What is an Empty Rectangle?
An Empty Rectangle is a box where all the spots for one number lie on one row and one column of that box, plus a row or column outside the box where the number has exactly two spots, one of them on the cross’s line. The number can be removed from the cell where the other end of that pair meets the cross’s other line.
The name comes from the box: the number avoids a rectangle of two or more cells in one corner of it. The pattern only works on one number at a time, so in Marble Sudoku you look for it one marble color at a time.
Use one in 4 moves
- 1Pick. One number at a time.
- 2Cross. Find a box where all its spots lie on one row and one column.
- 3Pair. Find a line outside the box with exactly two spots, one on the cross’s line.
- 4Clear. Remove the number where the pair’s other end meets the cross’s other line.
Then look for singles where you removed it.
TRY IT · GUIDED EXAMPLE
Spring the trap
Only Purple is shown in color. Faded Purple marbles mark every hole where Purple could still go. Find Purple’s cross in the glowing box, add the two-spot column, and see which hole it rules out.
This is the real board from Marble Sudoku. The nine marble colors stand for the numbers 1 to 9.
- Box
- Pair
- Clear
- Place
Spring the trap
Read the example in words
In the top-middle box, Purple can only go in row 1, column 4, row 2, column 4 and row 2, column 6. All three lie on row 2 or column 4: that is the cross. Column 1 has exactly two spots for Purple, in rows 2 and 9. Suppose row 9, column 4 were Purple. Then row 9, column 1 could not be, so row 2, column 1 would be Purple. That clears row 2 of the box, and row 9, column 4 clears column 4 of the box, so the box would have no Purple at all. So row 9, column 4 can’t be Purple, and it is left with a single color: Orange. With numbers turned on, Purple is 7 and Orange is 1.
YOUR TURN · PRACTICE
Spot the Empty Rectangles
Each round shows the spots for one color. Find a box whose spots form a cross and a two-spot line that ends on it, then tap the spot they rule out. On Easy the box glows. On Hard you find it yourself.
Round color:
Why an Empty Rectangle works
The box must hold a 7 somewhere, and every spot left for it is on row 2 or column 4. So whatever happens, the box’s 7 is on row 2 or on column 4.
Now test the target cell. If it were 7, it would rule out column 4 of the box and the far end of the pair. The pair must still place its 7, so the near end on row 2 takes it, and that rules out row 2 of the box. Both lines of the cross are gone and the box has no 7. That is impossible, so the target cell is not 7.
How to spot Empty Rectangles faster
- Scan boxes with few spots. Pick one number and look for boxes where it has two to five spots. Check whether they all fit on one row plus one column.
- Ignore one-line boxes. If all the spots are on a single row or column, that is a pointing pair, not an Empty Rectangle.
- Follow the cross’s lines out. Look along the cross’s row and column outside the box for a column or row with exactly two spots for the number.
- Find the meeting cell. The target is where the pair’s other end lines up with the cross’s other line, outside the box.
Empty Rectangle vs. 2-String Kite vs. Skyscraper
| Empty Rectangle | 2-String Kite | Skyscraper | |
|---|---|---|---|
| Built from | A box plus one two-spot line | A two-spot row plus a two-spot column | Two two-spot rows (or columns) |
| Where they join | On the box’s cross | In one box | In one column (or row) |
| Removes | From one cell | From one cell | From cells that see both tops |
| Difficulty | Advanced | Advanced | Advanced |
Common mistakes
- Spots off the cross. Every spot in the box must lie on the cross’s row or column. One stray spot breaks the pattern.
- A pair inside the box’s band. The two-spot line must cross the box’s band from outside. If its far end is in the same band as the box, the target falls inside the box.
- Pairs with three spots. The line outside the box needs exactly two spots for the number.
- Clearing the wrong corner. The target is on the cross’s other line, in the same line as the pair’s far end. It is never on the line the pair touches.
Empty Rectangle FAQ
What is an Empty Rectangle in Sudoku?
It is a single-number pattern. In one box, all the spots for a number lie on one row and one column. Combined with a row or column outside the box that has exactly two spots for the number, one of them on the cross, it removes the number from one cell.
Why is it called an Empty Rectangle?
Inside the box, the number avoids a block of cells in one corner, an empty rectangle. Only the cross of one row and one column can still hold it.
Does the cross cell itself need a spot?
No. The cell where the cross’s row and column meet may or may not hold a spot. What matters is that no spot lies off the cross.
How is it different from a kite?
A 2-String Kite joins a two-spot row and a two-spot column in a box. An Empty Rectangle uses the box itself as one link, so it works even when the box has three or four spots.
Does this work in Marble Sudoku?
Yes. Pick one marble color, find a box where its holes form a cross, and look along the cross’s lines for a row or column with only two holes for that color.
When should I look for one?
After singles, pairs, box/line reductions and X-Wings stop finding moves. Empty Rectangles are common in hard and expert puzzles.
PRACTICE IN THE GAME
Find Empty Rectangles in a real puzzle
When a hard puzzle stalls, pick one color and scan its boxes for a cross. One two-hole line in the right place often breaks the puzzle open.
Next techniqueW-Wing
