What is a Unique Rectangle?
A Unique Rectangle uses four cells at the corners of a rectangle, in two rows, two columns and exactly two boxes. If three corners have only the candidates X and Y, and the fourth has X, Y and more, the fourth corner can’t be X or Y. If it were, all four corners could swap X and Y, and the puzzle would have two solutions.
This is the simplest uniqueness technique, called type 1. It only works in puzzles with exactly one solution, which every well-made puzzle has, including the ones in Marble Sudoku. On the board you look for three holes with the same two small marbles at the corners of a rectangle.
Use one in 3 moves
- 1Corners. Find three cells with the same two candidates, X and Y, on three corners of a rectangle across two boxes.
- 2Fourth. Find the fourth corner. It has X, Y and at least one more candidate.
- 3Clear. Remove X and Y from the fourth corner.
If one candidate is left, place it.
TRY IT · GUIDED EXAMPLE
Build the Unique Rectangle
Small marbles show the colors left in each hole. Find three holes that can only be Teal or Purple on the corners of a rectangle, find the fourth corner, and watch why it can’t take the pair.
This is the real board from Marble Sudoku. The nine marble colors stand for the numbers 1 to 9.
- Corners
- Fourth
- Clear
- Place
Build the Unique Rectangle
Read the example in words
Row 7, column 9, row 9, column 5 and row 9, column 9 can only be Teal or Purple. With row 7, column 5 they make a rectangle in two rows, two columns and two boxes. If row 7, column 5 were Teal or Purple too, the four corners could swap Teal and Purple and still fit every row, column and box, so the puzzle would have two solutions. It has one, so row 7, column 5 can’t be Teal or Purple, and only Pink is left. With numbers turned on, Teal is 4, Purple is 7 and Pink is 3.
YOUR TURN · PRACTICE
Spot the Unique Rectangles
Each round is a real one-solution puzzle with its small marbles shown. Find three corners with the same two colors and tap the fourth corner. On Easy the three corners are ringed. On Hard you find them yourself.
Round color:
Why a Unique Rectangle works
Picture the four corners filled with only X and Y: X and Y in the top row, Y and X in the bottom row. Swap them and every row, column and box still has one X and one Y, because the rectangle covers exactly two rows, two columns and two boxes. That gives two solutions, a so-called deadly pattern.
A proper puzzle has only one solution, so this can’t happen. Three corners are already stuck with X or Y. The only way out is for the fourth corner to be something else. So X and Y can be removed from it.
How to spot Unique Rectangles faster
- Look at two-candidate cells. Mark cells with exactly two candidates and look for the same pair appearing three times.
- Check the shape. The four cells must form a rectangle: two rows, two columns, and only two boxes. Four boxes don’t count.
- Mind the fourth corner. The fourth corner has the pair plus at least one more candidate. If it has exactly one more, that is the answer.
- Trust only real puzzles. Use this only when the puzzle is known to have one solution. Puzzles from good apps and books always do.
Unique Rectangle vs. X-Wing vs. Naked Pair
| Unique Rectangle | X-Wing | Naked Pair | |
|---|---|---|---|
| Based on | One solution only | Where one number can go | Two cells with the same two candidates |
| Shape | Rectangle across two boxes | Rectangle, any boxes | Two cells in one unit |
| Removes | The pair from the fourth corner | The number from two columns or rows | The pair from the rest of the unit |
| Difficulty | Advanced | Advanced | Intermediate |
Common mistakes
- Rectangles across four boxes. If the corners are in four different boxes, a swap breaks a box, so there is no deadly pattern and nothing to remove.
- Clearing the wrong corner. Only the corner with extra candidates loses the pair. The three pair corners stay as they are.
- Puzzles with several solutions. The logic needs exactly one solution. In a homemade or broken puzzle it can give a wrong answer.
- Givens at a corner. All four corners must be empty. If one is already filled, the swap is impossible anyway.
Unique Rectangle FAQ
What is a Unique Rectangle in Sudoku?
It is four cells at the corners of a rectangle in two rows, two columns and two boxes. If three corners can only be X or Y, the fourth corner can’t be X or Y either, because then the puzzle would have two solutions.
What is a deadly pattern?
A deadly pattern is a group of cells that could swap their numbers and still fit every rule. Four corners holding only X and Y in two boxes are the smallest one. A puzzle with one solution can never end in one.
Is using uniqueness cheating?
No. It is a standard technique on solving sites and in competitions. It relies on the puzzle having one solution, which every proper puzzle promises.
Are there other types?
Yes. Type 1, shown here, is the simplest. Types 2 to 6 handle corners with extra candidates in other ways. They all start from the same rectangle.
Does this work in Marble Sudoku?
Yes. Every Marble Sudoku puzzle has exactly one solution. Look for three holes with the same two colors on rectangle corners in two boxes, and remove that pair from the fourth corner.
When should I look for one?
Whenever you see the same pair of candidates three times. Unique Rectangles are quick to check and often break a stuck hard puzzle.
PRACTICE IN THE GAME
Find Unique Rectangles in a real puzzle
When a hard puzzle stalls, look for a pair of colors that shows up in three holes. If they sit on rectangle corners in two boxes, the fourth corner gives you a marble.
Next techniqueNaked Quads
