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Unique Rectangle Technique: Advanced Sudoku Method

Introduction

The Unique Rectangle (UR) technique is one of the most important advanced strategies in Sudoku solving. Unlike many techniques that help you find the right number, Unique Rectangle exists to prevent invalid patterns—specifically the situation where the puzzle accidentally becomes ambiguous and allows multiple solutions.

A well-designed Sudoku puzzle must always have one unique solution. Unique Rectangle ensures you stay aligned with that rule and avoid configurations that could break uniqueness. This technique is especially useful in harder puzzles where cells start forming symmetrical patterns that look deceptively correct but lead to contradictions.

In this expanded guide, you'll learn what Unique Rectangles are, why they occur, and how to use them to eliminate candidates and unlock stuck puzzles.

What Is a Unique Rectangle?

A Unique Rectangle occurs when four cells form a rectangle across two rows and two columns, and these four cells contain the same pair of candidates.

Example pattern:

A | B
-----
C | D

Cells A, B, C, and D might all contain only {3, 7}.

Why is this a problem?

Because:

If all four cells can be either 3 or 7, you can swap the numbers across rows or columns, resulting in two valid solutions.
That violates Sudoku's uniqueness rule.

Key Points

Understanding Unique Rectangle is essential for advanced solving:

Uniqueness preservation: Prevents puzzles from having multiple valid solutions
Pattern recognition: Identifies four-cell rectangular patterns with identical candidate pairs
Candidate elimination: Removes unsafe candidates that would create ambiguity
Advanced technique: Most useful in harder puzzles with symmetrical candidate distributions
Logical foundation: Based on Sudoku's fundamental rule of having one unique solution

🎯 The Purpose of the Unique Rectangle Technique

Unique Rectangle is used to:

Prevent ambiguous solutions
Eliminate unsafe candidates
Maintain puzzle solvability
Detect hidden contradictions

It doesn't place numbers directly. Instead, it removes candidates that would allow a rectangle to remain unresolved.

In essence:

If forming a complete rectangle would cause multiple solutions, then one of the candidate options must be eliminated.

📐 Types of Unique Rectangle Patterns

There are several variations, but the most common forms include:

UR Type 1 — Basic Avoidance

Four cells contain only two candidates

→ Eliminate candidates to prevent ambiguity.

UR Type 2 — One Cell Has an Extra Candidate

Three cells contain {X, Y}

One contains an additional candidate Z

→ Z must be the correct elimination target.

UR Type 3 — Rectangle With a Strong Link

A strong link connects one pair of corners

→ A specific candidate gets eliminated.

UR Type 4 — Conjugate Pair Interaction

More complex structures involving chains

→ Still leads to targeted elimination.

This guide focuses on the foundational concepts that apply to all variations.

🧠 Why Unique Rectangles Work

Sudoku must have only one valid solution.

If you allow two pairs of cells in two rows and columns to contain the same two candidates, the puzzle becomes ambiguous:

3–7 in row 1
7–3 in row 2
Or vice-versa.

Since both arrangements satisfy Sudoku rules, the grid is no longer unique.

Thus, the solver must intervene:

The puzzle cannot allow the rectangle pattern to fully form, so one of the rectangle's candidate possibilities must be removed.

This logical contradiction allows safe elimination.

How It Works (Step-by-Step)

Here's how to apply the Unique Rectangle technique:

Step 1: Identify Rectangle Pattern

Look for four cells forming a rectangle (two rows, two columns) where all four cells contain the same two candidates. This creates a potential uniqueness violation.

Step 2: Check for Extra Candidates

Examine if any of the four cells contains additional candidates beyond the two shared candidates. Extra candidates break the rectangle pattern and prevent ambiguity.

Step 3: Apply Elimination Logic

If all four cells contain only the two candidates, eliminate one candidate from a cell that would prevent the rectangle from forming. The elimination ensures puzzle uniqueness.

Step 4: Verify Elimination

Confirm that the elimination doesn't create contradictions and maintains puzzle solvability. The elimination should reveal new placements or enable further progress.

Step 5: Continue Solving

After applying Unique Rectangle, update pencil marks and continue with other techniques. The elimination often reveals new singles or pairs.

📌 Step-by-Step Example

Let's say we're working with candidates 5 and 9.

You find:

Row 4, Column 3 = {5, 9}
Row 4, Column 7 = {5, 9}
Row 8, Column 3 = {5, 9}
Row 8, Column 7 = {5, 9}

These four cells form a perfect rectangle.

If left untouched, the solver could swap 5 ↔ 9 across the rectangle. Because Sudoku forbids multiple solutions, one of these cells must contain an extra candidate, or an elimination must occur.

Step 1: Check if any cell has more candidates

Suppose cell R4C7 actually has {5, 9, 2}.

Step 2: Remove {5, 9} from that cell?

No — instead:

Step 3: Candidate 2 must remain

Why?

Because:

If R4C7 is {5, 9} only, the rectangle becomes ambiguous.
So the extra candidate 2 is necessary.
And you can eliminate candidates elsewhere based on that constraint.

This prevents the deadly UR pattern.

🛠 When to Look for Unique Rectangles

URs typically appear:

In medium to very hard puzzles
When multiple boxes mirror each other
When candidate pencil marks are dense
When symmetric candidate patterns form

A fully marked-up grid helps identify them faster.

✔ Practical Tricks to Spot UR Patterns

1. Scan for identical pairs

Look for any pair like {4, 6}, {2, 8}, or {1, 9} repeating in two rows.

2. Check the opposite row or column

If the same pair appears in the same columns, you might have a UR.

3. Confirm the rectangle

Two rows + two columns = four cells.

4. Check extra candidates

If one corner has an extra number, it gives you leverage.

5. Test for contradictions

Ask yourself:

Would this lead to multiple solutions?

If yes → elimination is safe.

📊 Why Unique Rectangles Are Important

Unique Rectangle is one of the few strategies based purely on solution uniqueness, not on Sudoku's placement rules.

It helps avoid:

Unsolvable states
False assumptions
Ambiguity loops
Misleading candidate chains

In advanced puzzles, UR is often the missing step between:

"Everything is stuck."
and
"The grid suddenly opens up."

🧩 Common Variants at a Glance

| UR Type | Pattern | Key Action | |---------|---------|------------| | Type 1 | Plain rectangle | Prevent full UR formation | | Type 2 | One cell has extra candidate | Keep extra candidate, eliminate others | | Type 3 | Strong link aligns | Target linked candidate | | Type 4 | Conjugate chain involvement | Eliminations ripple outward |

🙋 FAQ

Q1: Does this technique place numbers?

Not directly.

It only eliminates candidates.

Q2: Can beginners use this?

It's more of an intermediate-to-advanced technique, but beginners can learn it with practice.

Q3: Why do some puzzles have many URs?

Because they are designed with symmetrical candidate layouts.

Q4: What happens if I ignore a UR?

You may accidentally allow multiple solutions or reach a contradiction later.

Q5: How many types of Unique Rectangle patterns exist?

There are four main types: Type 1 (basic avoidance), Type 2 (one cell with extra candidate), Type 3 (strong link interaction), and Type 4 (conjugate pair involvement). Each has specific elimination rules.

Q6: Do I need pencil marks to identify Unique Rectangles?

Yes, complete pencil marks are essential for spotting rectangular patterns with identical candidate pairs across multiple cells.

Examples

Example 1: Basic Unique Rectangle

Four cells form a rectangle: R4C3, R4C7, R8C3, R8C7 all contain {5, 9}. If cell R4C7 has an extra candidate 2, making it {5, 9, 2}, then the puzzle maintains uniqueness. We can use this to eliminate candidates accordingly.

Example 2: UR Type 2 Pattern

Three cells contain {X, Y} while one cell contains {X, Y, Z}. The extra candidate Z in the breaking cell prevents ambiguity, allowing targeted eliminations based on this constraint.

Summary

The Unique Rectangle technique is essential for solving advanced Sudoku logically and preventing invalid multi-solution states. This strategy identifies four-cell rectangular patterns with identical candidate pairs that would create multiple valid solutions, violating Sudoku's uniqueness rule. By recognizing these patterns and applying elimination logic, you make precise eliminations that keep puzzles solvable and unique.

Mastering UR strengthens your ability to analyze candidate structures deeply and helps you progress through high-level puzzles with more confidence. This technique is based purely on solution uniqueness rather than placement rules, making it a powerful tool when puzzles become stuck.

Ready to master Unique Rectangle? Try our Sudoku puzzles and apply this advanced technique!