Empty Rectangle

Level 5 · Orange Belt

Strategy:

Single-Digit Patterns

Requires:

Locked Candidates

Overview

Empty Rectangle uses the absence of a candidate in a rectangular area within a box. When four cells form an empty rectangle for a candidate, combined with a strong link, eliminations become possible.

How It Works

  1. Find a box where a candidate forms an "L" shape (all instances in one row and one column of the box).
  2. The remaining 4 cells form an empty rectangle for that candidate.
  3. Find a strong link outside the box that shares a line with the non-empty row/column.
  4. Eliminate from the intersection of the other strong link cell's line and the box's non-empty column/row.

Example

Look for Empty Rectangle.

1

Empty Rectangle

Check digit 2 in box 5. All candidates form an 'L' or 'T' shape.

2

Empty Rectangle

This 'Empty Rectangle' points along row 4 and column 6 (the hinge is at r4c6).

3

Empty Rectangle

Row 2 has a conjugate pair for 2 - one endpoint is at the ER's column.

4

Empty Rectangle

Either the ER's row or the conjugate's other end must contain 2. So r4c1, which sees both, cannot keep 2.

5

Empty Rectangle

Remove 2 from r4c1.

6

Result after applying the technique.

Tips & Patterns

  • Empty Rectangles are easier to find when you look for the "L" pattern first.
  • The technique exploits that the candidate must be in one of the L's arms.
  • Practice identifying Empty Rectangle Intersections (ERIs).