Rectangle Elimination in Sudoku

Specialist S9 Chains

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.

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

  • Pick one digit. Find a row or column with exactly two possible cells for it, in different boxes. Call these cells H, the hinge, and S, its linked partner.
  • If H and S share a row, look down H's column for a target T in another box. If they share a column, look along H's row. This second line must have more than two possible cells for the digit in Sudokuno's form.
  • Find the fourth box completing the rectangle of boxes. For a row link, it is in the same group of three rows as T and the same group of three columns as S. Reverse rows and columns for a column link.
  • Check every candidate for the digit in that box. Each must see T or S. If even one sees neither, this target cannot be eliminated by the pattern.

Why it works

Suppose T contains the digit. T sees H, so H cannot contain it.

H and S are the only possible cells for the digit in their shared row or column. With H ruled out, S must contain it.

T and S would now both contain the digit. Every possible cell for it in the fourth box sees at least one of them, so the box would have no place left for the digit.

Every box needs that digit. The assumption about T is therefore impossible: remove this candidate from T. Check the same argument separately for any other highlighted targets.

A real example from Sudokuno

The board below is a real Sudokuno puzzle, shown exactly as it stands just before Rectangle Elimination applies. Everything up to this point was solved with ordinary techniques.
8
2 7
2
7 8
6
7
8
5
1
5 7 4 6 3 1 2 8 9
6 8 2
1 5 3
3 1 9 8 2 5 6 7 4
4
1
8
2 9 7 3 1 8 4 6 5
8 5
4
1 2

Click a candidate to cross it out; click it again to restore it. Remove the 1 candidates ruled out by this technique, then press Check. This exercise leaves your game unchanged.

Example: Rectangle Elimination - Sudokuno - not yet published in the puzzle picker

  • Candidate 3
  • Houses: row 2, column 3 and box 1
  • Hinge H: R2C8
  • Strong mate S: R2C3
  • Fourth-box candidates: R7C2 and R9C3
  • Pattern: R2C8, R2C3, R7C2, R9C3 and R7C8
  • Removes 3 from R7C8
  • Step 27 of 56 in the solve

Key: hinge H strong mate S fourth-box candidates target cells

How the hint helps you

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

  1. Hint 1 H and S are marked. Exactly one must hold the digit because they are the only two places for it in their row or column.
  2. Hint 2 The targets and fourth-box candidates appear. Assume a target holds the digit: H cannot hold it, so S must. Check every place that the target and S would rule out in the fourth box.
  3. Hint 3 The target would leave the box with no place for the digit. The target candidate is marked for removal.

The useful check is coverage, not appearance: trace every candidate in the fourth box to T or S. If even one survives both, there is no Rectangle Elimination.

Ready to try it yourself?

You will get a real Sudokuno puzzle at Specialist level S9, where Rectangle Elimination is needed to finish it.

Practise Rectangle Elimination

What Sudokuno covers

Sudokuno covers the standard four-box pattern with one row or column strong link and more than two candidate positions on the perpendicular line. It uses ordinary Sudoku constraints, not uniqueness.

Where to go next

  • Chute Remote Pairs Chute Remote Pairs uses two cells with the same candidate pair in one band of three boxes or one stack of three boxes. A third group of cells, the bridge, rules out both endpoints holding one digit. At least one endpoint must hold the other, so cells seeing both lose that other digit.
  • 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.
  • W-Wing A W-Wing connects two cells holding the same candidate pair through a strong link on one digit. This proves that at least one of the pair cells holds the other digit. Remove that other digit from any cell seeing both pair cells.

All Sudoku techniques