XYZ-Wing

What is an XYZ-Wing?

An XYZ-Wing is an XY-Wing whose pivot has a third candidate. The pivot holds three digits, X, Y and Z. It sees two pincers with two candidates each: one holds X and Z, the other Y and Z. There are three ways things can go: the pivot is X and the first pincer is Z, the pivot is Y and the second pincer is Z, or the pivot is Z itself. In every case one of the three cells is Z, so a cell that sees all three cannot be Z.

Because the pivot can be Z, the reach is narrower than an XY-Wing’s: a cell must see the pivot as well as both pincers.

How to spot an XYZ-Wing

Start from a cell with three candidates. Look for two two-candidate cells it sees, each made only of the pivot’s digits and sharing exactly one digit with each other. The cells that can lose that digit are the ones that see all three, which in practice means cells in the pivot’s box that also share a line with the pincer outside it.

With one pincer in the pivot’s box and the other in its row or column, there are only ever one or two such cells, so the check is quick.

A worked example

Cells are named by row and column: r4c1 is row 4, column 1. Rows count from the top, columns from the left, and boxes are numbered 1 to 9 reading like a book, so box 1 is top-left and box 9 is bottom-right.

This position is from an Expert puzzle our game dealt, with every candidate pencilled in from the digits on the board; the diagram shows candidates only in the pattern and the cell it clears. Nothing simpler moves it forward: none of the earlier techniques in this guide, up to and including the Swordfish, removes anything.

The pivot is r4c4, which holds 6, 8 and 9. It sees r1c4 (6 and 9) up its column and r4c6 (8 and 9) along its row, inside box 5. The pincers share the 9. If the pivot is 6, r1c4 must be 9. If it is 8, r4c6 must be 9. If it is 9, the pivot is the 9.

So one of those three cells is 9. r6c4 sees all three: the pivot and r1c4 along column 4, and r4c6 in box 5. It cannot be 9, and as it held only 2 and 9, r6c4 is 2.

Why it works

The pivot has three possibilities, and every one of them puts a 9 into the pivot or a pincer. A 9 at r6c4 would share column 4 with the pivot and r1c4, and box 5 with r4c6, so it would clash with whichever of them is the 9.

Which difficulty levels need it

Expert only. Our game's own solver stops at naked pairs, and any puzzle it cannot finish with those is rated Expert, so no Easy, Medium or Hard puzzle needs an XYZ-Wing.

We measured 3,000 Expert puzzles our generator dealt. A solver that always tries the simplest technique first made a move with an XYZ-Wing in 507 of them. For 58 (1.9%), the XYZ-Wing was the hardest technique the puzzle needed: everything simpler stalls, and adding it finishes the grid. Of the 719 puzzles that the twelve techniques in this guide finish on their own, 58 cannot be finished without it. The rest of our Expert puzzles need techniques this guide does not cover yet, almost all of them chains.

Related: XY-Wing .