Unique Rectangle is a fascinating technique that uses the principle of uniqueness: a valid Sudoku has exactly one solution. By identifying patterns that would create multiple solutions, we can eliminate candidates.
Quick Summary
Premise: Valid Sudokus have exactly one solution
Pattern: Rectangle of 4 cells in 2 rows, 2 columns, 2 boxes
Deadly Pattern: If all 4 cells had only the same 2 candidates → 2 solutions!
Result: Eliminate candidates that would create the deadly pattern
The Concept
A properly constructed Sudoku puzzle has exactly one solution. If a pattern would allow two or more valid solutions, that pattern cannot exist in a valid puzzle.
Unique Rectangles identify potential "deadly patterns" — configurations that would create multiple solutions — and eliminate candidates to prevent them.
The Deadly Pattern
💀 The Deadly Pattern
Four cells forming a rectangle across 2 rows, 2 columns, and 2 boxes, where all 4 cells contain only the same two candidates, creates a deadly pattern — the candidates can be swapped diagonally, giving 2 solutions.
Deadly Pattern (must be avoided!):
Swapping the pair diagonally creates two solutions — the deadly pattern.
→ If all four cells are only 3/7, the puzzle would be invalid.
Types of Unique Rectangles
There are several types of Unique Rectangles, based on how many "extra" candidates appear in the rectangle cells:
Type 1
3 cells have only the pair, 1 cell has extra candidates.
→ Eliminate the pair from the "extra" cell.
Type 2
2 cells have extra candidates (same extra candidate).
→ Eliminate that extra from cells seeing both.
Types 3-6 exist but are increasingly rare and complex. Type 1 accounts for most real-world Unique Rectangles.
Type 1 (Most Common)
In Type 1, three corners of the rectangle are "locked" — they contain only the deadly pair (e.g., 3,7). The fourth corner has additional candidates.
Type 1 Unique Rectangle:
Three corners are locked to 3/7; the fourth corner has an extra candidate.
→ Eliminate 3 and 7 from the extra corner, leaving 9.
Often Gives Direct Placement
Type 1 Unique Rectangles often leave the "extra" cell with just one candidate after elimination — a direct placement!
How to Find Unique Rectangles
1
Find bi-value cells
Look for cells with exactly 2 candidates.
2
Look for matching pairs
Find bi-value cells with the same two candidates that could form a rectangle (same 2 rows, same 2 columns).
3
Check the boxes
Verify the 4 cells span exactly 2 boxes (not all in one box).
4
Count locked corners
How many corners have ONLY the deadly pair? 3 corners locked = Type 1.
5
Apply the elimination
Remove the deadly pair candidates from the "extra" cell(s).
Detection Tips
Start with Common Pairs
Look for bi-value cells with the same candidates appearing multiple times. Common pairs like (1,2), (3,7), etc., are good starting points.
Must Span 2 Boxes
The rectangle must span exactly 2 different boxes. If all 4 cells are in one box, or in more than 2 boxes, it's not a valid Unique Rectangle pattern.
Assumes Valid Puzzle
This technique assumes the puzzle has a unique solution. If you're solving a puzzle that might have multiple solutions, Unique Rectangle logic doesn't apply.
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