Unique Rectangles in Sudoku

Specialist S4 Uniqueness

A proper Sudoku has exactly one solution. That is a fact about the puzzle you can reason from, and Unique Rectangles is the technique that does. Four unsolved cells lying in two rows, two columns and exactly two boxes, all four holding the same two candidates, would be a disaster: the two digits could be swapped corner to corner and the grid would still be valid, so the puzzle would have two solutions rather than one. That arrangement is called a deadly pattern, and it can never occur. Whatever candidate would complete it can be removed.

How to spot it

  • Look for two cells holding exactly the same two candidates, then look for the two cells that complete a rectangle with them - same two rows, same two columns. This is a pattern you notice among cells that are already there, not one you set out to build.
  • Check the boxes. The four corners must fall in exactly two of them. Four boxes is the common mistake: such a rectangle can be swapped in its rows and columns but not in its boxes, so it is not deadly and proves nothing.
  • Check that all four corners still hold both digits of the pair. A corner that has already lost one of them cannot complete the pattern, and there is nothing to forbid.
  • Now look at what the corners hold beyond the pair. Three corners at exactly the pair and one with something extra is the basic case. Two corners at exactly the pair and two holding the pair plus the same one extra digit is the other.

Why it works

Start from the swap. Suppose all four corners ended up holding just the two digits, alternating round the rectangle. Exchange the two digits in all four cells. Each of the two rows still has one of each. Each of the two columns still has one of each. And because the four cells fall in exactly two boxes, each of those boxes still has one of each. The exchanged grid is a valid solution too.

So a grid containing that arrangement has an even number of solutions - never exactly one. This Sudoku has exactly one, so the arrangement must be unreachable. Anything that would produce it is therefore impossible, and that is the only thing this technique ever says.

The basic case follows immediately. If three corners hold exactly the pair and the fourth holds the pair plus something else, then giving the fourth corner either of the pair's digits would complete the deadly arrangement. Both of those candidates go from that corner. Its extra candidates are untouched - the technique says nothing about them, and one of them is where the answer is.

The other case is the same argument pointed outwards. Two corners hold exactly the pair; call them the floor. The other two - the roof - hold the pair plus the same one extra digit. If neither roof cell took that extra digit, both would fall back to the pair, all four corners would hold just the pair, and the deadly arrangement would be complete. So at least one roof cell holds the extra digit.

That is a claim about two cells at once, and it is what makes the elimination possible. Any cell outside the rectangle that sees both roof cells cannot hold the extra digit, because whichever roof cell turns out to have it, that cell would clash. The digit is removed from every such cell.

Where the roof sits changes which cells those are, and nothing else. The two roof cells may share a box as well as their line, in which case the eliminations can be in either; or they may be split between the rectangle's two boxes, in which case only the line they share can hold eliminations. Textbooks number these cases separately. The proof is identical, so this page does not: find the cells that see both roof cells, whatever their geometry, and remove the digit from them.

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 away the candidates the technique removes - 2 of them - then press Check. Nothing is saved, and your own game is untouched.

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 Sudokuno marks the four cells, without yet saying what is special about any one of them. They lie in two rows, two columns and exactly two boxes, and all four still hold the same two candidates.
  2. Hint 2 The roles appear. Either three corners hold exactly the pair and one has something extra, or two corners hold the pair and the other two hold the pair plus the same one extra digit. In both cases the question is the same: what stops all four corners from ending up at just the pair?
  3. Hint 3 The eliminations are marked. In the first case both of the pair's digits go from the one corner that had something extra. In the second, the extra digit goes from every cell outside the rectangle that sees both roof cells.

The premise is doing the work, so it is worth saying out loud: this reasoning is only valid because the puzzle is known to have exactly one solution. On a grid with two solutions a deadly pattern is perfectly possible, and every elimination here would be wrong.

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 implements the foundational Unique Rectangle family on the standard grid: four unsolved cells in two rows, two columns and exactly two boxes, one deadly pair, and either the basic single-corner escape or a roof of two cells sharing one extra candidate - conventionally Types 1, 2 and 2B. The harder members of the family, which combine the deadly-pattern premise with locked sets, pseudo-cells, conjugate links and larger candidate structures - conventionally Types 3, 3B, 4, 4B and 5 - are deliberately outside S10, as are Hidden and Extended Unique Rectangles, AIC formulations, arbitrary unavoidable sets, Avoidable Rectangles and Jigsaw or diagonal variants. Those advanced variants are a candidate for a separate, harder level later; S10 is where the deadly-pattern idea itself is learned.

Where to go next

  • BUG+1 BUG stands for Bi-Value Universal Grave: a position where every unsolved cell has exactly two candidates left and every digit has exactly two places left in every row, column and box. A position like that always has an even number of solutions, never one, so a proper Sudoku can never reach it. BUG+1 is that position with one exception - a single cell holding three candidates. One of those three is the only thing standing between the puzzle and the grave, and it can be placed straight away.
  • Rectangle Elimination Rectangle Elimination follows one candidate around four boxes. Start with a hinge H whose row or column has exactly one other place for the digit, S. A possible target T lies on the perpendicular line through H. If T held the digit, H would be off and the strong side would force S on. When T and S then see every remaining place for that digit in the diagonally opposite box, the box would have nowhere to put it. T therefore cannot hold the digit.
  • Simple Colouring Simple Colouring looks at one digit at a time. When a row, column or box has exactly two places left for that digit, the two cells are strongly linked: exactly one of them holds it. Join such pairs together and you get a network that can be coloured alternately in two colours, and then exactly one of the colours is the true one. That is enough to remove candidates, even though you do not know which colour is which.

All Sudoku techniques