Empty Rectangle

Level 8 · Blue 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

Starting position

Look for Empty Rectangle.

1

Scan box 7 for 9

Check digit 9 in box 7. All candidates form an 'L' or 'T' shape.

2

Empty Rectangle hinged at r7c3

This 'Empty Rectangle' points along row 7 and column 3 (the hinge is at r7c3).

3

Conjugate pair for 9 in row 1

Row 1 has a conjugate pair for 9; one endpoint lies on the Empty Rectangle column.

4

r7c7 sees both options for 9

Either the Empty Rectangle row or the conjugate pair tail must contain 9, so r7c7 sees both and cannot keep 9.

5

Remove 9 from r7c7

Remove 9 from r7c7.

6

Result

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).