What is the unique rectangle technique in sudoku?

A unique rectangle is a pattern of four cells forming the corners of a rectangle that span exactly two rows, two columns, and two boxes, where the same two candidates appear. If all four cells held only those two candidates, the puzzle would have two valid solutions. Since a proper sudoku has exactly one solution, you eliminate candidates to prevent that ambiguous configuration.

What is the deadly pattern in a unique rectangle?

The deadly pattern is four cells at the corners of a rectangle, sitting in two rows, two columns, and two boxes, each containing only the identical pair of candidates such as 3 and 7. Those values could be swapped between the two diagonals to give two different valid completions. Because a well-formed sudoku cannot have two solutions, this pattern is forbidden and must be broken.

Why does the unique rectangle strategy require a unique solution?

The technique works by assuming the puzzle has exactly one solution, then eliminating any candidate placement that would create a second solution. If a puzzle were allowed to have multiple solutions, the deadly pattern would be legal and the eliminations would be invalid. For that reason unique rectangles should not be used on puzzles that are not guaranteed to be uniquely solvable.

When can I use a unique rectangle and what does the extra candidate do?

You use it when three corners contain only the matching pair and the fourth corner contains that pair plus one or more extra candidates. Because the fourth cell must ultimately take one of the extra candidates to avoid the deadly pattern, you can eliminate the two pair candidates from that cell (a Type 1 unique rectangle). Other types handle cases where the extra candidates are spread across two corners.

Unique Rectangle: Sudoku Technique with Examples

Unique rectangle is the most famous uniqueness-based technique. It leans on a fact that lives outside the puzzle itself: a valid sudoku has exactly one solution. Any configuration that would allow two solutions cannot occur in a real puzzle.

How it works

A unique rectangle is four cells at the corners of a rectangle that lies inside exactly two 3x3 boxes (two rows of one box-pair, or two columns of one box-pair). Three of the corners hold the same two candidates {X, Y}. The fourth corner holds {X, Y}plus one or more extra candidates.

Imagine the fourth corner also held only {X, Y}. Then the digits X and Y could swap across the rectangle's diagonals and produce two valid completions of the puzzle. Sudoku is supposed to have one solution. So the fourth corner cannot be {X, Y}. The X and Y candidates must be the wrong choices for that cell. Erase them, leaving whatever else the cell carries.

Avoiding the deadly pattern forces the extras → erase {5, 7} from the fourth corner.

When to look for it

Expert and evil puzzles, where bivalue cells are common. Scan for rectangles where three corners share a bivalue candidate set. Confirm the rectangle spans exactly two boxes; a rectangle crossing four boxes does not have the required structure.

Only use this technique on puzzles guaranteed to have one solution. Most published puzzles and quality digital generators meet that requirement. If you cannot guarantee uniqueness, fall back on forcing chains.

Step-by-step example

Pick a bivalue cell {X, Y}. Look for another bivalue cell {X, Y} in the same row.
Check the column that the second cell shares with the first. You need two cells in that column, both of which are inside the same pair of 3x3 boxes.
The four corners must lie in exactly two boxes. Two rows of three columns, or two columns of three rows.
Three of the corners hold {X, Y} exactly. The fourth holds {X, Y} plus extras.
Erase X and Y from the fourth corner. Only its extra candidates can be the answer.

Tips for spotting it

Type 1 is the classic version above. Higher types (2, 3, 4) extend the same logic when more corners carry extras.
List all bivalue cells with the same candidate set; rectangles form between them.
Confirm the two-box constraint before applying. Four-box rectangles do not work.
Eliminations almost always cascade. Re-scan immediately for singles.

Common mistakes

Eliminating from the wrong corner. The target is the one corner with extras.
Using the technique on puzzles without guaranteed uniqueness. The argument breaks.
Counting four corners across four boxes. Two boxes is the rule.
Forgetting that the three "bivalue" corners must all be exactly {X, Y} with no extras.

Practise it

Try evil sudoku. Most generators produce uniqueness-friendly puzzles. After full pencil marks, scan for bivalue cells with matching candidate sets in the same row or column. Rectangles appear faster than you think. Pair this with BUG, another uniqueness-based technique for late-game positions.

Unique Rectangle