Unique Rectangle
What's a Unique Rectangle?
A Unique Rectangle is an expert-level Sudoku technique that uses a powerful logical principle: every properly-constructed Sudoku puzzle has exactly one solution. When you spot a rectangular pattern that would create multiple solutions if completed, you know that pattern cannot exist—allowing you to make eliminations.
Specifically, if you find four cells arranged in a rectangle (spanning two rows, two columns, and two boxes) where the same two candidates appear in all four cells, completing this pattern would allow you to swap those candidates and create a second valid solution. Since this violates the uniqueness constraint, one or more of those cells must contain additional candidates or resolved values to prevent the deadly pattern.
This technique builds on the foundation of naked pairs and pattern recognition, but introduces the concept of uniqueness logic—a step beyond pure constraint-based deduction.
Why is it called "Unique Rectangle"?
It's called "Unique Rectangle" because it exploits the uniqueness of Sudoku solutions. The "rectangle" part comes from the geometric shape formed by the four cells involved. The technique identifies rectangles that, if allowed to complete with just two candidates, would create non-unique (multiple) solutions.
The "deadly pattern" name is also used because these configurations are "deadly" to puzzle uniqueness—they would kill the one-solution guarantee if they existed in their pure form.
Why It Matters
Unique Rectangle is a powerful technique for expert-level puzzles. It's somewhat controversial because it relies on the puzzle having a unique solution (meta-knowledge about puzzle construction) rather than pure logical deduction from the given clues. However, since standard Sudoku puzzles guarantee unique solutions, it's a valid and powerful tool.
For very difficult puzzles, Unique Rectangles can provide breakthroughs when all other logical techniques have been exhausted. It's often the difference between solving a diabolical puzzle logically versus resorting to trial and error.
Step-by-Step: How to Spot a Unique Rectangle
- Find four cells in a rectangle — They must span exactly two rows, two columns, and two boxes (forming a rectangle).
- Check for the same two candidates — All four cells should contain the same two candidates (e.g., all have {2,7}).
- Identify the pattern type — Look at which cells have only the two candidates versus which have extras.
- Apply elimination rules — Different types have different rules (Type 1, Type 2, etc.).
- Make the elimination — Remove candidates that would create the deadly pattern.
Unique Rectangle Types
Type 1: One Cell Has Extras
Three corners contain only the two candidates {X,Y}, and one corner contains {X,Y} plus extra candidates. You can eliminate X and Y from the cell with extras (because if it were X or Y, the deadly pattern would complete).
Example: Corners at R1C1, R1C5, R3C1, R3C5 all contain {3,8}. Three cells are bi-value {3,8}, but R3C5 contains {3,8,6,9}. You can eliminate 3 and 8 from R3C5, leaving {6,9}.
Type 2: Two Cells in Same Unit Have Extras
Two opposite corners contain only {X,Y}, and the other two corners (in the same row, column, or box) contain {X,Y} plus extra candidates. The extra candidates in those two cells form a naked pair, allowing eliminations in their shared unit.
Example: R2C2 and R2C6 contain only {4,9}, while R5C2 and R5C6 contain {4,9,1,7}. Since R5C2 and R5C6 are in the same row, {1,7} forms a naked pair—eliminate 1 and 7 from other cells in Row 5.
Type 3: Extra Candidates Form a Subset
Similar to Type 2, but the extra candidates form other patterns like naked triples or hidden subsets, allowing broader eliminations.
Type 4: Extra Candidates in Two Cells Create Strong Links
Two diagonal corners have only {X,Y}, the other two have extra candidates. If the extra candidates create strong links or X-Wing patterns, you can make eliminations based on those patterns.
Visual Example
Imagine four cells in a rectangle all containing candidates {2,7}:
- R1C3: {2,7}
- R1C8: {2,7}
- R4C3: {2,7}
- R4C8: {2,7,5}
Analysis: Three cells are pure {2,7}, one cell (R4C8) has an extra candidate 5. This is Type 1.
Elimination: If R4C8 were 2 or 7, the rectangle would complete as a deadly pattern. Therefore, R4C8 must be 5. We can eliminate 2 and 7 from R4C8, solving it as 5.
Strategies for Spotting Unique Rectangles Quickly
- Focus on bi-value cells — Cells with only two candidates are prime corners for Unique Rectangles.
- Look for matching pairs — Scan for cells with the same two candidates that form rectangles.
- Check box boundaries — Unique Rectangles span two boxes, so focus on cells near box edges.
- Use candidate highlighting — In digital solvers, highlight candidates to visually spot rectangles.
- Start with common candidates — Some candidates (like 1,2,7,8,9) appear more frequently in rectangles.
Common Pitfalls
- Incorrect rectangle geometry — The four cells must span exactly two rows, two columns, and two boxes. Don't confuse other patterns with Unique Rectangles.
- Missing the configuration type — Different types have different elimination rules. Misidentifying the type leads to incorrect eliminations.
- Applying to invalid puzzles — Unique Rectangle only works if the puzzle has a unique solution. Don't use it on puzzle variants with multiple solutions.
- Forgetting box constraint — The rectangle must span exactly two boxes. Four cells in the same box or spread across three boxes don't form valid Unique Rectangles.
Practice: Find the Unique Rectangle
Scenario: You have four cells:
- R2C4: {6,9}
- R2C7: {6,9,3}
- R6C4: {6,9}
- R6C7: {6,9}
Question: What type of Unique Rectangle is this, and what can you eliminate?
Answer: This is Type 1. Three cells contain only {6,9}, and one cell (R2C7) has an extra candidate 3. To prevent the deadly pattern, R2C7 cannot be 6 or 9. Therefore, R2C7 must be 3. You can eliminate 6 and 9 from R2C7, solving it as 3.
Why Unique Rectangle Matters
Unique Rectangle is a bridge between pure logical techniques and uniqueness-based reasoning. It's particularly valuable for:
- Breaking through deadlocks in expert-level puzzles
- Avoiding trial and error in very difficult puzzles
- Understanding puzzle construction principles
- Complementing other expert techniques like simple coloring and X-Wing
While some purists debate its validity, Unique Rectangle is widely accepted in the Sudoku community and is essential for solving the most difficult puzzles without guessing.
Quick Recap
| Technique | How it Works | Difficulty |
|---|---|---|
| Unique Rectangle | Uses uniqueness principle to eliminate candidates that would create multiple solutions | Expert |
| Naked Pairs | Two cells with same two candidates eliminate from unit | Intermediate |
| X-Wing | 2x2 pattern allows column/row eliminations | Advanced |
Final Thought
Unique Rectangle is a powerful tool in your expert Sudoku arsenal. While it requires understanding uniqueness logic beyond pure constraint deduction, it's essential for conquering the hardest puzzles without resorting to guessing. Master this technique, and you'll unlock solutions to puzzles that seem impossible.
Frequently Asked Questions
What is a Unique Rectangle in Sudoku?
A Unique Rectangle is an expert-level Sudoku technique based on the uniqueness principle: every valid Sudoku puzzle has exactly one solution. If you find a rectangular pattern of four cells across two boxes, two rows, and two columns where the same two candidates appear, completing it as-is would create multiple solutions. This impossibility allows you to eliminate candidates.
Why does a Unique Rectangle work?
Unique Rectangles work because they exploit the uniqueness constraint. If four cells in a rectangle pattern all contain only the same two candidates, you could swap those candidates and create a second valid solution. Since valid Sudoku puzzles have only one solution, this configuration is impossible, allowing you to eliminate candidates that would create it.
What are the different types of Unique Rectangle?
Common Unique Rectangle types include: Type 1 (one cell has extra candidates), Type 2 (two cells in same unit have extra candidates), Type 3 (extra candidates form a naked subset), Type 4 (extra candidates create strong links), Type 5 (combined with other techniques), and Type 6 (hidden rectangle). Each type has specific elimination rules.
Is Unique Rectangle controversial in Sudoku?
Yes, Unique Rectangle is somewhat controversial because it assumes the puzzle has a unique solution, which goes beyond pure logic deduction from the given clues. Some purists avoid it, preferring techniques that don't rely on uniqueness. However, it's widely accepted for standard Sudoku puzzles which guarantee unique solutions.
When should I look for Unique Rectangles?
Look for Unique Rectangles when stuck on expert-level puzzles after exhausting other techniques. They're particularly useful in very difficult puzzles where candidates are heavily constrained. Focus on bi-value cells (cells with only two candidates) as these often form the corners of Unique Rectangles.
Practice Unique Rectangle
Looking for more advanced techniques? Try Simple Coloring or Multi-Coloring.