Unique Rectangle Sudoku: Avoiding Deadly Patterns
A Unique Rectangle uses the promise that a standard Sudoku has one solution. Four cells at the corners of a rectangle span exactly two rows, two columns, and two boxes. If all four became the same pair {X,Y}, the digits could swap and create two solutions. In Type 1, an extra candidate in one corner must remain, so X and Y can be removed from that corner.
Unique Rectangle logic prevents a two-solution swap; it depends on a trustworthy uniqueness assumption.
What is a Type 1 Unique Rectangle?
Three rectangle corners contain exactly {X,Y}, while the fourth contains {X,Y,Z...}. The four corners must occupy two rows, two columns, and two boxes. If the fourth cell also resolved to X or Y, the rectangle could swap X and Y without changing any outside clue. Therefore the fourth cell cannot be X or Y.
How to verify a Unique Rectangle
- Confirm puzzle uniqueness. Use the technique only on a standard puzzle intended to have exactly one solution.
- Check the geometry. The four cells must form two rows, two columns, and exactly two boxes.
- Verify three pure pairs. Three corners must contain only the same two candidates X and Y.
- Protect the extra candidate. Remove X and Y from the fourth corner, leaving its extra candidate or candidates.
Example: Type 1 rectangle on {4,9}
Three corners are {4,9}; the fourth is {2,4,9}. The rectangle spans two rows, two columns, and two boxes in a unique puzzle.
- Test the deadly pair. If the fourth corner were 4 or 9, all four cells could become the pair {4,9}.
- Recognize the swap. The 4s and 9s could exchange positions while every row, column, and box stayed valid.
- Keep uniqueness. Remove 4 and 9 from the fourth corner, so it must be 2.
Common Unique Rectangle mistakes
Using it on a multi-solution puzzle
The method is not valid when uniqueness is unknown or deliberately absent.
Spanning four boxes
A standard Unique Rectangle must occupy exactly two boxes, not four.
Misreading impure corners
Type 1 requires three corners to contain only the same two candidates.
Unique Rectangle practice checklist
- Search repeated bivalue pairs across two rows and columns.
- Count the boxes before analyzing candidates.
- State the uniqueness assumption before making a removal.
Unique Rectangles are powerful but conditional. Verify the two-box geometry and the puzzle’s uniqueness before using the pattern; otherwise choose a technique that does not depend on that assumption.
Unique Rectangle FAQ
Does Unique Rectangle work on every Sudoku?
No. It relies on the puzzle having exactly one solution.
Why must the rectangle span two boxes?
That geometry allows the two digits to swap while preserving row, column, and box counts.
What is a deadly pattern?
It is a candidate arrangement that could support two interchangeable solutions unless another candidate breaks the symmetry.
