The Empty Rectangle Technique in Sudoku
What Is an Empty Rectangle?
For one candidate digit, an Empty Rectangle occurs when every candidate in a block lies along one row or one column of that block, forming an L-shaped set, while their intersection does not contain the candidate. This block pattern can combine with an external strong link to force an elimination.
How to Find an Empty Rectangle
- Choose one candidate and inspect a block.
- Confirm that all its positions in the block lie on one row or one column arm.
- Check that the arms' intersection is empty of that candidate.
- Find an external strong link connected to one arm.
- Eliminate the candidate from a cell that sees the forced endpoint and the other arm.
In block 1, candidate 7 lies along row 2 or column 2, while R2C2 does not contain 7. Row 8 has a strong link between R8C2 and R8C8. If R8C2 is 7, it controls column 2; if it is not 7, R8C8 must be 7 and the block forces a 7 into row 2. Either way, R2C8 cannot contain 7.
Why the Elimination Is Valid
The L-shaped block candidates guarantee that removing one arm forces the other. The external strong link supplies the complementary branch. Both branches make the target cell see a confirmed candidate, so its candidate can be deleted.
Empty Rectangle and Locked Candidates
Locked Candidates uses a direct row-block or column-block intersection. Empty Rectangle adds an external strong link, allowing an elimination beyond the immediate block.
What Are Common Empty Rectangle Mistakes?
The intersection cell must not contain the candidate. All candidate positions inside the block must belong to the two arms, and the external pair must be a true strong link.
Checking the Example Step by Step
First verify the L shape in block 1 and the empty intersection R2C2. Next confirm that R8C2-R8C8 is the only pair for 7 in row 8. If R8C2 is true, the column arm controls the block; if it is false, R8C8 is true and the block forces the row arm. Both branches eliminate 7 from R2C8.
Frequently Asked Questions
Must the candidates draw a literal rectangle?
No. The name refers to the empty logical intersection; the visible candidates usually form an L.
Can a block support more than one Empty Rectangle?
Possibly, but each candidate and external strong link must be verified separately.
Summary
An Empty Rectangle needs an L-shaped block restriction, an empty intersection, and an external strong link. Visual resemblance alone is never sufficient.