Unique Loop Type 1 is an advanced "uniqueness" strategy that extends the concept of Unique Rectangle (Type 1) to larger patterns. While Unique Rectangles work with exactly 4 cells, Unique Loops handle patterns of 6, 8, 10, or more cells that form a closed loop.
This strategy relies on a fundamental Sudoku rule: every valid puzzle must have exactly one unique solution.
Interactive Example
Click "Apply Logic" to see the strategy in action.
Real Example Walkthrough
In the example puzzle above, the solver identifies a 6-cell Unique Loop:
The Loop Cells:
| Cell | Position | Candidates Before |
|---|---|---|
| 1 | R1C2 | {3, 9} |
| 3 | R1C4 | {3, 9} |
| 10 | R2C2 | {3, 9} |
| 13 | R2C5 | {3, 9} |
| 57 | R7C4 | {3, 9} |
| 58 | R7C5 | {3, 5, 9} ← Rescue Cell |
Step-by-Step Analysis:
- Identify the loop pair: Candidates {3, 9} appear in all 6 cells
- Count cells with extras: Only R7C5 has an extra candidate (5)
- Apply Type 1 logic: R7C5 cannot be 3 or 9, because that would complete the deadly pattern
- Eliminate: Remove 3 and 9 from R7C5
Result: R7C5 is reduced to {5}, which places the value 5 in that cell.
What Is a Unique Loop?
A Unique Loop is a series of cells that:
- Forms a closed loop (the chain returns to its starting cell)
- Contains 4 or more cells (most commonly 6 cells)
- All cells share the same two candidate values (the "loop pair")
- The loop alternates through rows, columns, and boxes in a specific pattern
If all cells in the loop contained only the loop pair (e.g., {3, 9}), the puzzle would have two valid solutions—you could swap the values around the loop and both arrangements would work. This is called a Deadly Pattern.
The Deadly Pattern Problem
Imagine a 6-cell loop where every cell contains exactly {3, 9}:
| Cell Position | Candidates |
|---|---|
| R1C2 | {3, 9} |
| R1C4 | {3, 9} |
| R2C2 | {3, 9} |
| R2C5 | {3, 9} |
| R7C4 | {3, 9} |
| R7C5 | {3, 9} |
In this state, you could place either: - 3 in the odd positions and 9 in the even positions, OR - 9 in the odd positions and 3 in the even positions
Both arrangements satisfy all Sudoku rules—creating two solutions. Since a proper Sudoku cannot have multiple solutions, this pattern is impossible and must be prevented.
Type 1: The Single Rescue Cell
Unique Loop Type 1 applies when:
- All cells in the loop contain the loop pair
- Exactly one cell has extra candidates beyond the pair
This cell is called the "rescue cell" because it "rescues" the puzzle from having a deadly pattern.
The Logic
Since the deadly pattern cannot exist: - The rescue cell must NOT be one of the loop pair values - It must be one of its extra candidates - Therefore, we can eliminate the loop pair values from the rescue cell
How the Loop Forms
Unlike Unique Rectangles which use a simple 2×2 shape, Unique Loops follow a path through the grid:
Loop path (simplified):
R1C2 → (Row 1) → R1C4 → (Column 4) → R7C4 → (Row 7) → R7C5
↑ ↓
└── (Column 2) ← R2C2 ← (Row 2) ← R2C5 ← (Column 5) ←───┘
Key validation rules: - Each row/column/box visited by the loop must contain exactly 2 cells from the loop - These 2 cells must be at alternating positions in the loop (odd/even) - The loop must span exactly 2 boxes (not 1, not 3+)
How to Spot Unique Loop Type 1
- Find bivalue cells: Look for cells with exactly 2 candidates
- Look for matching pairs: Find multiple cells sharing the same pair of candidates
- Check for a loop: Verify the cells form a closed path through rows/columns
- Count rescue cells: For Type 1, exactly ONE cell should have extra candidates
- Eliminate: Remove the loop pair values from the rescue cell
Common Loop Sizes
| Loop Size | Cells | Shape Description |
|---|---|---|
| 4 cells | 4 | Standard Unique Rectangle |
| 6 cells | 6 | This example (most common loop) |
| 8 cells | 8 | Extended loop pattern |
| 10+ cells | 10+ | Rare but possible |
When Type 1 Doesn't Apply
Type 1 specifically requires exactly one rescue cell. If you find:
- Zero rescue cells: The puzzle is invalid (has multiple solutions)
- Two rescue cells with same extra: See Unique Loop Type 2
- Two rescue cells with different extras: See Unique Loop Type 3
- One value locked in rescue cells: See Unique Loop Type 4
Comparison with Unique Rectangle
| Feature | Unique Rectangle | Unique Loop |
|---|---|---|
| Cell count | Exactly 4 | 4, 6, 8, 10... |
| Shape | Rectangle (2 rows × 2 cols) | Any closed loop |
| Box constraint | Spans 2 boxes | Spans 2 boxes |
| Complexity | Easier to spot | Harder to spot |
Tips for Beginners
- Start with rectangles: Master Unique Rectangle (Type 1) first
- Use notation: Mark bivalue cells with their candidates
- Look for pairs: Cells with the same two candidates are your starting point
- Trace carefully: Loops can be tricky—trace the path step by step
- Verify the loop: Ensure each region has exactly 2 loop cells
Why This Strategy Works
The uniqueness principle states that a properly constructed Sudoku puzzle has one and only one solution. If a deadly pattern could exist, the puzzle would have multiple solutions—violating this principle.
By identifying potential deadly patterns before they form, we can deduce which candidates are impossible and eliminate them. The rescue cell's extra candidates are the only values that can prevent the deadly pattern, so the loop pair values must be false.
Related Strategies
Unique Loop Family
- Unique Loop Type 2 — Two rescue cells share the same extra candidate
- Unique Loop Type 3 — Rescue cells form a Naked Subset with other cells
- Unique Loop Type 4 — One loop value is locked to rescue cells
Unique Rectangle Family
- Unique Rectangle (Type 1) — 4-cell version with single rescue cell
- Unique Rectangle (Type 2) — 4-cell version with shared extra
- Unique Rectangle (Type 3) — 4-cell version with Naked Subset
- Unique Rectangle (Type 4) — 4-cell version with locked value
Other Uniqueness Strategies
- BUG+1 — Bivalue Universal Grave with single extra cell