Empty Rectangle

Hard8 min

Overview

An Empty Rectangle uses one digit in a box where every place for it lies on one row and one column of the box, which cross. Together with a strong link outside the box, it removes the digit from one cell.

How It Works

Suppose every place for a digit in a box lies on one row and one column of that box, which cross: the four cells off that row and column, the empty rectangle, can't hold it. Now take a column that doesn't pass through the box, where the digit fits only two cells: one on the box's row, the other outside the box's rows. Check the cell where the other end's row meets the box's column. If the far end holds the digit, that cell shares its row. If the near end holds it, the box's row is taken, so the box's digit is in the box's column, which that cell shares. Either way, the cell can't hold the digit. The same works with rows and columns swapped.

When To Use It

Look for an Empty Rectangle when a box has only a few places left for a digit, all along one row and one column of the box, and the digit has a strong link in a row or column outside the box.

How To Detect It

For each digit, find boxes where every place for it is on one row and one column of the box. Then look for a column (or row) that doesn't pass through the box, where the digit has exactly two places: one on the box's row (or column), the other outside the box's rows (or columns). Remove the digit from the cell where the link's other end lines up with the box's column (or row).

Visual Example

Step 1 of 9

One Box, One Row, One Column

An Empty Rectangle works with one digit. In one box, every place for the digit lies on one row and one column of the box, which cross. A strong link outside the box then removes the digit from one cell.

Where to look

Pattern cells

Eliminations

Removed

Ruled out here

7Solved

Key candidates

This step's candidates

Supposed, not known

One of them must be

Can't both be

From an earlier step

Candidates already ruled out by earlier techniques aren't shown.

Practice

Your goal: fill the highlighted cell. Select a cell, then pick a number.

An Empty Rectangle is hiding here. Find a box whose places for a digit fit on one row and one column, find the strong link, then eliminate and fill the Naked Single.Candidates already ruled out by earlier techniques aren't shown.

Use it in a real game

Hard puzzles are where you'll need techniques like this one.

Play a Hard puzzle