Unique Rectangles: Using Puzzle Uniqueness

Unique Rectangles is a fascinating technique that uses a fundamental property of valid Sudoku puzzles: they must have exactly one solution. This constraint allows for powerful eliminations!

The Foundation: Puzzle Uniqueness

A properly constructed Sudoku puzzle has:

• Exactly one valid solution
• No ambiguity
• No guessing required

This uniqueness principle is the key to this technique.

What is a Unique Rectangle?

A Unique Rectangle is a deadly pattern that would create multiple solutions if allowed to exist. It consists of:

• Four cells arranged in a rectangle
• Two rows and two columns
• Same two candidates in all four cells

If all four cells had only these two candidates, the puzzle would have at least two solutions (swapping the values in the rectangle). Since valid puzzles can't have multiple solutions, we can use this to eliminate candidates!

The Deadly Pattern

Row A:  (X,Y)  ...  (X,Y)
Row B:  (X,Y)  ...  (X,Y)
        Col 1      Col 2

If all four cells contained only (X, Y), you could swap them and have two solutions. This can't happen in a valid puzzle!

How to Use Unique Rectangles

When you find a near-unique rectangle where three corners have (X, Y) and the fourth has (X, Y, Z):

You can eliminate X and Y from the fourth cell!

Why? If the fourth cell were X or Y, it would complete the deadly pattern and create multiple solutions.

The Four Types

Type 1: Classic Unique Rectangle

• 3 cells have exactly (X, Y)
• 1 cell has (X, Y, Z)

Elimination: Remove X and Y from the fourth cell, leaving only Z

Type 2: Row/Column Elimination

• 2 cells in one row/column have (X, Y)
• 2 cells in another row/column have (X, Y, ...other candidates)
• The extra candidates are in different cells

Elimination: Remove X and Y from the row/column where they appear in only those two cells

Type 3: Naked Pair Creation

• The two "extra candidate" cells are in the same row/column
• Both have additional candidate Z

Result: Those two cells form a naked pair for Z, eliminate Z elsewhere

Type 4: Conjugate Pair

• One candidate appears in only two places in the rectangle

Result: Use this conjugate pair for further eliminations

Requirements for Unique Rectangles

The pattern MUST:

1. Be in exactly 2 rows and 2 columns
2. Involve exactly 2 candidates as the deadly pair
3. Have cells in two different boxes (bi-value cells in same box don't create multiple solutions)
4. Be found in a valid puzzle (puzzle must have unique solution)

When to Look for Unique Rectangles

Search when you see:

• Multiple bi-value cells with the same two candidates
• Cells arranged in rectangular patterns
• Stuck puzzles that need advanced techniques

Common Mistakes

• Using in same-box rectangles: The four cells must span at least two boxes
• Forgetting the uniqueness assumption: Only works if the puzzle is properly constructed
• Misidentifying the pattern: All four corners must have the same two candidates as candidates
• Wrong elimination: Make sure you're eliminating from the right cells
• Using in puzzles with multiple solutions: This technique requires unique-solution puzzles

The Controversy

Some solvers debate whether Unique Rectangles are "pure" logic since they rely on puzzle construction rather than just Sudoku rules. However:

• They're accepted in competitions
• They work on all properly-made puzzles
• They're a legitimate solving technique
• They're necessary for some expert puzzles

Example Walkthrough

Cells:

• R2C3: (4, 7)
• R2C8: (4, 7)
• R6C3: (4, 7)
• R6C8: (4, 7, 9)

This is a near-deadly rectangle. If R6C8 were 4 or 7, the puzzle would have multiple solutions.

Elimination: Remove 4 and 7 from R6C8, leaving 9 Result: R6C8 = 9

Recognition Tips

1. Look for bi-value cells with matching candidates
2. Check if they form rectangles (2 rows × 2 columns)
3. Verify they're in different boxes
4. Identify which type of Unique Rectangle it is
5. Apply the appropriate elimination

Advanced Unique Rectangles

Once you master the basics, explore:

• Avoidable Rectangles: More complex versions
• Unique Loops: Circular extensions
• Hidden Rectangles: Less obvious patterns
• BUG (Bi-value Universal Grave): Extended uniqueness technique

Frequency

Unique Rectangles:

• Appear in hard/expert puzzles
• Are less common than X-Wing or XY-Wing
• Often provide crucial breakthroughs
• Become easier to spot with practice

Practice Strategy

1. Mark all bi-value cells in the puzzle
2. Look for cells with matching candidate pairs
3. Check if any four form a rectangle
4. Verify they span multiple boxes
5. Identify the type and apply elimination

Integration with Other Techniques

Unique Rectangles work well with:

• Naked Pairs: Often creates or uses them
• X-Wing: Similar rectangular thinking
• Forcing Chains: Can be starting points
• Coloring: Can highlight patterns

The Uniqueness Principle

This technique teaches an important lesson: in Sudoku, the puzzle construction itself provides information. The requirement for a unique solution is a constraint just as important as the row/column/box rules!

When Not to Use

Avoid Unique Rectangles when:

• Puzzle might have multiple solutions
• You're not sure about puzzle validity
• Simpler techniques haven't been exhausted
• You're practicing "pure logic" solving

The Expert Edge

Mastering Unique Rectangles:

• Separates expert from advanced solvers
• Opens up otherwise-impossible puzzles
• Develops pattern recognition skills
• Provides deep satisfaction when correctly applied

Next Steps

Continue your expert journey with:

• BUG (Bi-value Universal Grave) - Extended uniqueness
• Sue de Coq: Complex intersections
• Advanced Coloring: Multi-color chains

You're now among the elite Sudoku solvers. Well done!