Empty Rectangle (Rectangle Elimination)
The Empty Rectangle, also called Rectangle Elimination, is a sudoku technique that 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 one row plus one column of the box (a cross), 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.
| Link meets the hinge's row | Link meets the hinge's column | |
|---|---|---|
| Near end of the link | on the hinge's row, outside the box | on the hinge's column, outside the box |
| Far end | on some row of its own | on some column of its own |
| Target cell | that row crossed with the hinge's column | that column crossed with the hinge's row |
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
- Choose a digit and a box in which it is still unplaced.
- 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.
- 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.
- Look along the hinge's row, outside the box, for a cell in a strong link on the same digit.
- Follow the link to its far end and note the line running through it.
- 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 5 only. Solvers and software also call this pattern an Empty Rectangle: it works like a two-link chain whose one end is a whole box.
Start with box 8, the bottom middle box. It holds a 5 in only three cells: G4, G6 and H5.
Now look for the cross. G4 and G6 lie in row G, and H5 lies in column 5, so all three sit on row G or column 5. These lines cross at G5, the hinge (it is already filled with 9). The cells H4, H6, J4 and J6 hold no 5, and that is the empty rectangle.
The cross gives a link inside the box. G4 and G6 count as one group, because they share row G and a 5 in either one has the same effect. If neither is a 5, box 8 must put its 5 in H5. That is a strong link between the group G4 + G6 and H5.
Next, find a second link on column 5. H5 sees E5 along column 5, a weak link: they cannot both be 5. Row E has a 5 in only two cells, E5 and E9, which is a strong link.
Chain them: if G4 and G6 are not 5, then H5 is 5, so E5 is not, so E9 is. Either way, G4, G6 or E9 is always 5. The target is G9: it sees G4 and G6 along row G and E9 along column 9. Whichever of the three holds the 5, G9 cannot.
The result is a single change: 5 is removed from G9.
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.