Sudoku Solver

Expert

Unique Rectangle (Type 4)

When one UR candidate is "locked" in a row or column, the other candidate can be eliminated.

Unique Rectangle (Type 4) is an advanced uniqueness strategy. Like Type 1, Type 2, and Type 3, it uses the rule that a valid Sudoku has exactly one solution to eliminate candidates that would create a "Deadly Pattern."

Type 4 is special because it uses locked candidates within the rectangle to determine which digit must stay.

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
6
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

Remember the Deadly Pattern from Unique Rectangle (Type 1): four cells in a rectangle, all containing the same pair (like {4, 8}), would allow two solutions.

In Type 4, we notice that one of the UR candidates is locked in a row or column: - If digit X can ONLY appear in the UR cells within that row/column... - Then X must go in one of those UR cells. - This "rescues" us from the Deadly Pattern. - Therefore, the other candidate can be safely eliminated from those cells.

Real Example Explanation

In the example above:

  1. The Rectangle: Cells R5C7, R5C9, R8C7, R8C9 form a UR with candidates {4, 8}.

  2. The Lock: Look at Column 9 (or Column 7).

    • Where can 8 go in this column within the UR cells?
    • The digit 8 is "locked" into only appearing in the UR cells for this column.

  3. The Logic:

    • Since 8 is locked in the UR cells of this column, at least one UR cell MUST contain 8.
    • This prevents the Deadly Pattern (we won't have four identical {4, 8} pairs).
    • The "rescue" is guaranteed by digit 8.

  4. The Elimination:

    • Since 8 rescues us, we don't need 4 to rescue us.
    • We can safely eliminate 4 from the UR cells in that column.

How to Spot It

  1. Find a UR: Four cells forming a rectangle in 2 rows, 2 columns, spanning 2 boxes.
  2. Identify the Pair: All four cells share candidates like {A, B}.
  3. Check for a Lock: In any row/column containing two UR cells:
    • Is one of the candidates (A or B) found ONLY in those UR cells?
    • If yes, that candidate is "locked."
  4. Eliminate: Remove the OTHER candidate from those two UR cells.

Comparison with Other UR Types

Type Condition Elimination
Type 1 3 cells are bivalue, 1 has extras Remove pair from the 4th cell
Type 2 2 cells have same extra Remove extra from common peers
Type 3 Extras form Naked Subset Standard subset elimination
Type 4 One candidate locked in unit Remove other candidate from UR cells

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