Unique Loop Type 3 is an advanced "uniqueness" strategy that combines the concept of a Unique Loop with Naked Subset logic. This powerful technique handles situations where multiple rescue cells have different extra candidates that together form a Naked Subset with other cells in a shared region.

This strategy uses the fundamental Sudoku rule: every valid puzzle has exactly one unique solution.

Real Example Walkthrough

In the example puzzle above, the solver identifies a 6-cell Unique Loop:

The Loop Structure:

The Key Insight:

1. The loop pair: Cells share candidates that could form a deadly pattern
2. Rescue cells with different extras: R1C4 and R1C5 have extra candidates (e.g., 3 and 7)
3. Shared region: Both rescue cells are in the same row (Row 1)
4. Partner cell: R1C2 (index 1) also contains {3, 7}
5. Naked Pair formed: The extras from rescue cells + R1C2 form a Naked Pair {3, 7}
6. Elimination: Remove 3 and 7 from other cells in Row 1 (specifically R1C6)

Result: Eliminate 3 and 7 from R1C6.

Understanding the Logic

The "Virtual Cell" Concept

In Type 3, we treat the combined extras from all rescue cells as a single "virtual cell":

• Rescue cell 1 has extra: 3
• Rescue cell 2 has extra: 7
• Virtual cell contains: {3, 7}

This virtual cell can then participate in Naked Subset logic!

How the Naked Subset Forms

Together, these form a Naked Pair in Row 1. Therefore: - One of the rescue cells must be 3 or 7 (to break the loop) - R1C2 must be the other value (3 or 7) - No other cell in Row 1 can be 3 or 7

Why This Works

The loop would become a deadly pattern without the extras. So: - At least one rescue cell must become its extra value - Combined with the partner cell, all extras are "locked" to specific cells - Other cells in the region can be eliminated

Step-by-Step: How to Apply Type 3

1. Identify a Unique Loop: Find 4+ cells forming a closed loop with the same two candidates
2. Find rescue cells: Locate cells with extras beyond the loop pair
3. Check extras are different: Type 3 requires DIFFERENT extras (same extra = Type 2)
4. Find shared region: Do rescue cells share a row, column, or box?
5. Combine the extras: Union of all extra candidates forms a "virtual cell"
6. Find partner cells: Look for cells in the shared region with only these values
7. Form Naked Subset: Virtual cell + partners = Naked Pair/Triple/Quad
8. Eliminate: Remove subset values from other cells in the region

Comparison with Other Types

Visual Pattern

Row 1: R1C2 R1C4 R1C5 R1C6 ┌─────┬─────────┬─────────┬─────────┐ │{3,7}│ Loop+3 │ Loop+7 │{3,7,8,9}│ │ │ (rescue)│ (rescue)│ (target)│ └─────┴─────────┴─────────┴─────────┘ ↑ ↑ ↑ ↓ │ └────┬────┘ │ │ │ │ Partner Virtual Cell Eliminate Cell {3,7} 3 and 7! │ │ └─────────────┘ Naked Pair

The logic: R1C2 + (extras from R1C4 and R1C5) = Naked Pair {3, 7}. Therefore, R1C6 cannot contain 3 or 7.

Common Subset Sizes

The most common case is a Naked Pair (2 values in 2 cells), but larger subsets are possible.

Common Misconceptions

"Rescue cells must have the same extra"

That's Type 2! Type 3 specifically handles different extras. In Type 2, we eliminate the shared extra from common peers. In Type 3, we combine different extras into a virtual cell.

"I can only use 2 rescue cells"

Yes, typically. The current implementation focuses on 2 rescue cells forming the virtual cell, combined with 1+ partner cells.

"The partner cell must be bivalue"

Not necessarily. The partner cell(s) just need to contain only values from the combined extras. For a Naked Pair with extras {3, 7}, a partner could be {3, 7} or even {3} or {7}.

When Type 3 Doesn't Apply

• Same extras: Both rescue cells have the same extra → Use Type 2
• No shared region: Rescue cells don't share row/column/box → Type 3 can't apply
• No partner cells: No cell in the shared region matches the extras → No subset
• No eliminations: Other cells don't contain the subset values → Nothing to eliminate

Tips for Beginners

1. Master the prerequisites: Learn Naked Pairs and Unique Loop Type 1 first
2. Look for different extras: When rescue cells have different extras, think Type 3
3. Scan shared regions: Focus on the row, column, or box containing both rescue cells
4. Combine extras mentally: Imagine rescue cells as one "virtual" cell with combined candidates
5. Apply Naked Subset logic: Once you see the virtual cell, standard subset rules apply

Connection to Naked Subsets

Type 3 is essentially a hybrid strategy:

If you understand both prerequisites, Type 3 becomes intuitive!

Why This Strategy Works

The uniqueness principle guarantees one and only one solution.

1. The loop creates a potential deadly pattern
2. Rescue cells prevent it by containing extra candidates
3. At least one rescue cell must become its extra (or the pattern forms)
4. Combined extras behave like a single cell in subset logic
5. Partner cells complete the subset
6. Other cells in the region are excluded by standard Naked Subset rules

It's the intersection of two powerful principles: uniqueness + subset elimination.

Related Strategies

Unique Loop Family

• Unique Loop Type 1 — Single rescue cell
• Unique Loop Type 2 — Multiple cells share same extra
• Unique Loop Type 4 — One loop value is locked

Unique Rectangle Family

• Unique Rectangle (Type 1) — 4-cell deadly pattern
• Unique Rectangle (Type 2) — Shared extra in roof cells
• Unique Rectangle (Type 3) — 4-cell version of this strategy
• Unique Rectangle (Type 4) — Locked value in rectangle

Naked Subset Strategies

• Naked Pair — 2 cells, 2 candidates
• Naked Triple — 3 cells, 3 candidates
• Naked Quad — 4 cells, 4 candidates

BUG Strategies

• BUG Type 1 — Single extra cell prevents bivalue grid
• BUG Type 3 — Multiple cells form Naked Subset (grid-wide)