Interactive Sudoku Technique
Follow the Empty Rectangle
One candidate bends through a box. A Strong Link waits outside. Together they reach a victim that neither possible outcome allows to survive.
First, Clear the Name
This Is Not a Unique Rectangle
An Empty Rectangle is a single-candidate strategy. It does not compare two digits, create a deadly pattern, or depend upon uniqueness. Its logic comes from one candidate confined to one row and one column inside a box, joined to a Strong Link outside that box.
One candidate. One box. One row arm. One column arm.
Recognize the Footprint
The Candidate Bends Through One Box
Choose one digit and inspect a single box. Every possible home for that digit in the box must fit along the union of only one row and one column. Depending upon which cells remain available, the footprint may resemble a T, an L, or a plus.
Sideways T
The verified example below uses this footprint.
L Shape
The same confinement rule can leave a shorter arm.
Plus Shape
The row and column arms may extend in both directions.
The letters are memory aids, not the proof. The governing fact is that all candidates in the box are covered by one chosen row and one chosen column.
A Real Sudoku Paradise Case
An Empty Rectangle on 7
Read the colors in order
-
Green: In Box 3, every candidate 7 lies
along Row 2 or Column 7. The given 2 at
r2c7occupies their intersection and makes the sideways T easy to see. -
Blue X:
r4c7continues the column arm outside Box 3. -
Gold Y: Row 4 contains only two possible
homes for 7:
r4c7andr4c4. X and Y form a Strong Link. -
Red victim:
r2c4lies where Y's column meets the Empty Rectangle's row arm.
Interactive Proof
Test Both Possible Values of X
X at r4c7 is either 7 or it is not. Explore both
possibilities. The victim receives the same verdict either way.
Both roads agree: remove candidate 7 from
r2c4.
The Two Roads
Why the Victim Cannot Be 7
If X is 7
X removes 7 from the Column 7 arm inside Box 3. The box must
then place its 7 somewhere along Row 2. The victim at
r2c4 sees that row and cannot be 7.
If X is not 7
X and Y are a conjugate pair on Row 4, so Y at
r4c4 must be 7. The victim at r2c4
sees Y in Column 4 and cannot be 7.
Knowing whether X or Y is true is unnecessary. Both possibilities eliminate the same candidate from the victim.
Interactive Recognition
Does the Box Qualify?
Judge whether every candidate 7 shown in each miniature box can be covered by one row and one column. Shape names are hints; the confinement rule decides.
0 of 4 examples mastered
Before You Eliminate
Check Every Connection
- Use one candidate throughout the entire pattern.
- Inside the chosen box, all occurrences must fit within one row and one column.
- X must align with one arm outside the Empty Rectangle box.
- X and Y must form a genuine Strong Link in their shared line.
- The victim must see Y and lie on the line supplied by the other arm of the box pattern.
- Do not invoke uniqueness. If the box-and-line proof is absent, this is not an Empty Rectangle elimination.
Common Questions
Empty Rectangles Clarified
Must the pattern look like a perfect letter?
No. T, L, and plus are visual aids. Candidate confinement to one row and one column inside the box is what matters.
Is this related to a Unique Rectangle?
No. They share the word rectangle, but not their proof. Empty Rectangles use one candidate, box-line confinement, and a Strong Link. Unique Rectangles compare two candidates and avoid deadly ambiguity.
Why are X and Y useful if I do not know which is true?
Their Strong Link guarantees that one is true. Each possibility independently removes the same candidate from the victim.
Continue Practicing
Follow One Candidate Across the Grid
Generate a fresh puzzle, enter one from another source, or review the Strong-Link patterns that prepared the way for this lesson.