Fish

Empty rectangle

An empty rectangle is a box where a digit's candidates all sit on one row and one column of it, in a cross or an L, while the rest of the box has none. Wherever the digit lands, the box puts it on that row or that column. Add a strong link outside the box, a column with only two homes for the digit, one of them on the box's row, and the digit can be removed from the cell where the link's other end lines up with the box's column.

Written by Georgi Nedyalkov

Level
Advanced

The pattern

Empty rectangle: what it does.

The name comes from the empty part: the cells of the box outside that row and column, which hold no candidate for the digit. Because they're empty, the box's digit is forced onto its cross.

Now take a column outside the box with exactly two homes for the digit, one of them on the box's row. If that cell is the digit, the box can't use its row, so its digit is on its column. If it isn't, the column's other home is the digit.

Look at the cell where the far home's row crosses the box's column. In the first case the box's digit sees it down the column; in the second, the far home sees it along the row. Either way it can't be the digit. Turned on its side, the same argument works with a row link and the box's column.

Example

Empty rectangle: a worked example.

  1. Find a box with a cross

    Follow one digit and look for a box where every candidate for it lies on one row and one column of that box. They mustn't all lie on just one of them, or it's a pointing pair instead.

  2. Find a strong link that touches the cross

    Look outside the box for a column with exactly two homes for the digit, one of them on the box's row. Or for a row with exactly two homes, one of them on the box's column.

  3. Follow the link to its far end

    The link's other home is the far end. Follow its row, or its column for a row link, to where it crosses the box's other line.

  4. Remove the digit at the crossing

    That crossing cell, outside the box, can't hold the digit. Remove it.

Spotting it

Empty rectangle: how to spot it.

  • Boxes with only two or three candidates for a digit are where empty rectangles hide. Check whether those candidates fit on one row and one column.
  • The cross is like a pointing pair that points two ways at once. On its own it removes nothing; it needs the strong link to choose between them.
  • The strong link must be outside the box and have one end exactly on the box's row or column, not just somewhere in the same band of boxes.
  • Try both directions: a column link against the box's row, and a row link against the box's column. One box can give two removals.

Where it stops working

An empty rectangle removes exactly one candidate, and only if the crossing cell still has the digit. It needs the box's candidates to be complete, because one missing mark can make a cross where there isn't one, and a link that really has two homes, not three.

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 Empty rectangle 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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