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Unique Rectangle

Exploits the fact that a valid puzzle has one solution to avoid a deadly four-cell rectangle.

Unique Rectangle techniques rely on the guarantee that a proper Sudoku has exactly one solution. If four cells form a rectangle spanning two rows, two columns and two boxes, and they share the same two candidates, the puzzle would have two solutions (the pair could swap). Since that is impossible, any extra candidate that breaks the deadly pattern must be the answer.

These are the only techniques that use the uniqueness assumption rather than pure logic.

How to spot it

Look for a rectangle of four cells in two boxes where three corners show exactly {X,Y} and the fourth shows {X,Y,Z}. To avoid the unsolvable double, the fourth cell must be Z — so X and Y are removed from it.

  • Four cells, two rows, two columns, two boxes.
  • Three corners are bi-value {X,Y}.
  • The fourth corner must take its extra candidate.
Four identical corners would mean two solutions — the extra digit stands.

Worked example

  1. Cells at (r1,c1),(r1,c4),(r6,c1) all show {3,8}.
  2. Cell (r6,c4) shows {3,8,5}.
  3. If it were 3 or 8, the four cells could swap — two solutions.
  4. A unique puzzle forbids that, so (r6,c4) must be 5.
  5. Place 5.

Try it yourself

2
9
5
2
6
7
1
3
8
4
6
1
9
3
4
5
7
1
7
9
5
6
4
9
7

Tap a cell, then a number, to practise.

Frequently asked questions

Is it safe to assume uniqueness?
For published puzzles, yes — they are guaranteed unique. Avoid uniqueness techniques only on unverified or hand-made grids.
Are there several unique-rectangle types?
Yes, types 1–6 cover different extra-candidate arrangements; type 1 above is the most common.

Related techniques

Practice: Unique Rectangle

Put the Unique Rectangle to work on a live board — free puzzles with notes, hints and four difficulty levels.

Try it on a live board

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