Techniques / Wings

Wings: XY-Wing, XYZ-Wing, W-Wing and remote pairs

A pivot and its wings, forcing one digit out wherever both are seen.

8 techniques in this family, introduced at tier 5, 4 with a worked example from a real board. How to read a sudoku grid explains the notation used below.

What these have in common

A handful of cells, usually holding two candidates each, arranged so that however the pivot resolves, one particular digit ends up in one of the wings. You never learn which wing, and you do not need to.

Why the logic holds

Take a cell holding exactly X and Y. It sees one cell holding X and Z, and another holding Y and Z. If the pivot is X then the first wing must be Z; if it is Y then the second must be Z. Either way a Z appears in one of the wings, so no cell that sees both wings can hold Z. That is the XY-Wing, and every other wing is a variation on which digits sit where.

When to reach for it

Once pencil marks are complete and the grid is full of bivalue cells. Wings are found by scanning for two candidate cells and checking what each one sees, which is fast, so they are usually tried before chains.

How the members differ

XYZ-Wing lets the pivot carry Z as well, which costs you something: the elimination now has to see all three cells rather than two. WXYZ and VWXYZ extend the same trade further. W-Wing is two identical bivalue cells joined by a strong link in one of their digits. Remote Pairs is a chain of identical bivalue cells where position alternates, so the ends are opposite and anything seeing both loses both digits. The letter named wings are catalogue entries for particular shapes, and Bent Naked Subset is the general frame that most of them turn out to be instances of.

The mistake to avoid

Eliminating from a cell that only sees one wing. The elimination target must see every wing, and on a grid where the wings sit in different boxes that is easy to get wrong by one cell.

Every technique in this family

• Bent Naked SubsetA group of squares bent across two rows, columns or boxes.
• Letter-named wings (M, S, L, H, Split, Hybrid, Local)Short chains that people gave letters to.
• Remote PairsA string of squares that all hold the same two numbers, each one linked to the next.
• VWXYZ-WingThe same idea stretched to five squares and five numbers.
• W-WingTwo squares far apart hold the same two numbers.
• WXYZ-WingFour squares holding four numbers between them.
• XY-WingThree squares, each with two numbers left.
• XYZ-WingLike the XY-Wing, but the middle square has three numbers instead of two.