XY-loop
A closed XY-chain — the endpoints meet rather than going off into eliminations. Every step in the loop is constrained from both sides at once, surfacing extra eliminations.
An XY-loop is what you get when an XY-chain closes back on itself — the chain's last cell sees its first cell, and the candidate that would have been eliminated outside the chain is now constrained inside it. Loops are stronger than the equivalent open chain because every step is constrained from both sides simultaneously, and the eliminations happen on every shared-unit relationship around the loop, not just at one pair of endpoints.
What a closed loop produces
In an open XY-chain on candidate a, only cells that see both endpoints lose a. In a closed XY-loop, every adjacent pair of cells in the loop shares some candidate, and every "weak edge" of the loop produces an elimination.
A worked layout. Imagine four bivalue cells forming a loop: C1 {a, b}, C2 {b, c}, C3 {c, d}, C4 {d, a}, where C4 sees C1. The trial-cascade in either direction shows that the loop must alternate: either (C1=a, C2=b, C3=c, C4=d) or (C1=b, C2=c, C3=d, C4=a). Either way, C1 and C3 don't share a candidate-value; C2 and C4 don't either; and each "non-passing" candidate of each cell is forced.
The eliminations: on each weak edge of the loop, the candidate not used as the link can be removed from any cell outside the loop that sees both ends of that edge. Closed XY-loops typically produce three or four eliminations from a single pattern, where an open XY-chain of the same length would produce one.
Why they're rarer than open chains
Closed loops require the chain to complete its circuit — the last bivalue cell has to share a unit with the first. Most puzzle states with chain potential have open chains that drift across the grid; the closed condition is restrictive. When a loop does form, though, it usually solves a stuck state in one move where simpler chain techniques would have needed several.
A useful spotting habit: when scanning for an XY-chain, watch whether the chain you're tracing keeps coming back to its starting region. If the trace bends back toward the first cell, check whether it can complete the loop — those are the configurations where XY-loop fires.
See also
- XY-chain— A chain of bivalue cells linked by shared candidates. Eliminates a digit from any cell that sees both endpoints — the workhorse intermediate-to-advanced chain technique.
- Y-wing (XY-wing)— Three bivalue cells where the pivot shares one candidate with each wing — eliminating the third candidate from any cell that sees both wings.
- Alternating Inference Chain (AIC)— The general-purpose chain technique. Alternates strong and weak links along a sequence of candidates, eliminating a digit from any cell that sees both endpoints' candidates.
- Candidate— A digit (1–9) a cell could still legally hold — one not yet ruled out by anything in its row, column, or 3×3 box. Every empty cell has between one and nine.
Read more
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