Empty Rectangle
Using Box Geometry for Eliminations
By Minimal Sudoku Team•Last updated:
The Empty Rectangle is a single-digit technique that uses a specific pattern within a box — a 2×2 area empty of a candidate — combined with strong links to create eliminations.
Quick Summary
- Pattern: A box where a candidate forms an "L" or "+" shape (leaving a 2×2 empty rectangle)
- Requirement: A strong link connecting to the box pattern
- Result: Eliminate from the intersection of the empty rectangle's lines
- Difficulty: Advanced — requires understanding strong links and box geometry
The Concept
Within a 3×3 box, a candidate might be distributed in a way that leaves a 2×2 corner completely empty of that candidate. This "empty rectangle" creates a useful constraint when combined with strong links from outside the box.
The Empty Rectangle Pattern
⬜ Empty Rectangle Structure
A box contains an empty rectangle when a candidate appears only in cells forming an L-shape or cross pattern — never in a 2×2 corner region.
Empty Rectangle in a box (candidate X):
How It Works
The empty rectangle interacts with strong links outside the box:
- Find a strong link for the candidate (exactly 2 cells in a row/column)
- One end of the strong link should "see" the ER box
- The elimination happens at the intersection of the ER's defining lines and the strong link
Complex Geometry
Empty Rectangles are among the more geometric techniques in Sudoku. They're easier to understand with visual examples than text descriptions.
Similar to X-Cycle
Empty Rectangle can be understood as a type of discontinuous X-Cycle. If you understand X-Cycles, ERs will make more sense.
Detection Tips
Find the Empty Corner First
Scan boxes for candidates that don't appear in any 2×2 corner region. These are potential Empty Rectangles.
Look for Strong Links
Once you find an ER pattern, look for strong links (candidate in exactly 2 cells of a line) that connect to the box.
Use Software
Empty Rectangles are difficult to spot manually. Many solvers use software highlighting to find them efficiently.
Related Techniques
Single-Digit Patterns
ExpertX-Cycle
A more general technique that subsumes Empty Rectangle patterns.