Wings

WXYZ-Wing

A WXYZ-Wing is four cells, linked by seeing one another, that hold only four digits between them: W, X, Y and Z. For three of those digits, every cell of the four that holds it sees every other cell that holds it, so each of them can fill at most one of the four cells. Three digits can't fill four cells, so the fourth digit, Z, must be in one of them, and Z can be removed from any cell that sees every Z in the pattern.

Written by Georgi Nedyalkov

Level
Advanced

The pattern

WXYZ-Wing: what it does.

It's the XYZ-Wing grown by one cell: four cells and four digits instead of three and three, often a pivot holding several of the digits with wings around it.

The argument is a count. A restricted digit, one whose cells in the pattern all see each other, can go in the four cells at most once. With three restricted digits filling three cells at most, the fourth cell has to be Z, wherever it lands.

That's why the removal is narrow: Z could be in any of the pattern's Z cells, so only a cell that sees all of them is safe to clear.

Example

WXYZ-Wing: a worked example.

  1. Find four cells and four digits

    Look for four cells, linked by seeing each other, whose candidates together are only four digits. They usually include a cell with three or four candidates and a few pairs.

  2. Check each digit

    For every digit, look at the cells of the four that hold it. If those cells all see each other, the digit is restricted.

  3. Find the one exception

    Exactly one digit, Z, must be unrestricted, sitting in two cells that don't see each other. If every digit is restricted, the pattern removes nothing; if two aren't, it proves nothing.

  4. Clear Z

    Remove Z from every cell outside the pattern that sees all of the pattern's cells holding Z.

Spotting it

WXYZ-Wing: how to spot it.

  • Start from a cell with three or four candidates and look at the pairs it sees. Wings grow around a crowded cell.
  • Keep a running set of the digits as you add cells. When it reaches five, the group is too wide.
  • Expect a small result: one or two removals, in the cells that see every Z.
  • If all four cells sit in one row, column or box, you're looking at a naked quad instead, which removes more.

Where it stops working

A WXYZ-Wing takes careful checking, digit by digit, and it removes only Z, only from cells that see every Z in the pattern. One missing candidate can make a digit look restricted when it isn't, and the removal then becomes false.

Keep going

Related topics.

New to sudoku? The rules take two minutes to read, and the first-puzzle walkthrough covers the basics that come before any of these.

Beyond the lessons.

The game's lessons stop at X-Wing, so there's no WXYZ-Wing lesson. The eight it does teach will get you through almost any puzzle; this page is here for when you run into the pattern anyway.

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