Unique Rectangle

Also called UR, Uniqueness Test, Unique Rectangle (general).

New to this shorthand? How to read a sudoku grid explains r4c7, houses, candidates and the rest.

This technique assumes the puzzle has exactly one solution. It reasons about what would create a second solution, so it is only valid on a properly constructed puzzle. Do not use it to check whether a grid is well formed.

What it means

Four squares in a rectangle across two boxes, all holding the same two numbers. If nothing else were in them you could swap the numbers round and get a second answer. A good puzzle has one answer, so something must break the swap.

The exact rule, for stronger players

The rule, stated exactly: Four cells forming a rectangle across exactly two rows, two columns, and two boxes, where all four share the same two candidates {x,y}. Such a configuration is impossible in a uniquely solvable puzzle, so at least one cell must take a different value.

What it removes: Depends on type. Every type removes candidates that would complete the deadly rectangle.

Why it works: If all four cells held only x and y, the two diagonal assignments would both be valid, giving two solutions. This technique is only valid when the puzzle is known to have exactly one solution. It is not a deduction about Sudoku rules, it is a deduction about the puzzle setter's promise. On a grid with multiple solutions it produces wrong answers.

Worked examples

These come from real Proven Sudoku boards. Nobody drew them by hand to make the pattern look neat. Each one is checked first, so the board is one you could really meet, the pattern is really there, and every digit it rules out is really wrong.

Sudoku grid showing Unique Rectangle. r2c5, r2c6, r4c5, r4c6 form a rectangle on 8 and 9 in two boxes. r4c5 and r4c6 each hold one extra digit, 2. One of them must be 2, or the rectangle could be swapped for a second solution, so no cell seeing both can be 2. This argument assumes the puzzle has exactly one solution. — r2c5, r2c6, r4c5, r4c6 form a rectangle on 8 and 9 in two boxes. r4c5 and r4c6 each hold one extra digit, 2. One of them must be 2, or the rectangle could be swapped for a second solution, so no cell seeing both can be 2. This argument assumes the puzzle has exactly one solution.

Sudoku grid showing Unique Rectangle. r5c2, r5c5, r6c2, r6c5 form a rectangle in two boxes, and three of them hold only 7 and 8. If r6c2 were 7 or 8 the four could be swapped for a second solution, so it is neither. This argument assumes the puzzle has exactly one solution. — r5c2, r5c5, r6c2, r6c5 form a rectangle in two boxes, and three of them hold only 7 and 8. If r6c2 were 7 or 8 the four could be swapped for a second solution, so it is neither. This argument assumes the puzzle has exactly one solution.

Learn these first

Deadly PatternA shape that would give the puzzle two answers if it ever appeared.
Naked PairTwo squares in the same group both hold only the same two digits.

Related uniqueness

Unique Rectangle Type 1Three corners of the rectangle hold only the two numbers.
Unique Rectangle Type 2Two corners hold only the pair.
Unique Rectangle Type 3Two corners hold the pair.
Unique Rectangle Type 4Two corners hold only the pair.
Unique Rectangle Type 5Like Type 2, but the extra number sits in corners that sit across from each other rather than side by side.

How often does it come up?

Our rough estimate: common on hard grids. The most frequently used uniqueness family by a wide margin.

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