Sudoku Solver

Expert+

Unique Loop Type 4

When one loop value is locked to rescue cells in a shared region, eliminate the other loop value from them.

Unique Loop Type 4 is an advanced "uniqueness" strategy that uses the concept of locked candidates within a Unique Loop. When one of the loop's two values can only appear in the rescue cells within a shared region, we know one rescue cell MUST be that value—allowing us to eliminate the other loop value.

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

Interactive Example

5
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
6
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
6
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
8
1 2 3 4 5 6 7 8 9
2
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
9
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9
4
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
5
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
8
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
8
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
7
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
3
1 2 3 4 5 6 7 8 9

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 Structure:

Cell Index Position Candidates Role
30 R4C4 {6, 9} Loop cell
32 R4C6 {6, 9} Loop cell
54 R7C1 {6, 9} Loop cell
59 R7C6 {6, 9} Loop cell
63 R8C1 {6, 9} + extras Rescue cell
66 R8C4 {6, 9} + extras Rescue cell

The Key Analysis:

  1. Loop values: The deadly pattern uses values {6, 9}
  2. Rescue cells: R8C1 and R8C4 both have extra candidates (breaking the deadly pattern)
  3. Shared region: Both rescue cells are in Row 8
  4. Locked value check: Where can 9 appear in Row 8?
  5. Scanning Row 8: Value 9 appears ONLY in R8C1 and R8C4!
  6. The logic: Since 9 is locked to the rescue cells, one of them MUST be 9
  7. Elimination: If one rescue cell must be 9, neither can be 6 (the other loop value)

Result: Eliminate 6 from both R8C1 and R8C4.

Understanding the Logic

What Does "Locked" Mean?

A value is locked to certain cells in a region when it can ONLY appear in those cells within that region.

In our example: - Row 8 needs a 9 somewhere - Looking at all cells in Row 8, only R8C1 and R8C4 contain candidate 9 - Therefore, 9 is locked to {R8C1, R8C4} - One of these cells must be 9

The Deadly Pattern Connection

We already know from the Unique Loop that: - At least one rescue cell must NOT be a loop value (to break the pattern) - The rescue cells have extras for this reason

Type 4 adds another constraint: - One rescue cell must be the locked loop value (9) - Therefore, neither can be the other loop value (6)

Why Eliminate the Other Loop Value?

If R8C1 = 9 If R8C4 = 9
R8C4 cannot be 9 or 6 (loop broken) R8C1 cannot be 9 or 6 (loop broken)
R8C4 must be an extra R8C1 must be an extra

In BOTH scenarios, neither rescue cell is 6. So we can safely eliminate 6 from both!

Step-by-Step: How to Apply Type 4

  1. Identify a Unique Loop: Find 4+ cells forming a closed loop with the same two candidates (e.g., {6, 9})
  2. Find rescue cells: Locate the 2 cells with extra candidates
  3. Find shared region: Do rescue cells share a row, column, or box?
  4. Check for locked value: In that shared region, is one loop value ONLY in the rescue cells?
  5. Eliminate: Remove the OTHER loop value from both rescue cells

Visual Pattern

``` Row 8: R8C1 R8C2 R8C3 R8C4 R8C5 ... ┌─────────┬─────────┬─────────┬─────────┬─────────┐ │{6,9,x,y}│ filled │ filled │{6,9,z} │ filled │ │ rescue │ 5 │ 2 │ rescue │ 8 │ └─────────┴─────────┴─────────┴─────────┴─────────┘ ↑ ↑ └───────────────┬─────────────┘ │ Where can 9 go in Row 8? ONLY R8C1 and R8C4!

                ∴ One of them MUST be 9
                ∴ Neither can be 6

                → Eliminate 6 from both

```

Comparison with Other Types

Type Condition Elimination Target What's Eliminated
Type 1 1 rescue cell The rescue cell Loop values
Type 2 Same extra in rescue cells Common peers The shared extra
Type 3 Different extras + subset Other region cells Subset values
Type 4 Loop value locked to rescues Rescue cells The OTHER loop value

Type 4 vs Type 1

Aspect Type 1 Type 4
Rescue cells Exactly 1 Exactly 2
What's proven Rescue must NOT be loop values One rescue MUST be locked value
Eliminated from The rescue cell Both rescue cells
What's eliminated Both loop values One loop value

Type 1: "The rescue cell must use its extras, not loop values" Type 4: "One rescue must be the locked value, so neither can be the other loop value"

Common Misconceptions

"I need to find which rescue cell is the locked value"

No! We don't need to know WHICH rescue cell is 9. We only know ONE of them is. Either way, 6 is eliminated from both.

"The locked value must be in ALL rescue cells"

Yes, but that's automatic. For Type 4, we need exactly 2 rescue cells, and the locked value must be a loop value that's locked to exactly those cells.

"I should eliminate the locked value"

No! We eliminate the OTHER loop value. The locked value stays because one rescue cell needs it.

When Type 4 Applies

Type 4 requires a specific combination: - ✅ Exactly 2 rescue cells with extras - ✅ Both rescue cells share a region (row, column, or box) - ✅ One loop value appears ONLY in the rescue cells within that region - ✅ The rescue cells still contain the other loop value (something to eliminate)

When Type 4 Doesn't Apply

  • 1 rescue cell: Use Type 1 instead
  • Same extra in rescues: Try Type 2 first
  • No shared region: Rescue cells in different rows/columns/boxes → Type 4 can't apply
  • No locked value: Both loop values appear elsewhere in the shared region
  • Already eliminated: Rescue cells don't contain the other loop value

Tips for Beginners

  1. Look for 2 rescue cells: Type 4 requires exactly 2
  2. Check shared regions: Row, column, OR box—any will work
  3. Scan for locked candidates: Count where each loop value can appear in the shared region
  4. Remember the logic: Locked value → One rescue has it → Neither has the other
  5. Eliminate confidently: This is a strong deduction!

Connection to Locked Candidates

Type 4 combines two powerful concepts:

Concept Application
Unique Loop Identifies cells that must break pattern
Locked Candidates Shows one loop value must be in rescue cells
Type 4 Combines both → eliminate other loop value

If you understand Intersection (Pointing/Claiming) strategies, the "locked candidate" idea will feel familiar!

Why This Strategy Works

The uniqueness principle guarantees one and only one solution.

  1. The loop creates a potential deadly pattern with values {6, 9}
  2. Rescue cells prevent it by containing extra candidates
  3. Within their shared region, value 9 is locked to the rescue cells
  4. Therefore, one rescue cell must be 9 (standard Sudoku logic)
  5. If one is 9, the other must be an extra (to break the loop)
  6. Neither rescue cell can be 6

It's the intersection of uniqueness logic and standard locked candidate logic.

Related Strategies

Unique Loop Family

Unique Rectangle Family

Related Intersection Strategies

BUG Strategies