Empty rectangle

An empty rectangle is the third of three single-digit patterns sharing the same underlying logic. The shape is a box whose candidate cells for a particular digit are confined to one row and one column inside the box — never to a cell that's outside both the chosen row and the chosen column. That row-column intersection cell may or may not itself hold the digit's candidates; what matters is that all the digit's box candidates lie on the chosen row or the chosen column.

The argument with a strong link

The empty-rectangle box on its own doesn't fire any eliminations. It needs a strong link on the same digit elsewhere on the matching row or column. Suppose the box has its digit-candidates confined to row R and column C intersecting at cell (R, C). Now find a strong link in some other row, say row R', between two cells, one of which is in column C. The two cells of the strong link sit in column C and some other column C'.

Trial: if the strong-link cell at (R', C) is the digit, then column C has the digit at (R', C), so the box's digit must come from row R outside column C. Trial: if the strong-link cell at (R', C') is the digit, then column C' has the digit at (R', C'), and the box's digit comes from column C, somewhere along the box. Either way, every cell on row R outside the box and outside column C' loses the digit. Any cell on column C between row R and the box also loses the digit, depending on the geometry.

The full elimination set takes the geometry seriously. The simple version: cells on the same row as the box's row-confined candidates, but outside both the box and the strong link's columns, lose the digit.

The shape's "empty" name

The "empty" in empty rectangle refers to the box's interior: the four cells at the corners of a 2×2 sub-rectangle inside the box are all empty of the digit. The candidates for the digit are restricted to the two-cell row and two-cell column that exclude that empty corner, plus the cell where they intersect.

When it fires

Empty rectangles appear regularly on hard puzzles and feel a little harder to spot than the other two single-digit patterns because the diagnosis requires looking at a box and asking whether a 2×2 sub-region is empty. Once you've trained the eye on the pattern, it shows up across boxes that have already been worked on with pointing pairs — the reduced candidate distributions create empty-rectangle conditions naturally.

The Sudoku Mountain solver detects empty rectangles, skyscraper, and 2-string kite under one shared detector. Find any of the three and the eliminations are mechanical.