Unique Rectangles in Sudoku

Specialist S4 Uniqueness

Unique Rectangles uses the fact that a puzzle has exactly one solution. Four cells in two rows, two columns and exactly two boxes cannot all be restricted to the same candidate pair. At least one corner must use an extra digit, which can rule out candidates.

A candidate is a digit that can still go in an empty cell. Cells see each other when they share a row, column or 3×3 box. A strong link means a digit has exactly two possible cells in one such group: exactly one must contain it.

How to spot it

  • Find four unsolved cells forming a rectangle in two rows, two columns and exactly two boxes. Every corner must contain both digits of the same candidate pair, possibly with extras.
  • Basic case: three corners contain only the pair; the fourth contains the pair plus one or more extra candidates.
  • Shared-extra case: two corners contain only the pair. The other two share a row or column and contain the pair plus exactly the same one extra candidate. These two are called the roof; the exact-pair cells are the floor.
  • Check the boxes and the full candidate sets. A rectangle spread over four boxes does not support this deduction. The puzzle must be known to have exactly one solution.

Why it works

First, why must there be an extra digit? Suppose the four corners used only 1 and 2 in a completed solution. Swapping 1 and 2 in all four cells would leave each row, column and box valid, giving a second solution. This cannot happen in a puzzle with exactly one solution.

Basic case: three corners have only 1/2, and the fourth has 1/2/3. The fourth must be 3 to avoid that swap. Remove 1 and 2 from it. If it has several extra candidates, keep them all; this technique does not choose between them.

Shared-extra case: two corners have only 1/2, and the roof cells have 1/2/3. At least one roof cell must be 3. Any cell outside the rectangle that sees both roof cells therefore cannot be 3.

Use the same test whether the roof cells share a box or lie in separate boxes: remove the extra digit only from outside cells seeing both roof cells.

A real example from Sudokuno

The board below is a real Sudokuno puzzle, shown exactly as it stands just before Unique Rectangles applies. Everything up to this point was solved with ordinary techniques.

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

Click a candidate to cross it out; click it again to restore it. Remove the 2 candidates ruled out by this technique, then press Check. This exercise leaves your game unchanged.

Example: Unique Rectangles - Sudokuno - not yet published in the puzzle picker

  • Houses: row 3, column 7 and box 3
  • Rectangle: R3C7, R3C9, R4C7 and R4C9
  • Removes 1 from R3C7
  • Removes 4 from R3C7
  • Step 32 of 59 in the solve

Key: corner holding only the pair the corner with extra candidates

How the hint helps you

Sudokuno's hint reveals the structure in three steps, without giving away the digit straight away.

  1. Hint 1 The four corners are marked. They lie in two rows, two columns and exactly two boxes, and all include the same candidate pair.
  2. Hint 2 The corner roles appear. The hint describes the case on this board: one corner with extras, or two roof cells with the same extra digit.
  3. Hint 3 The removals are marked. In the basic case the pair leaves the extra corner; in the shared-extra case, the extra digit leaves outside cells that see both roof cells.

Both forms depend on the puzzle having exactly one solution. The basic case removes the pair from one corner; the shared-extra case removes the extra digit from cells outside the rectangle.

Ready to try it yourself?

You will get a real Sudokuno puzzle at Specialist level S4, where Unique Rectangles is needed to finish it.

Practise Unique Rectangles

What Sudokuno covers

Sudokuno covers the basic one-corner case and the two-roof-cell case with one shared extra candidate, commonly called Types 1, 2 and 2B. More advanced Unique Rectangle forms are outside this page.

Where to go next

  • BUG+1 BUG+1 applies when exactly one unsolved cell has three candidates and all the others have two, with a specific count pattern across the whole board. One candidate appears three times in that cell's row, column and box. In a puzzle known to have one solution, place that candidate.
  • Rectangle Elimination Rectangle Elimination tests whether a candidate would leave a box with no place for a digit. Assuming the target contains the digit forces another cell to contain it too. If together they rule out every place in the opposite box, remove the candidate from the target.
  • Simple Colouring Simple Colouring follows one digit through a network of strong links. Colour linked cells alternately: either every cell of colour A holds the digit and none of B does, or the reverse. Those two possibilities can rule out candidates.

All Sudoku techniques