ALS-XY-wing
Three Almost Locked Sets in a Y-wing-like configuration. Generalises ALS-XZ to a longer chain and surfaces eliminations that a single-pair ALS interaction would miss.
ALS-XY-wing is the three-set generalisation of ALS-XZ. Three Almost Locked Sets — A, B, and C — connect through restricted-common candidates in a Y-wing-like shape. The eliminations run from the candidate that the two outer sets share, on the cells that see both.
The shape
Three ALSes A, B, C, where A and B share a restricted common on one candidate (call it y), B and C share a restricted common on another candidate (call it z), and A and C share a third common candidate (x) that isn't restricted. The middle set B is the "pivot." A and C are the "wings."
The argument runs in two cases. If A contains y, then B doesn't (the restricted-common condition), so B contains z, so C doesn't (again restricted), so C contains its non-z candidates including x. If A doesn't contain y, then A contains x. Either way, x ends up in either A or C — at least one of them. So any cell outside A, B, and C that sees every x in both A and C can have x eliminated.
The Y-wing shape comes through clearly: A and C are the pincers; B is the pivot; x is the candidate that gets eliminated.
Why it generalises further
The structure extends to longer chains: ALS-XYZ-wing uses four sets, ALS-VWXYZ uses five. As the chain length grows, the pattern starts to look more like an AIC traversing ALSes than a separate technique. Most published treatments stop the named series at ALS-XY-wing and treat longer chains as AIC variants.
When you'll see it
ALS-XY-wing is rare in practice — three ALSes with the right shared-candidate structure is restrictive — but on extreme puzzles where every wing and every fish has been exhausted, it sometimes appears as the move that breaks the deadlock. A solver who can't readily construct ALSes by eye will struggle to find these patterns; software solvers find them efficiently because the search is structurally regular.
For a foundational ALS treatment, see ALS-XZ rule. The reasoning is the same in spirit; the bookkeeping just runs across one more set.
See also
- ALS-XZ rule— An interaction between two Almost Locked Sets sharing a common candidate. Eliminates a second shared candidate from cells outside both sets that see all instances.
- 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.
- 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.
Read more
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How to read a fresh hard Sudoku and predict which intermediate technique will break it open before you've placed a single digit.