Sudoku techniques · Expert
Empty Rectangle in Sudoku
An Empty Rectangle is a box in which all the remaining spots for one number lie on a single row and a single column that cross inside it. Combined with a strong link on the same number elsewhere, the geometry forces an elimination: the candidate that would starve the box of every placement can be removed.
Why it works
When a box's only spots for a number lie on one row and one column, placing that number in an outside cell can block the row and eliminate one end of a strong link, forcing the other end to block the column. This would leave the box with nowhere to put the number, so the candidate is removed from the outside cell. The same reasoning applies with rows and columns swapped.
How to spot it
The name is misleading: the box is not empty, and there is no rectangle drawn on the board.
- Look for a box where the number's candidates occupy one row and one column of the box, leaving a 2×2 corner of the box with none.
- The crossing cell of that row and column is the hinge, even if the number is not a candidate there.
- Then find a strong link on the same number in a line that meets the hinge's row or column outside the box.
Builds on
Worked example
These are the app's own lesson boards. Step through them the same way a hint does.
- 1
In the outlined box, every occurrence of candidate 4 lies in either the same row or the same column, or at their intersection. Whichever of the two finally holds 4, the box always claims one of those two lines.
- 2
The connected line outside is a strong link: it holds 4 in exactly two cells, one of them on one of the box's two lines. The red cell lies on the box's other line and sees the far end of the strong link. If the red cell were 4, it would clear its own line inside the box and remove 4 from the far end, so the strong link's other cell would have to be 4, clearing the box's remaining line. The box would have nowhere left for 4, so 4 can be removed from the red cell.