Sudoku Wing Techniques Compared: XY, XYZ, W and M

The single idea behind all Sudoku wing techniques

Sudoku wing techniques all rest on bi-value cells, cells that hold exactly two candidates. In a wing there is a pivot and one or two pincers, sometimes called the wings. The pivot is linked to each pincer because they share a candidate and see each other, meaning they sit in the same row, column or box. The magic is that no matter which of its two values the pivot takes, it forces one of the pincers to become a specific target digit. If both pincers can be pushed to that same target from opposite outcomes, then the target must appear in one of them. Every cell that can see both pincers therefore cannot hold the target, so you erase it there. That is the entire mechanism, and each named wing is just a different arrangement of how the pivot and pincers share their candidates.

XY-Wing, the one to learn first

The XY-Wing uses three bi-value cells whose candidates read like a loop: one holds X and Y, the next X and Z, the last Y and Z. Put the pivot at r5c5 holding a 1 and a 2. Let one pincer be r5c8 with 2 and 3, and the other be r2c5 with 1 and 3. The pivot sees both because they share its row and column. If r5c5 is 1, then r2c5 must be 3; if r5c5 is 2, then r5c8 must be 3. Either way a 3 lands in one of the two pincers, so any cell seeing both pincers loses its 3. Here r2c8 sees r5c8 down its column and r2c5 along its row, so you delete 3 from r2c8. Reading that loop of shared digits is the key skill for every other wing.

XYZ-Wing folds the pivot into the firing line

The XYZ-Wing is a close cousin where the pivot carries all three candidates, a 1, a 2 and a 3, while the pincers are 1 with 3, and 2 with 3. Now the target 3 can come from the pivot as well as from either pincer, so the eliminations are stricter. A cell can only be cleared if it sees all three cells at once, pivot included. That usually means the victim shares a box with the pivot and lines up with both pincers. XYZ-Wings fire less often than XY-Wings for that reason, but when a stubborn box refuses to resolve they are often the answer that finally breaks it open.

W-Wing and M-Wing use a link, not a shared cell

W-Wing and M-Wing drop the shared pivot cell and use a strong link instead. A W-Wing takes two identical bi-value cells, say both holding 2 and 7, that do not see each other, and bridges them through a candidate. If the digit 2 has only two homes in some row and each of those homes sees one of your 2-7 cells, that row acts as the bridge. If both 2-7 cells were 2, the bridge row would have nowhere to put its 2, which is impossible, so at least one of them is 7. Any cell seeing both 2-7 cells then loses 7. The M-Wing rearranges the same parts with one bi-value cell and a conjugate pair, producing a similar squeeze. They feel abstract at first, yet they are simply the wing idea carried over a link rather than through a single pivot. In Shapedoku the matching pincers are easy to scan because each candidate is a distinct glowing shape, so a pair of identical bi-value cells jumps out by shape and colour together; the free solver at shapedoku.com can highlight one when you want to see it live.

• XY-Wing: three bi-value cells in an X-Y, X-Z, Y-Z loop; erase the shared Z from cells seeing both pincers.
• XYZ-Wing: pivot holds all three digits, so the victim must see the pivot and both pincers.
• W-Wing: two matching bi-value cells bridged by a strong link on the other digit.
• M-Wing: a bi-value cell plus a conjugate pair arranged to force one common digit.