ALS-XY-Wing

Three almost-locked sets chain through a pivot to trap a digit.

The ALS-XY-Wing sudoku technique generalises the familiar XY-Wing by letting each of its three parts be an almost-locked set instead of just a single bivalue cell. A central pivot set shares a restricted common candidate with each of two wing sets, and both wings also hold a further digit Z. Whichever way the pivot eventually resolves, one of its restricted commons is forced out, driving Z into one wing or the other — so any cell that can see every Z in both wings can never hold Z itself, and Z is eliminated from that cell.

Worked example

Pivot set R1C1={1,2} joins wing R1C5={1,3} on 1 and wing R5C1={2,3} on 2; both wings share 3, so a 3 is forced into one wing and R5C5 — seeing both — loses 3.12132339
Pivot set R1C1={1,2} joins wing R1C5={1,3} on 1 and wing R5C1={2,3} on 2; both wings share 3, so a 3 is forced into one wing and R5C5 — seeing both — loses 3.

The pivot here is R1C1={1,2}. It links to wing A, R1C5={1,3}, through the restricted common 1, and to wing B, R5C1={2,3}, through the restricted common 2. Both wings also carry a 3 — the shared digit Z. If R1C1 is 1 then R1C5 loses its 1 and must be 3; if R1C1 is 2 then R5C1 loses its 2 and must be 3 — either way a 3 is forced into one of the wings. R5C5={3,9} sees both R1C5 and R5C1, so it can never be that 3; 3 is eliminated from R5C5, leaving it as 9.

When to use it

Reach for it deep in a hard puzzle, once basic wings and single chains have dried up. Hunt for a pivot set that shares a restricted common with two separate wings, where both wings also hold a common extra digit Z, then scan for any cell that sees all of Z in both wings. In its simplest form all three sets are bivalue cells, so on the grid it can look exactly like an ordinary XY-Wing until you notice a set spanning more than one cell.

How to use it

  1. Find the pivot and two wings. Take three almost-locked sets: a pivot set linked to each of two wing sets by a restricted common candidate — X to the first wing, Y to the second, with X and Y different.
  2. Follow the forced chain. If a wing does not take the shared digit Z, the restricted commons march through the pivot and force the other wing to take Z — so Z must appear in one wing or the other.
  3. Eliminate Z outside the wings. Because Z is trapped in the two wings, remove Z from any cell outside the three sets that sees every Z candidate in both wings.
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Frequently asked questions

What is an ALS-XY-Wing in Sudoku?

It is the almost-locked-set version of the XY-Wing. A pivot set shares a restricted common candidate with two wing sets, the two wings share an extra digit Z, and Z can then be removed from any cell that sees every Z in both wings. In the worked example the pivot R1C1={1,2} feeds wings R1C5={1,3} and R5C1={2,3}, which forces a 3 into one wing and eliminates 3 from R5C5={3,9}.

How do I spot an ALS-XY-Wing on the grid?

Look for it deep into a hard solve, when wings and single chains have stalled. Find a pivot that shares a restricted common candidate with two separate sets, confirm both of those wings hold the same extra digit Z, then look for a cell that sees every copy of Z in both wings. That cell is where Z is eliminated — in the example R5C5 loses its 3.

How is an ALS-XY-Wing different from a plain XY-Wing?

A plain XY-Wing is exactly three bivalue cells. The ALS-XY-Wing lets any of the three roles — the pivot or either wing — be a multi-cell almost-locked set, so it catches eliminations a bivalue-only wing would miss. When all three sets happen to be single bivalue cells, as with R1C1, R1C5 and R5C1 above, the two techniques coincide.