LeafyOrb

Sudoku techniques · Fiendish level

XYZ-Wing: an XY-Wing with a three-digit pivot

An XYZ-Wing has a pivot with three candidates, X, Y and Z, that sees two pincers: one with X and Z, one with Y and Z. One of the three cells must be Z, so any cell that sees all three can lose Z.

How it works

Say the pivot can be 4, 7 or 9, one pincer 4 or 9, and the other 4 or 7. If the pivot is 9, the first pincer is 4. If it is 7, the second pincer is 4. If it is 4, the pivot holds the 4 itself. Every way, one of the three cells is 4.

Because the pivot can be Z too, a cell has to see the pivot as well as both pincers to lose Z. That usually means a cell in the pivot's box, on the line it shares with one pincer. So the XYZ-Wing removes less than the XY-Wing, but you find it in the same way.

  • The row, column or box the move uses
  • The cells that make the pattern
  • Candidates the move removes
  • Links from the pivot to its pincers
XYZ-Wing example on a Sudoku board: One of R7C2, R6C2 and R7C3 must be 4. A cell that sees all three can't be 4.

Example 1 · from Fiendish puzzle #24

The pivot R7C2 can be 4, 7 or 9. It sees R6C2 (4 or 9) and R7C3 (4 or 7). If R7C2 is 9, R6C2 must be 4; if it is 7, R7C3 must be 4; otherwise R7C2 is 4 itself. One of the three is always 4, so a cell that sees all three cannot be. Remove 4 from R9C2.

Play this position in Ninebloom: Home › Your own puzzle, then paste the code. NG-25VQ23-HQ1DMH-M0NCHN-JPMG2Q-KIANZ6-4ZZXRX-CR9FOO-MOZT0M-J5JUS

Cells are named by row and column: R4C3 is row 4, column 3. Boxes are numbered 1 to 9, left to right and top to bottom.

How to spot it

  1. Fill in all pencil marks and look for cells with exactly three candidates. Each is a possible pivot.
  2. Among the two-candidate cells the pivot sees, look for two whose digits are both inside the pivot's three and share one digit, Z.
  3. Find the cells that see the pivot and both pincers.
  4. Remove Z from those cells.

More examples

XYZ-Wing example on a Sudoku board: One of R7C2, R3C2 and R7C3 must be 5. A cell that sees all three can't be 5.

Example 2 · from Fiendish puzzle #301

The pivot R7C2 can be 5, 6 or 9. It sees R3C2 (5 or 6) and R7C3 (5 or 9). If R7C2 is 6, R3C2 must be 5; if it is 9, R7C3 must be 5; otherwise R7C2 is 5 itself. One of the three is always 5, so a cell that sees all three cannot be. Remove 5 from R8C2.

Play this position in Ninebloom: Home › Your own puzzle, then paste the code. NG-172JMN-CLIE6F-P0KY4N-46285Y-ELJZCM-69TTCL-0HENAT-KX8KT5-RXFMQ

Common mistakes

  • Removing Z from a cell that sees the pincers but not the pivot. The pivot can be Z, so the cell must see all three.
  • A pincer with a digit the pivot does not have. Both pincers must use only the pivot's digits.
  • Two pincers with the same two digits. They must share Z and differ in their other digit.

Questions

What is the difference between an XYZ-Wing and an XY-Wing?

In an XY-Wing the pivot has two candidates and cannot be Z, so cells that see both pincers lose Z. In an XYZ-Wing the pivot also holds Z, so a cell must see the pivot as well.

Why does the XYZ-Wing remove so little?

Few cells can see three cells at once. In practice it is usually one or two cells in the pivot's box.

Which Ninebloom level uses the XYZ-Wing?

Fiendish, alongside the Swordfish, XY-Wing, simple colouring and the unique rectangle.