- A Unique Rectangle (UR) exploits the fact that every valid Sudoku has exactly one solution
- If a pattern would create two possible solutions, the puzzle maker must have prevented it — which you can use as an elimination
- The classic UR type involves 4 cells in 2 rows, 2 columns, and 2 boxes forming a rectangle
- URs are controversial in pure logic circles but widely used in competitive solving
- Only use URs when you trust the puzzle is well-formed (has exactly one solution)
The Logic Behind Unique Rectangles
A well-formed Sudoku puzzle has exactly one solution. If two cells in the same two boxes, same two rows, and same two columns could swap two digits to create a second valid solution, the puzzle would be broken. Therefore, in any valid puzzle, that "deadly pattern" must be prevented — and the way it's prevented tells you something about the candidates.
The Classic Type 1 UR
The simplest form: four cells forming a rectangle (two rows, two columns, two boxes) all contain the same two candidates X and Y. This is called a "deadly pattern" — if those four cells could each hold either X or Y, the puzzle would have multiple solutions.
Since the puzzle has only one solution, one of those four cells must have a different digit — meaning it has an additional candidate beyond X and Y. That additional candidate must be placed in that cell to prevent the deadly pattern. You can therefore eliminate X and Y from that cell if they would complete the rectangle.
UR Types at a Glance
| Type | How to Spot It | What You Can Do |
|---|---|---|
| Type 1 | 3 cells have only XY; 1 cell has XY + extras | Eliminate X and Y from the extra cell |
| Type 2 | 2 cells have XY; 2 cells have XY + same extra Z | Eliminate Z from cells that see both extra cells |
| Type 3 | 2 cells have XY + different extras | Treat the extras as a naked pair in the unit |
| Type 4 | One of XY is locked in two cells across the roof | Eliminate the other of XY from those two cells |
When to Apply URs
Look for UR patterns when you have exhausted Naked/Hidden Singles, Pairs, and Locked Candidates but the puzzle is still stalled. URs are most common in Hard and Expert level puzzles. The techniques library contains visual examples of each UR type with annotated grids.