WXYZ Wing

WXYZ-Wing is an extension of the XYZ-Wing strategy used to eliminate candidates. It is sometimes called ‘Bent Quadruple’ or ‘Bent Quad’ and is considered an advanced strategy.

What is WXYZ-Wing?

• Four cells and four different digits.
• One digit out of the four (Z) cannot see all the other instances of itself in the four cells. For the other three digits, each instance of a digit can see all the other instances of itself in the four cells.

In WXYZ-Wing, the digit Z must appear in one of the cells of the WXYZ-Wing.

The Action:

Eliminate Z from all the cells that see all the Z’s in the WXYZ-Wing.

Scenarios

Scenario 1:

The four cells below form a WXYZ-Wing with digits W, X, Y, and Z. The digit Z in cell 4 cannot see the Z in cell 2 and cell 3. Each of the other digits (W, X, Y) can see all the instances of themselves in the four cells.

• Y: Cell 1 sees cell 2.
• X: Cell 1 sees cell 3.
• W: Cell 1 see_ cell 4.

According to WXYZ-Wing, the correct answer for one of the four cells must be the digit Z. If none of the cells is Z, cell 1 would have no candidates left. Thus, Z must be in one of the cells, and Z can be eliminated from all the cells that see all the cells with Z in the WXYZ-Wing (highlighted in red).

Scenario 2:

This setup also meets the WXYZ-Wing definition. Four cells and four digits are present, with the digit Z unable to see all instances of itself. One cell (cell 4) has three candidates. According to WXYZ-Wing, one of the cells must be Z. If none of the cells is Z, cell 1 would have no candidates left, leading to a contradiction. Therefore, Z can be eliminated from all cells that see all instances of Z in the WXYZ-Wing (highlighted in red).

Below are additional WXYZ-Wing scenarios. Verify they meet the definition and understand why Z must be in one of the cells containing it.

Examples

Example 1:

The highlighted cells below form a WXYZ-Wing with digits 1, 5, 8, 9. The digit 8 cannot see all instances of itself; for instance, B9 cannot see the 8 in F7 and I7. According to WXYZ-Wing, 8 must be in one of these cells. Therefore, 8 can be eliminated from A7.

Why is this true?

• If C7 = 1, then B9 = 8.
• If C7 = 9, then F7 = 8.
• If C7 = 5, F7 and I7 must be 8 & 9.

No matter the value in C7, 8 must be in one of the other cells, allowing elimination of 8 from A7.

Example 2:

The highlighted cells below form a WXYZ-Wing with digits 1, 4, 8 & 9. The digit 9 doesn’t see all instances of itself; for instance, F7 cannot see E4 and E6. According to WXYZ-Wing, all 9’s that see all instances of 9 in the WXYZ-Wing can be eliminated. The only 9 that fits this criteria is in F5.

Example 3:

This board has two variants (Thermo and Chess Knight Sudoku) alongside Classic Sudoku, though these variants are irrelevant for this example. The highlighted cells below form a WXYZ-Wing with digits 1, 3, 4 & 6. The digit 3 doesn’t see all its instances, so it must be in one of the WXYZ-Wing cells. Therefore, 3 can be eliminated from cells D1, D3, and F8, which see all instances of 3 in the WXYZ-Wing.