-digit case, which makes them a useful bridge between Simple Coloring and full AIC work.
Explore all 28 solving techniques in our complete technique guide.