Rectangle Elimination in Sudoku

Specialist S9 Chains

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.

How to spot it

  • Choose one digit and find a row or column where it has exactly two candidate cells. Call the cell where the pattern turns H, and the other end of the strong side S.
  • From H, look along the perpendicular line for a target candidate T in another box. For Sudokuno's S7 form this line must have more than two candidate positions, so it is not another strong link.
  • The fourth-corner box is in T's band and S's stack for a row-strong pattern; transpose that description for a column-strong pattern.
  • List every remaining candidate for the digit in that fourth box. Each one must see T or S. A rectangle-like picture alone proves nothing, and one uncovered candidate invalidates the elimination.

Why it works

Assume T contains the digit. Because T and H share the perpendicular line, H cannot contain it.

H and S are the only two positions in their strong row or column. With H off, S is forced on.

Now T and S are both on under the assumption. If their sight covers every candidate in the fourth-corner box, they remove every possible location for the digit there. A Sudoku box cannot be left without that digit, so the assumption is impossible.

Only the candidate digit is removed from T. Nothing in this proof licenses another candidate removal or a placement.

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 away the candidates the technique removes - 1 of them - then press Check. Nothing is saved, and your own game is untouched.

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 Sudokuno marks H, S and their row or column. The digit has exactly two places in that line, so switching H off must switch S on.
  2. Hint 2 The target, fourth-corner box and its candidate witnesses appear. Suppose the target were on: H would turn off, S would be forced on, and T plus S would cover every place for the digit in the fourth box.
  3. Hint 3 That assumption would empty the fourth box of the digit, so the highlighted candidate can be removed from the target.

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 implements the classical standard-grid, single-digit form with one genuine row or column strong side and a non-strong perpendicular side. Double and Extended Rectangle Elimination, the two-strong-link variation, Empty Rectangle as a separate technique, X-Cycles, grouped or AIC forms, Jigsaw and diagonal variants, Unique Rectangles are outside S7.

Where to go next

  • Chute Remote Pairs A chute is three boxes in a line - either three side by side or three stacked. If you find two cells in the same chute holding exactly the same candidate pair, and they share no row, column or box, then the three bridge cells in the unused line settle the matter. If one of the pair candidates is missing from the bridge, the other can be removed from the cells that see both ends.
  • 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.
  • W-Wing A W-Wing starts with two cells that have exactly two candidates each, and the same two: say 7 and 9. Somewhere else on the board, one of those digits - say 7 - has exactly two places left in a single row, column or box, and each of the two cells sees one of those two places. That is enough to prove that at least one of the two cells holds 9, so any cell that sees both of them cannot.

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