Sudoku Solver

Expert

Unique Rectangle (Type 3)

When extra candidates in the "roof" cells form a Naked Subset with neighboring cells.

Unique Rectangle (Type 3) is the most flexible UR type. While Type 1 and Type 2 have specific conditions, Type 3 can work with many different configurations.

The key idea: the "extra" candidates in the roof cells form a Naked Subset with other cells in the same unit.

Interactive Example

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

Click "Apply Logic" to see the strategy in action.

The Core Concept

In a Unique Rectangle, we have: - Floor cells: Contain only the UR pair (e.g., {5, 6}) - Roof cells: Contain the UR pair PLUS extra candidates

In Type 3, the extra candidates in the roof cells combine with neighboring cells to create a Naked Pair, Triple, or Quad. This allows us to eliminate those candidates from other cells in the unit.

Why Does This Work?

Remember the Deadly Pattern: if all four UR cells ended up with just the UR pair, we'd have two solutions.

To avoid this: - At least one roof cell MUST contain its "extra" candidate (not the UR pair) - The roof cells act as a "virtual cell" containing all their extras - If this virtual cell + real neighbors form a Naked Subset, standard elimination rules apply

Real Example Explanation

In the example above:

  1. The Rectangle: Cells R3C8, R3C9, R4C8, R4C9 form a UR.

    • Floor cells (R3C8, R3C9): contain the UR pair
    • Roof cells (R4C8, R4C9): contain UR pair + extras {5, 9}

  2. The Virtual Cell: The roof cells' extras combine to form {5, 9}.

  3. The Neighbor: Cell R3C3 (index 29) in the same row also contains {5, 9}.

  4. The Naked Pair: The "virtual cell" (roof extras) + R3C3 = two cells with {5, 9}.

    • This is a Naked Pair in Row 4!

  5. The Elimination: Since {5, 9} are "claimed" by this Naked Pair, we can eliminate 5 and 9 from other cells in Row 4 (like R3C1, index 27).

Versions of Type 3

Type 3 can form different subsets:

Subset Configuration
Naked Pair Roof extras + 1 neighbor share 2 candidates
Naked Triple Roof extras + 1-2 neighbors share 3 candidates
Naked Quad Roof extras + 2-3 neighbors share 4 candidates

The logic is identical—only the size of the subset changes.

How to Spot It

  1. Find a UR: Four cells with a common pair spanning 2 rows, 2 columns, 2 boxes.
  2. Identify Floor/Roof: Floor = bivalue cells. Roof = cells with extras.
  3. Check Roof Extras: What extra candidates do the roof cells have?
  4. Look for Partners: In the same row/column as the roof cells, are there other cells that share those extras?
  5. Form a Subset: If roof extras + neighbors = a Naked Pair/Triple/Quad, make the elimination.

Comparison with Other UR Types

Type How It Eliminates
Type 1 Remove UR pair from the 4th cell (with extras)
Type 2 Remove shared extra from common peers of roof
Type 3 Roof extras + neighbors form Naked Subset
Type 4 Locked candidate in unit forces elimination

Related Strategies