How it works
Unique rectangles are the most famous uniqueness-based technique. They use a fact outside the puzzle itself: every valid sudoku has exactly one solution. If a configuration would allow two solutions, that configuration is impossible.
The setup is four cells at the corners of a rectangle, occupying exactly two boxes. Three of the corners hold the same two candidates {X, Y}. If the fourth corner also held {X, Y}, you could swap X and Y across the rectangle to get two valid completions. Since that breaks uniqueness, the fourth corner must hold something else.
The elimination removes X and Y as candidates from the fourth corner, leaving its remaining candidates. Often that creates an immediate single.
When to look for it
Expert and evil puzzles. Scan for rectangles where three corners share a bivalue candidate set.
Tips for spotting the pattern
- Type 1 (the classic) has three corners exactly bivalue with {X, Y} and the fourth has extras. Eliminate X and Y from the fourth.
- More elaborate types (2, 3, 4) extend the same logic with more corners showing extras.
- Some puzzles are uniqueness-resistant; the technique only works on puzzles that already have one solution.
Common mistakes
- Misidentifying the rectangle. The four corners must span exactly two boxes, not four.
- Eliminating from the wrong corner. The eliminations target the one corner with extras.
- Using it on non-unique puzzles. Some puzzle generators do not guarantee uniqueness; check first.