WXYZ-Wing
A WXYZ-Wing is a wing of four cells that together hold exactly four candidates, W, X, Y and Z, laid out across two crossing units: a row or column and a box, or a row and a column. Three of the digits are confined: each appears only in cells that all see each other, so each can be placed in at most one of the four cells. The fourth digit, Z, appears in cells that do not all see each other. Four cells need four different digits, and the confined digits fill at most three of them, so at least one cell holding Z is Z. Z can therefore be removed from every cell that sees all the cells holding Z. The WXYZ-Wing extends the XYZ-Wing by one cell and one digit, and it is an Expert technique in Bare Sudoku.
When to use it
When the XYZ-Wing and the simpler wings find nothing and the puzzle has a cluster of cells with two to four candidates in one box and the row or column through it. WXYZ-Wings are less common than XYZ-Wings. They need complete notes in the cells involved.
How to spot it
Start from a cell with three or four candidates and collect cells in its box, row or column whose candidates are all among its own, until you have four cells with exactly four candidates between them, lying in two crossing units. For each digit, check whether all the cells holding it see each other. If exactly one digit fails that test, it is Z: every cell outside the four that sees all the cells holding Z loses Z. In the classic form one cell, the pivot, holds all four digits and the other three hold two each, but the pivot may hold fewer, and a wing may hold three.
Example
| c1 | c2 | c3 | c4 | c5 | c6 | c7 | c8 | c9 | |
|---|---|---|---|---|---|---|---|---|---|
| r1 | 29 | 5 | 6 | 4 | 1 | 8 | 7 | 239 | |
| r2 | 8 | 24 | 1 | 7 | 9 | 3 | 246 | 26 | 5 |
| r3 | 479 | 3 | 79 | 5 | 6 | 2 | 14 | 8 | 149 |
| r4 | 3 | 6 | 4 | 9 | 8 | 7 | 12 | 5 | 12 |
| r5 | 19 | 19 | 2 | 3 | 5 | 6 | 8 | 4 | 7 |
| r6 | 5 | 7 | 8 | 2 | 4 | 1 | 9 | 36 | 36 |
| r7 | 1247 | 124 | 57 | 8 | 3 | 45 | 2467 | 9 | 246 |
| r8 | 6 | 249 | 39 | 1 | 7 | 49 | 5 | 23 | 8 |
| r9 | 479 | 8 | 3579 | 6 | 2 | 459 | 347 | 1 | 34 |
Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.
- The four cells r2c7 (2 4 6), r2c8 (2 6), r3c7 (1 4), r4c7 (1 2) hold exactly four candidates between them: 1, 2, 4, 6.
- Digits 1, 4, 6 are each confined to cells that see each other, so each of them can occupy at most one of the four cells. Digit 2 is held by r2c7, r2c8, r4c7, which do not all see each other.
- The four cells need four different digits. The three confined digits fill at most three cells, so at least one cell is 2, and it is one of r2c7, r2c8, r4c7.
- A cell that sees all of r2c7, r2c8, r4c7 can never be 2. Remove 2 from r1c7.
Common mistakes
- Using four cells with five candidates between them. The count must be exactly four, or the cells could be filled without Z.
- Treating a digit as Z when its cells all see each other. Such a digit is confined and can occupy only one cell; Z is the one digit whose cells do not all see each other.
- Removing Z from cells that see only some of the cells holding Z. The eliminated cell must see every cell that holds Z, including the pivot if it has Z.
In Bare Sudoku
Needed from level: Expert
In the game, the hint says: WXYZ-Wing
Frequently asked questions
What is a WXYZ-Wing in Sudoku?
Four cells with exactly four candidates between them, bent across two crossing units, in which three digits are each confined to cells that see each other and the fourth, Z, is not. At least one of the cells holding Z is Z, so Z is removed from every cell that sees all of them.
How is a WXYZ-Wing different from an XYZ-Wing?
It has one more cell and one more digit. An XYZ-Wing has a three-candidate pivot and two wings; a WXYZ-Wing usually has a four-candidate pivot and three wings, but it also covers shapes where the digits are shared differently, as long as four cells hold exactly four candidates and only one digit is unconfined.
Does the pivot have to hold all four candidates?
No. Bare Sudoku accepts any four cells with four candidates in two crossing units where exactly one digit is unconfined. The classic pivot with all four digits is simply the most common form.
Is a WXYZ-Wing the same as a bent naked quad?
Yes. A bent naked subset is a naked subset whose cells lie in two crossing units instead of one. When exactly one digit is not confined to cells that see each other, that digit can be removed from cells that see all of its holders. The WXYZ-Wing is the four-cell case; the Y-Wing and the XYZ-Wing are three-cell cases.