Rectangle Elimination

Rectangle Elimination works on a single digit and uses the shape of one box to reach a cell far away from it. It is named after the empty rectangle of four cells that makes it work, and usually kills one candidate — often the one that unblocks the grid.

What the pattern is

Pick a digit and look at one box. If every remaining place for that digit inside the box lies in a single row or a single column of the box, the four cells outside both of those lines hold none of it. Those four are the empty rectangle. The cell where the chosen row and column cross is the hinge.

The hinge gives you a promise: wherever the digit lands in this box, it is on the hinge's row or on the hinge's column, and if it is not on one it must be on the other.

Now add a strong link on the same digit somewhere else — a unit with exactly two places left for it. Put one end of that link on the hinge's row, outside the box. The other end sits on some line of its own, and where that line crosses the hinge's column you get the target.

If the target held the digit, it would clear the hinge's column inside the box, forcing the digit onto the hinge's row inside the box. That kills the near end of the strong link, so the far end must take the digit — and the far end sees the target. Contradiction, so the target loses the candidate.

When to look for it

• When a digit has only two or three candidates left in a box and they cluster along one row and one column.
• When you already have a conjugate pair on that digit — the same strong links you use for colouring feed straight into this pattern.
• After a fish scan fails. Rectangle Elimination catches shapes that a clean X-Wing cannot.

How to apply it, step by step

1. Choose a digit and a box in which it is still unplaced.
2. Check whether every candidate for it in that box fits inside one row plus one column of the box. If four cells outside those lines are clear of it, you have an empty rectangle.
3. Mark the hinge where that row and column cross. Both arms should hold at least one candidate, otherwise the box is simply locked to one line and you only need box-line reduction.
4. Look along the hinge's row, outside the box, for a cell in a strong link on the same digit.
5. Follow the link to its far end and note the line running through it.
6. The target is where that line crosses the hinge's column. Delete the digit there, then repeat with the arms swapped.

A worked example

Digit 4 only. Box 1 covers rows 1 to 3 and columns 1 to 3:

         c1 c2 c3   ...   c9
  r1      4  .  .           .
  r2      4  4  .           4     <- near end of the strong link
  r3      4  .  .           .
  r7      ?  .  .           4     <- far end;  ? = the target r7c1

Inside box 1 every 4 sits in row 2 or column 1. The cells r1c2, r1c3, r3c2 and r3c3 hold no 4, so that is the empty rectangle, and the hinge is r2c1.

Column 9 has exactly two places for a 4, at r2c9 and r7c9. That is the strong link, and r2c9 lies on the hinge's row.

Suppose r7c1 were a 4. Column 1 would then be finished with 4, so box 1 would have to take its 4 in row 2, at r2c2. Row 2 would then be finished with 4, so r2c9 is not a 4. The strong link forces r7c9 to be the 4 instead — but r7c1 and r7c9 share row 7. Impossible, so r7c1 cannot be 4.

Common mistakes

• A rectangle that is not empty. All four off-line cells must be free of the digit. One stray candidate destroys the pattern.
• Using a weak link. The linking unit needs exactly two places for the digit, not three.
• Confusing the arms. The strong link starts on one arm and the elimination lands on the other. Using the same arm twice proves nothing.
• Expecting a placement. The pattern only removes a candidate; it never says what the target is.

Where you meet it

Rectangle Elimination belongs to the tough tier, the last single-digit pattern before chains take over. You will need it on hard classic Sudoku, where it often supplies the one elimination that restarts a stalled grid.