ALS-XY-Wing
The ALS-XY-Wing is the next rung above the ALS-XZ rule. Instead of two almost locked sets, it uses three — a central pivot joined to two outer sets by two different restricted-common digits. The payoff is an elimination on a digit shared by the two outer sets.
The idea
Call the three almost locked sets A, B (the pivot) and C. The pivot B shares a restricted common digit X with A — every X in A sees every X in B — and a different restricted common Y with C. Because a restricted common can be true in only one of its two sets, X ties B to A and Y ties B to C. Follow the cases: if B does not supply X, then A must, locking A; if B does not supply Y, then C must, locking C. Whatever B does, one of A or C ends up locked and forced to contain any digit Z the two outer sets share. So a cell outside them all that sees every Z in A and every Z in C cannot be Z.
A worked example
Let the pivot B be the bi-value cell r5c5 = {4,7}. Set A is the bi-value r5c1 = {4,6}: it shares the restricted common X = 4 with B along row 5, and also holds Z = 6. Set C is r5c9 = {6,7}: it shares the restricted common Y = 7 with B, and also holds Z = 6. Now follow the pivot. If B = 4 then A cannot be 4, so A locks to 6; if B = 7 then C cannot be 7, so C locks to 6. Either way a 6 lands in A or in C, so r5c3, which sees both r5c1 and r5c9 along row 5, can never be 6.
Pivot r5c5 = {4,7} (purple) ties to set A = {4,6} by X = 4 and to set C = {6,7} by Y = 7 (dashed, both restricted along row 5). Whichever the pivot takes, a 6 lands in A or C — so r5c3 loses its 6.
Step by step
- Find a pivot ALS and two other almost locked sets.
- Check two restricted commons — X between pivot and A, Y between pivot and C, with X ≠ Y.
- Find a shared digit Z in both outer sets A and C.
- Eliminate Z from cells outside all three that see every Z in A and in C.
Common mistakes
- Commons not restricted. Each linking digit must be fully visible between its two sets, or the lock never triggers.
- Same digit twice. The two restricted commons X and Y must differ; reusing one collapses the wing.
- Z inside a set. The elimination lands on outside cells only — the sets themselves carry the argument.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.