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.

464767✗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

  1. Find a pivot ALS and two other almost locked sets.
  2. Check two restricted commons - X between pivot and A, Y between pivot and C, with X ≠ Y.
  3. Find a shared digit Z in both outer sets A and C.
  4. 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.

Works Classic Sudoku Reaches eliminations beyond the ALS-XZ rule on the hardest Classic grids. → Works Killer Sudoku Cages provide ready almost locked sets to chain into a wing. → Works Sudoku X The diagonals add candidate sets to serve as the wing's arms. → Limited Samurai Sudoku Too intricate to be practical across five grids. →

Where to practise this

This technique starts paying off at Evil, which is where puzzles first stop yielding to anything simpler. Reading about a pattern and spotting it on a live grid are different skills - the second one only comes from playing.

Play Evil Sudoku About Evil Sudoku

Related techniques

ALS-XZ - Almost Locked Set Sudoku Technique Two almost locked sets sharing a restricted common digit X force a second common digit Z out of any cell that sees all its copies in both sets. Jellyfish - the Order-4 Sudoku Fish One digit confined to the same four columns across four rows (or vice versa) - eliminate it from the rest of those columns. Finned Swordfish - Sudoku Fish Variation An almost-Swordfish with one extra candidate; the digit is eliminated only from cover cells that also see the fin. BUG+1 - Bi-Value Universal Grave Technique When all cells but one are bi-value, the extra candidate in that one cell must be the solution, or the puzzle would have two answers.
All Articles Practice in the Game