Empty Rectangle

The Empty Rectangle is a compact single-digit technique that pairs a special box formation with a strong link elsewhere. It's part of the Turbot Fish family, alongside the Skyscraper and 2-String Kite, and it's worth learning for the eliminations the others can't reach.

What is an Empty Rectangle?

Look at a box for one digit. Sometimes its candidates all lie within a single row and a single column of that box - the other four cells (the "empty rectangle") hold none. That means the digit in this box is either on that row or on that column. Combine it with a strong link (a two-cell line) for the same digit elsewhere, and you can eliminate the digit at the point where the box's line and the strong link's free end meet.

In the top-left box the 3 lies only on row 2 and column 2 (blue). With a strong link for 3 in column 8 (amber), the 3 is eliminated where the box's column meets the link's free end - the red cell.

Step by step

Scan boxes for one digit. Find a box where its candidates share a single row and column.
Find a strong link for the same digit - a line where it has exactly two homes.
Locate the crossing point of the box's line and the link's far end.
Eliminate the digit there.

Common mistakes

Candidates outside the row/column. If any of the box's candidates for the digit fall outside the single row and column, there's no empty rectangle.
Missing the strong link. The technique needs a genuine two-cell conjugate pair elsewhere to fire.

Which Sudoku types is this best for?

The same logic applies across variants, but it pays off more in some than others.

Works Classic Sudoku A solid intermediate single-digit tool once fish and wings are familiar. → Limited Killer Sudoku Rare in Killer. → Works Sudoku X Applies as in Classic, with diagonals adding links. → Limited Samurai Sudoku Seldom the fastest path on Samurai. →

Where to practise this

This technique starts paying off at Expert, which is where puzzles first stop yielding to anything simpler. Reading about a pattern and spotting it on a live grid are different skills - the second one only comes from playing.

Play Expert Sudoku About Expert Sudoku

Related techniques

Skyscraper Sudoku Technique Explained Two rows where a digit sits in only two cells, sharing one column; the digit is cleared from cells that see both far ends. X-Wing Sudoku Technique Explained One digit confined to the same two columns across two rows (or vice versa) - eliminate it from the rest of those lines. XY-Wing Sudoku Technique (Y-Wing) Explained A pivot cell with two candidates links two pincer cells; the digit they share is removed from any cell both pincers see. Swordfish Sudoku Technique Explained One digit confined to the same three columns across three rows (or vice versa) - eliminate it from the rest of those columns.