Empty Rectangle
A digit boxed into one row and column, plus a conjugate pair.
An empty rectangle is a single-digit pattern that hides inside one box. When every candidate for a digit in that box lines up along just one row and one column — forming an L that crosses at a single cell, with the box's other four cells empty of the digit — you have the shape the empty rectangle sudoku technique is named for. Paired with a conjugate pair on the same digit in a crossing line, the L forces that digit out of one far cell, delivering an elimination that plain scanning would walk straight past.
Worked example
In box 1 the candidate 1s form an L: a row arm at R1C2 and a column arm at R2C1, meeting at row 1, column 1. Column 6 holds a conjugate pair on 1 — only R1C6 and R5C6 can be the 1 — and R1C6 sits on the L's row, row 1. If R1C6 is not the 1, then R5C6 must be; either way the box's 1 has to lie in row 1 or column 1. That rules out the far corner R5C1 — column 1 from the L, row 5 from the conjugate pair's far end — so 1 is eliminated from R5C1.
When to use it
Reach for the empty rectangle when a single digit refuses to place and one box has all its copies squeezed into a single row plus a single column. Spot the L by scanning each box for a digit whose candidates share one row and one column, leaving the remaining four cells empty of it. Then look for a conjugate pair on that digit in a line that crosses the L's row or column — the far corner of that geometry is your elimination.
How to use it
- Find the empty rectangle. In a box, find a digit whose candidates all lie in one row and one column of the box — an L with a corner cell.
- Find a conjugate pair. Find a line crossing the corner’s row or column where the digit appears exactly twice, one end on that row or column.
- Eliminate at the intersection. Remove the digit where the conjugate pair’s far end lines up with the rectangle’s other arm.
Frequently asked questions
What is an empty rectangle in Sudoku?
An empty rectangle is a single-digit pattern inside one box where every candidate for that digit occupies just one row and one column, forming an L that crosses at a single cell. The four box cells not on those two arms hold no copy of the digit — that empty block is what the name points to. In our board the 1s in box 1 sit only at R1C2 (the row arm) and R2C1 (the column arm), leaving the rest of the box empty of 1s.
How do I spot the L in a box?
Take one digit at a time and list its remaining candidate cells within a box. If they all fall on a single row and a single column, meeting at one crossing point, you have the L. In box 1 the 1s live at R1C2 and R2C1, which share row 1 and column 1, so the L crosses at R1C1 and the other four cells of the box are empty of 1.
How is the empty rectangle related to the skyscraper and two-string kite?
All three are single-digit techniques that join a small structure to a conjugate pair to force an elimination. The skyscraper and two-string kite use two line-based strong links, whereas the empty rectangle swaps one of them for an L-shape inside a box. Here the box-1 L combines with the conjugate pair on 1 in column 6 (R1C6 and R5C6) to eliminate 1 from R5C1 — the same far-corner deduction, driven by a box instead of a second line.