WXYZ-Wing Sudoku Technique

The WXYZ-Wing is the four-cell wing: four cells holding four digits between them, with one cell — the hinge — seeing the other three. It extends the same idea as the XYZ-Wing by one cell and one digit, and it fires far more often than its awkward name suggests.

What the pattern is

Take four cells whose candidates, pooled together, come to exactly four digits. Call them W, X, Y and Z. Now check each digit separately:

• A digit is restricted if every cell of the group that still holds it can see every other cell of the group that holds it. Those cells share a unit, so the digit can be used at most once inside the group.
• A digit is non-restricted if two of its cells do not see each other. It could, in principle, be used twice.

A WXYZ-Wing needs exactly one non-restricted digit. Call it Z. Since the other three digits can each appear only once, they can fill at most three of the four cells. The fourth cell has to be Z. So at least one cell of the group is Z, and any cell outside the group that sees every group cell containing Z cannot be Z itself.

When to look for it

Look after the three-cell wings have failed and you still have a cluster of bi-value and tri-value cells packed into one box. The hinge is usually a bi-value or tri-value cell in a box, with two partners in the same box and one further along its row or column.

How to apply it, step by step

1. Find a candidate hinge and the three cells it sees.
2. Pool all four candidate sets. Stop unless the union is exactly four digits.
3. For each digit, list the group cells that hold it and check whether they all see each other.
4. Confirm there is exactly one non-restricted digit, Z.
5. Collect the group cells that hold Z, and find every outside cell that sees all of them.
6. Remove Z from those cells.

A worked example

four cells, four digits {1,2,3,4}

r5c5 {1,2}   hinge, sees the other three
r5c1 {1,4}   same row as the hinge
r4c4 {2,4}   same box as the hinge
r6c6 {2,3}   same box as the hinge

digit 1 -> r5c5, r5c1        share row 5     restricted
digit 2 -> r5c5, r4c4, r6c6  share box 5     restricted
digit 3 -> r6c6 only                         restricted
digit 4 -> r5c1, r4c4        do NOT see each other -> non-restricted

=> one of r5c1 / r4c4 must be 4
=> 4 leaves r4c1 and r5c4, the cells that see both

Check the logic. Suppose none of the four cells is 4. Then all four take values from {1, 2, 3}, and each of those three digits can be used only once inside the group — so three digits would have to fill four cells. Impossible. One cell is therefore 4, and only r5c1 and r4c4 can supply it. Cell r4c1 sees r4c4 along row 4 and r5c1 down column 1; r5c4 sees r5c1 along row 5 and r4c4 down column 4. Neither can be 4.

Common mistakes

• Counting five digits. Four cells, four digits total — a fifth candidate anywhere in the group ends the pattern.
• Two non-restricted digits. Then the pigeonhole argument collapses and nothing follows.
• Eliminating a restricted digit. Only Z comes out, and only from cells outside the group.
• Missing a target because you checked visibility against the whole group. The target only has to see the group cells that contain Z, not all four.
• Forgetting the hinge must see all three partners. Without that, the digits cannot be pinned into one shared region.

Where you meet it

The WXYZ-Wing is diabolical tier. It first becomes necessary in Expert puzzles, where four-cell patterns start to replace the simpler wings. On Hard grids an XY-Wing, XYZ-Wing or a subset almost always gets there first, so keep this one in reserve.