Empty Rectangle
An Empty Rectangle is a single-digit pattern. In one box, every candidate for a digit lies in a single row or a single column of that box; the four cells outside those two lines hold no candidate, which gives the pattern its name. The box is joined to a strong link: a row or column outside the box's band or stack in which the digit has exactly two candidate cells, one of them in the box's row or column. Then the cell that sees the other end of that link and lies on the box's other line can never be the digit, and it is removed from that cell. It is an Expert technique in Bare Sudoku.
When to use it
When a digit's candidates in a box form an L or a T along one row and one column, and the digit also has a row or column with exactly two candidates nearby. Empty Rectangles are common in Expert puzzles, about as common as Finned X-Wings. They need complete notes for the digit.
How to spot it
Pick a digit and look at each box where it has at least two candidate cells. If all of them fit in one row plus one column of the box, you have an empty rectangle for that digit. Now find a column outside the box's stack where the digit has exactly two candidate cells, one of them in the box's row; the other end of that column is the far cell. The cell in the far cell's row and the box's column loses the digit. Try the same with a row outside the box's band and the box's column. In the game, tap the digit key and look for a box whose highlighted notes form an L or a T.
Example
| c1 | c2 | c3 | c4 | c5 | c6 | c7 | c8 | c9 | |
|---|---|---|---|---|---|---|---|---|---|
| r1 | 24 | 6 | 5 | 148 | 9 | 3 | 12 | 18 | 7 |
| r2 | 29 | 3 | 7 | 5 | 168 | 16 | 4 | 1689 | |
| r3 | 49 | 1 | 8 | 467 | 467 | 2 | 69 | 3 | 5 |
| r4 | 5 | 9 | 246 | 3 | 12468 | 16 | 126 | 7 | 1268 |
| r5 | 3 | 8 | 26 | 167 | 1267 | 9 | 5 | 4 | 126 |
| r6 | 1 | 7 | 246 | 468 | 2468 | 5 | 3 | 68 | 9 |
| r7 | 8 | 4 | 3 | 2 | 156 | 7 | 169 | 1569 | 16 |
| r8 | 7 | 2 | 9 | 16 | 156 | 4 | 8 | 156 | 3 |
| r9 | 6 | 5 | 1 | 9 | 3 | 8 | 7 | 2 | 4 |
Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.
- In box 6, every candidate for 1 lies in row 4 or column 9. The other four cells of the box hold no 1: they form the empty rectangle.
- In column 6, 1 has exactly two candidate cells, r4c6 and r2c6, and r4c6 lies in row 4. One of the two must be 1.
- If r2c6 is 1, then r2c9 is not 1, because it sees r2c6.
- If r4c6 is 1, then row 4 has its 1 outside box 6, so the box must place 1 in column 9. r2c9 lies in column 9 too, outside the box, so it is not 1 either.
- Both cases rule out the same cell. Remove 1 from r2c9.
Common mistakes
- Using a box where the digit has a candidate outside the chosen row and column. Every candidate in the box must lie on one of the two lines.
- Taking a link row or column with three candidate cells for the digit, or one that runs through the box's own band or stack. The link must be a strong link outside the box's band or stack, with one end in line with the box.
- Removing the digit from the wrong end. The eliminated cell lies on the box's other line, in line with the far end of the link, not next to the box.
In Bare Sudoku
Needed from level: Expert
In the game, the hint says: Empty Rectangle
Frequently asked questions
What is an Empty Rectangle in Sudoku?
A box in which all candidates for a digit lie in one row and one column, joined to a row or column elsewhere where the digit has exactly two candidate cells, one of them in line with the box. The cell that lies on the box's other line and sees the far end of the pair cannot be the digit.
Why is it called an Empty Rectangle?
Because the four cells of the box that lie outside the chosen row and column contain no candidate for the digit; they form a rectangle that is empty of it.
Why does an Empty Rectangle work?
The two-candidate line forces the digit into one of its two cells. If it is the far cell, the eliminated cell sees it. If it is the near cell, the box must put the digit on its other line, and the eliminated cell lies on that line too. Both cases rule out the same cell.
Is an Empty Rectangle a kind of fish?
It is a close relative: an Empty Rectangle is a grouped Turbot Fish, in which one strong link runs through the box's row and column instead of two single cells. The Skyscraper and the 2-String Kite are its single-cell cousins.
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