WXYZ-Wing Sudoku Technique
The WXYZ-Wing is the four-cell wing in Sudoku: 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.
| XYZ-Wing | WXYZ-Wing | |
|---|---|---|
| Cells | 3 | 4 |
| Digits in total | 3 | 4 |
| Hinge sees | both wings | all three partners |
| Digit removed | the one shared by all three cells | the single non-restricted digit |
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
- Find a candidate hinge and the three cells it sees.
- Pool all four candidate sets. Stop unless the union is exactly four digits.
- For each digit, list the group cells that hold it and check whether they all see each other.
- Confirm there is exactly one non-restricted digit, Z.
- Collect the group cells that hold Z, and find every outside cell that sees all of them.
- Remove Z from those cells.
A worked example
Four cells hold only the digits 3, 4, 6 and 7 between them: H1 , H3 , J3 and J4 . J3 is the hinge: it sees H1 and H3 through box 7 and J4 through row J.
Check each digit:
- 3 sits in H3 and J3, which share box 7: restricted.
- 4 sits in H1 and H3, which share row H: restricted.
- 6 sits in J3 and J4, which share row J: restricted.
- 7 sits in H1 and J4, which do not see each other: non-restricted.
The four cells need four values, and 3, 4 and 6 can supply at most three of them. So H1 or J4 is 7. H5 and H6 see H1 through row H and J4 through box 8, so 7 comes off both.
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 (in the example, H5 and H6 need to see only H1 and J4).
- 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.