Sudoku techniques · Fiendish level

Unique rectangle: a Sudoku has only one answer

A unique rectangle uses four cells at the corners of a rectangle, in two rows, two columns and two boxes. If three corners can only be the same two digits, the fourth corner cannot be either, or the puzzle would have two answers.

How it works

Imagine all four corners held only 4 and 8. They could be filled 4, 8 / 8, 4 or 8, 4 / 4, 8, and nothing else on the board could tell the two apart. That is a "deadly pattern": a puzzle with two answers. A proper Sudoku has exactly one, so the pattern cannot happen.

So when three corners hold only 4 and 8, the fourth must be something else. Remove 4 and 8 from it. The rectangle must sit in exactly two boxes; across four boxes, swapping would change what each box holds, so the rule does not hold. Every Ninebloom puzzle is checked to have one answer, so the move is always safe in the app.

The row, column or box the move uses
The cells that make the pattern
Candidates the move removes
The rectangle's corners

Unique rectangle example on a Sudoku board: Three corners of a rectangle hold only 4 and 8. If the fourth were 4 or 8 too, the puzzle would have two answers, so those digits come out of it. — Example 1 · from Fiendish puzzle #752

R3C7, R5C7 and R5C9 can only be 4 or 8, and R3C9 could be 4, 7 or 8. The four cells sit in rows 3 and 5, columns 7 and 9, and two boxes. If R3C9 were 4 or 8 too, the 4s and 8s could swap corners and the puzzle would have two answers. **Remove 4 and 8 from R3C9**, leaving 7.

Play this position in Ninebloom: **Home › Your own puzzle**, then paste the code. `NG-1N9PZD-IHICX6-SUO4QA-5HSBHJ-L49N76-1HW21Q-80LS2G-599HBJ-4YYCH`

Cells are named by row and column: R4C3 is row 4, column 3. Boxes are numbered 1 to 9, left to right and top to bottom.

How to spot it

Fill in pencil marks and look for two-candidate cells with the same pair of digits.
Find three of them on three corners of a rectangle, in two rows and two columns.
Check that the four corners sit in exactly two boxes.
Remove the two digits from the fourth corner.

More examples

Unique rectangle example on a Sudoku board: Three corners of a rectangle hold only 2 and 7. If the fourth were 2 or 7 too, the puzzle would have two answers, so those digits come out of it. — Example 2 · from Fiendish puzzle #578

R5C9, R7C8 and R7C9 can only be 2 or 7, and R5C8 could be 1, 2, 4 or 7. The four cells sit in rows 5 and 7, columns 8 and 9, and two boxes. If R5C8 were 2 or 7 too, the 2s and 7s could swap corners and the puzzle would have two answers. **Remove 2 and 7 from R5C8**, leaving 1 or 4.

Play this position in Ninebloom: **Home › Your own puzzle**, then paste the code. `NG-2AN704-TMUI3Q-QG3VYE-4Y049Z-5SOHQ0-GP1US9-N94I1X-KZ4MH9-R3UPF`

Common mistakes

Using a rectangle that spans four boxes. It must sit in exactly two.
Using it on a puzzle that might have more than one answer. The move relies on there being exactly one.
Using givens as corners. All four corners must be empty cells.

Questions

Is it fair to use uniqueness to solve a Sudoku?

Yes, if the puzzle has one answer, which every proper Sudoku does. Some solvers prefer not to rely on it, but the logic is sound.

Are there other kinds of unique rectangle?

Yes. This is type 1, the simplest. Other types use extra candidates in two corners, and Ninebloom's hints only use type 1.

Which Ninebloom level uses the unique rectangle?

Fiendish. It is the last technique on Ninebloom's Learn ladder.