sudokly
Advanced technique

Unique Rectangle

A unique rectangle is a four-cell pattern at the corners of a rectangle, spanning exactly two boxes, where three corners have the same two candidates {X, Y}. If the fourth corner were also {X, Y}, the puzzle would have two solutions. Since valid sudoku has one, the fourth corner cannot be {X, Y}.
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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.

2 72 72 72 5 7FOUR CORNERS, TWO BOXES. THE FOURTH CORNER LOSES 2 AND 7
Three corners hold {2, 7}. The fourth has extras. Eliminate 2 and 7 from it. only the extras can be the answer.

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

  1. Pick a bivalue cell {X, Y}. Look for another bivalue cell {X, Y} in the same row.
  2. 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.
  3. The four corners must lie in exactly two boxes. Two rows of three columns, or two columns of three rows.
  4. Three of the corners hold {X, Y} exactly. The fourth holds {X, Y} plus extras.
  5. 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.

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