Unique Rectangles

Unique Rectangles exploit a fundamental property of valid Sudoku puzzles: they have exactly one solution. If a pattern would allow multiple solutions, it can't exist — and we can use this to make eliminations.

The Deadly Pattern

Consider this configuration:

        Col A       Col B
Row 1:  [3,7]       [3,7]
Row 5:  [3,7]       [3,7]

Four cells forming a rectangle, each with candidates [3,7], spanning exactly two boxes.

The problem: This grid has TWO valid solutions!

Solution 1:

        Col A       Col B
Row 1:    3           7
Row 5:    7           3

Solution 2:

        Col A       Col B
Row 1:    7           3
Row 5:    3           7

Both satisfy all Sudoku constraints! This means a valid puzzle with a unique solution can NEVER reach this state.

The insight: If we see this pattern forming, something must break it. We can use this to eliminate candidates.

Unique Rectangle Types

There are several UR types based on how close we are to the deadly pattern.

Type 1: One Extra Candidate

Three cells have exactly [3,7], one cell has [3,7,X]:

        Col A       Col B
Row 1:  [3,7]       [3,7]
Row 5:  [3,7]       [3,7,9] ← extra candidate

The logic:

• If R5CB were [3,7], we'd have the deadly pattern
• Deadly patterns can't exist in valid puzzles
• Therefore, R5CB must NOT be reduced to [3,7]
• R5CB must contain 9

Elimination: Remove 3 and 7 from R5CB, leaving only 9.

Or equivalently: R5CB = 9

Type 2: Two Extra Candidates (Same Extra)

Two cells have an extra candidate, both the same:

        Col A       Col B
Row 1:  [3,7]       [3,7]
Row 5:  [3,7,9]     [3,7,9] ← same extra in both

The logic:

• If neither R5CA nor R5CB had 9, we'd have a deadly pattern
• At least one of them must have 9
• Therefore, 9 must be in one of these two cells in row 5

Elimination: Remove 9 from all OTHER cells in row 5 (outside this pair).

This creates a hidden single or pair scenario.

Type 3: Two Extra Candidates (Forming Pair with Outside Cell)

Two floor cells have different extras that form a naked pair with another cell:

        Col A       Col B
Row 1:  [3,7]       [3,7]
Row 5:  [3,7,6]     [3,7,9]

If row 5 has another cell with [6,9], we might form a naked triple or pair.

More complex scenario — analyze based on what the extras create.

Type 4: Strong Link on Extra Digit

        Col A       Col B
Row 1:  [3,7]       [3,7]
Row 5:  [3,7,9]     [3,7,9]

If 9 appears only in R5CA and R5CB within row 5 (strong link):

Logic:

• 9 must be in R5CA or R5CB (strong link)
• Whichever has 9 cannot have both 3 and 7
• So the deadly pattern is prevented

But we can go further:

• Consider the columns: if 3 or 7 has a strong link in columns A or B...
• Complex interactions can force eliminations

Type 5 and Beyond

There are additional UR types involving:

• Strong links within the rectangle
• External candidates that resolve ambiguity
• Combinations with other techniques

These become quite advanced and are often found by software.

Recognizing Unique Rectangles

Visual Indicators

Look for:

• Four cells forming a rectangle
• Spanning exactly two boxes (row-wise or column-wise)
• Containing the same two "core" candidates
• Some cells may have additional candidates

The "Almost Deadly" State

When you see [X,Y] appearing in multiple cells that form a rectangle:

• Count how many cells have ONLY [X,Y]
• Count how many have [X,Y] + extra
• Apply the appropriate UR type

Step-by-Step: Type 1 Example

Given grid (partial):

        Col 2       Col 8
Row 3:  [1,6]       [1,6]       ← Box 1 and Box 3
Row 9:  [1,6]       [1,6,5]     ← Box 7 and Box 9

Analysis:

1. Rectangle spans rows 3, 9 and columns 2, 8
2. Boxes involved: 1, 3, 7, 9 (two per row — valid UR setup)
3. Three cells have exactly [1,6]
4. One cell (R9C8) has [1,6,5]

Conclusion:

• R9C8 cannot be reduced to [1,6] (would create deadly pattern)
• R9C8 must be 5

Action: Place 5 in R9C8.

Step-by-Step: Type 2 Example

Given grid (partial):

        Col 3       Col 7
Row 2:  [4,8]       [4,8]
Row 6:  [4,8,2]     [4,8,2]

Analysis:

1. Rectangle: rows 2, 6; columns 3, 7
2. Two "floor" cells (R2) have [4,8]
3. Two "roof" cells (R6) have [4,8,2]

Conclusion:

• At least one roof cell must have 2 (to prevent deadly pattern)
• 2 must be in R6C3 or R6C7
• This acts as a hidden pair for 2 in row 6

Action: Eliminate 2 from all other cells in row 6.

When Unique Rectangles Don't Apply

Invalid setups:

• Rectangle doesn't span exactly two boxes
• Cells share a single box (no ambiguity arises)
• The "deadly" digits aren't actually the same pair

Already broken patterns:

• If a cell in the rectangle is already solved
• If the extra candidates are already forcing a unique solution

UR Strategy

When to Look

• After basic techniques plateau
• When you notice bi-value cells forming rectangles
• In medium-hard puzzles and above

What to Check

1. Are there four cells forming a rectangle?
2. Do they span exactly two boxes?
3. Do they share a common pair of candidates?
4. Are there extra candidates preventing the deadly pattern?

Order of Types

• Type 1: Look first — immediate placement
• Type 2: Look second — clear elimination rule
• Types 3-5: More complex, often software-assisted

Practice Exercise

Find the Unique Rectangle and resolution:

        Col 4       Col 9
Row 1:  [2,5]       [2,5]
Row 7:  [2,5,8]     [2,5]

Common Mistakes

Mistake 1: Applying UR to single-box rectangles

The deadly pattern only exists when the rectangle spans TWO boxes. Within one box, other constraints prevent ambiguity.

Mistake 2: Wrong candidate identification

The "deadly pair" must be the same in all four cells. Extra candidates are what break the pattern.

Mistake 3: Forgetting the uniqueness assumption

URs assume the puzzle has exactly one solution. Some non-standard puzzles intentionally have multiple solutions — URs don't apply there.

Mistake 4: Missing UR opportunities

Since URs are based on prevention logic (not direct forcing), they're easy to overlook. Actively scan for rectangles.

Quick Reference

Unique Rectangle setup:

• 4 cells forming a rectangle
• Spanning exactly 2 boxes
• Sharing a pair of candidates [X,Y]

Type 1 (one extra):

• Three cells: [X,Y]
• One cell: [X,Y,Z]
• Resolution: that cell = Z

Type 2 (same extra in two):

• Two cells: [X,Y]
• Two cells: [X,Y,Z]
• Resolution: Z must be in one of the [X,Y,Z] cells; eliminate Z elsewhere in their shared unit

Why it works:

• Deadly patterns would allow multiple solutions
• Valid Sudoku has exactly one solution
• Therefore, deadly patterns can't form

When to look:

• Bi-value cells forming rectangles
• Medium difficulty and above
• After scanning for simpler patterns