Unique Rectangles - Using the Single-Solution Property to Eliminate Candidates
The Unique Rectangle technique exploits the fact that a properly constructed Sudoku has exactly one solution. By identifying candidate patterns that would create multiple solutions, those candidates can be eliminated. An advanced technique essential for Expert and above.
The Single-Solution Principle
A properly constructed Sudoku puzzle has exactly one solution. This is a fundamental property and a quality guarantee. The Unique Rectangle technique uses this property as a solving tool. If a specific candidate arrangement would create a structure allowing multiple solutions, that arrangement cannot be correct. Therefore, those candidates can be eliminated.
Basic Form (Type 1)
The most basic Unique Rectangle occurs when four cells at the intersection of 2 rows and 2 columns span exactly 2 blocks, and three of those cells contain only the same 2 candidates (e.g., {2, 7}). If the fourth cell also had only {2, 7}, swapping 2 and 7 would create a deadly pattern with two valid solutions, violating the single-solution property. Therefore, the fourth cell must contain a digit other than 2 and 7, and {2, 7} can be eliminated from it.
Type 2 and Beyond
Type 2 applies when two of the four cells share exactly one additional candidate beyond the pair. That additional candidate must be correct (otherwise a deadly pattern forms), so it can be used to eliminate that digit from cells that see both of those cells. Types 3 and 4 handle more complex conditions, but the underlying logic is the same: if a candidate being wrong would violate the single-solution property, then that candidate must be correct.
Conditions for Discovery
To find Unique Rectangles, look for groups of 4 cells that: (1) form a rectangle across 2 rows and 2 columns, (2) span exactly 2 blocks, and (3) have at least 3 cells sharing the same 2 candidates. In practice, focus on cells with exactly 2 candidates and check whether cells with the same candidate pair form rectangles. Accurate pencil marks are an absolute prerequisite.
Limitations and Cautions
This technique depends on the puzzle having a unique solution. Hand-crafted puzzles or those from low-quality generators may not guarantee uniqueness. Applying Unique Rectangles to such puzzles risks incorrect eliminations. Puzzles from reliable sources (including this site) guarantee uniqueness and are safe. Use this technique as a last resort after exhausting all other logical techniques.
Why the Unique-Solution Premise Holds
The basis on which the unique rectangle works lies in the quality guarantee that a good Sudoku always has a unique solution. If somewhere on the board a rectangle of two rows, two columns, and two blocks forms in which swapping two numbers leaves both valid, two solutions arise at that instant. Because this violates the unique-solution principle, it cannot happen in a correctly designed puzzle. In other words, when you find a placement about to complete such a deadly pattern, you can work backward to conclude that the candidate avoiding it is the correct one. This sets it apart from other techniques in that it relies not only on the numbers on the board but on the design promise of a unique solution.
Where to Use It in Practice, and Cautions
The unique rectangle is powerful but also a dangerous technique if used wrongly. First, the major premise is that the target puzzle has a unique solution. Handmade or poorly generated puzzles may have multiple solutions, so applying the unique rectangle would make a wrong elimination. It can be used with confidence only on puzzles from a reliable source. Second, take care not to apply it mistakenly to placements that do not satisfy the condition that the rectangle spans exactly two blocks. After confirming these premises, using it as a last resort in situations that do not progress even after exhausting other basic techniques is the safest and most effective practice.
Combining with Other Techniques
The unique rectangle can move the board greatly on its own, but in many cases it shows its power combined with other techniques. One elimination by the rectangle produces a naked single or hidden single in another cell, from which a chain spreads. Conversely, after you pare candidates with a naked pair or pointing pair, the conditions for the deadly pattern may fall into place and make the unique rectangle applicable. Hard puzzles can be solved only when several techniques provide footholds for one another. Positioning the unique rectangle as part of that cooperation lets you use it most effectively.