ALS-XY-Wing Sudoku: Three Almost Locked Sets Working Together

ALS-XY-Wing Sudoku replaces three cells with three flexible sets

ALS-XY-Wing Sudoku is a generalization of familiar wing logic. An Almost Locked Set, or ALS, is a group of N cells in one house containing exactly N plus 1 different candidates. A single bivalue cell is therefore the smallest possible ALS. In an ALS-XY-Wing, three ALSes play the roles of two wings and a pivot. Call them A, B, and C. A and B share one restricted common candidate, meaning the candidate cannot be true in both sets because every occurrence in one set sees every occurrence in the other. B and C share a different restricted common candidate. Finally, the outer sets A and C share another candidate, which becomes the elimination digit. ALS-XY-Wing Sudoku may look large on the grid, but the logic is the same either-or structure that makes an ordinary XY-Wing work.

Follow the two restricted links in ALS-XY-Wing Sudoku

Suppose A and B are linked by candidate X, while B and C are linked by candidate Y. The outer sets A and C both contain candidate Z. If A does not contain X, then because A is almost locked it must contain its other required candidates, including Z. If A does contain X, B cannot contain X, which pushes B toward Y; because Y is restricted between B and C, C must then resolve so that Z is present there. The exact wording changes with set sizes, but the endpoint conclusion is stable: Z must occur in A or C. Therefore, any outside Z candidate that sees every possible Z in both outer sets can be eliminated. The target does not need to see the pivot set B. It must see all endpoint Z positions. That visibility check is the most important safeguard in ALS-XY-Wing Sudoku.

Find ALS-XY-Wing Sudoku by starting with small sets

Searching every possible ALS combination is exhausting, so look for compact sets first. Bivalue cells, two-cell ALSes with three candidates, and three-cell ALSes with four candidates are easiest to manage. Choose a small middle set and see whether it has restricted common candidates with two nearby ALSes in different units. Then check whether the two outer sets share a candidate that can attack a common target. If the notation feels abstract, label the sets A, B, and C on paper and write their candidate collections beside them. Shapedoku's Notes can help train the same set recognition because two cells containing three possible shapes form an ALS just as they do with digits. On difficult boards, this perspective can reveal structure that individual-cell scanning misses. For a calmer alternative built around grouping and connection, Rune Flow from SnailPixel asks you to connect matching crystals across a board: [https://play.google.com/store/apps/details?id=com.snailpixel.runeflow](https://play.google.com/store/apps/details?id=com.snailpixel.runeflow). In Sudoku, keep the sets small until the logic becomes natural.

The most common ALS-XY-Wing Sudoku error is assuming that a candidate shared by two sets is automatically restricted common. It is not. Every occurrence of that candidate in one ALS must see every occurrence in the other ALS for the restriction to hold. Check this for both pivot links before looking at the endpoint digit. Then perform the same visibility audit on the elimination target: it must see every possible endpoint occurrence in both outer sets. These two all-to-all checks are tedious at first, but they are the difference between a real ALS wing and three candidate groups that merely overlap in notation.

• Confirm that every proposed ALS has N cells and exactly N plus 1 candidates.
• Link A to B with one restricted common candidate and B to C with another.
• Check that the two outer sets share the intended elimination candidate.
• Require the target to see every occurrence of that candidate in both outer sets.
• Do not eliminate from the pivot merely because it participates in the chain.
• Start with bivalue cells and two-cell ALSes before attempting larger constructions.