Extended Unique Rectangles in Sudoku: 2x3 and 3x3 Patterns
Extended Unique Rectangles are deadly patterns larger than the classic 2x2 rectangle, applying uniqueness logic to 2x3 and 3x3 arrangements of cells. They are advanced solving techniques used to prevent puzzles from having multiple solutions by identifying and breaking larger, multi-cell configurations of bi-value cells. The core principle remains the same as a standard [Unique Rectangle](/blog/sudoku-unique-rectangle): if a deadly pattern is fully formed, the puzzle would have two valid solutions. Therefore, you must eliminate candidates to prevent that pattern from being completed. This article explains the 2x3, Triple, and 3x3 patterns and how their Type 1, Type 2, and Type 4 elimination rules work.
What is an Extended Unique Rectangle?
An Extended Unique Rectangle (EUR) is a deadly pattern involving more than four cells, typically six cells in a 2x3 rectangle or nine cells in a 3x3 square. Like its smaller counterpart, it relies on the uniqueness assumption—that a well-formed Sudoku puzzle has only one solution. The most common and practically useful form is the 2x3 pattern. While a full 3x3 pattern occupying three rows, three columns, and three boxes is theoretically possible, a 2x3 arrangement is the minimal deadly window needed, as removing any one column from a 3x3 still leaves a multi-solution pattern.
How do you spot a 2x3 Extended Unique Rectangle?
Look for six cells that form a 2x3 rectangle, meaning they span exactly two rows, three columns, and two or three boxes. Crucially, five of these six cells must be bi-value cells containing only two candidates from a set of three digits. For example, five cells might hold combinations of digits {1,2,3}, such as [1,2], [1,3], and [2,3]. The pattern is considered deadly if placing the sixth cell as a bi-value cell from the same set would complete a configuration with multiple solutions. Patterns can be oriented either vertically (2 rows x 3 columns) or horizontally (3 rows x 2 columns).
What is an Extended UR Triple?
An Extended UR Triple is a specific 2x3 pattern where three of the six cells are the critical bi-value cells. These three cells, together, contain exactly three different candidate digits. A classic example is three cells with candidates [1,2], [1,3], and [2,3]. They form the core of the deadly triple. The other three cells in the 2x3 rectangle provide the structural context, often sharing rows, columns, and boxes with the triple. Identifying this triple is the first step in applying elimination logic.
How does Extended UR Type 1 elimination work?
In an Extended UR Type 1, one of the cells in the deadly pattern is the 'odd cell out' because it contains extra candidates beyond the two or three digits defining the pattern. To prevent the deadly pattern from forming, this odd cell must use one of its extra candidates. Therefore, you can eliminate the pattern digits (the ones that would complete the deadly set) from that specific cell. For instance, if the deadly triple digits are {1,2,3} and a cell has candidates [1,2,3,4], you can remove 1, 2, and 3 from it, leaving 4 as the solution. This logic directly extends from the standard Unique Rectangle Type 1.
How do Type 2 and Type 4 eliminations extend to larger patterns?
The logic for Types 2 and 4 extends similarly from the classic rectangle to the larger pattern. In a Type 2 scenario, one of the pattern digits appears as a candidate in all cells of the deadly set across two rows or two columns. This digit then becomes a 'guardian' and can be eliminated from all other cells in those rows or columns outside the pattern. Type 4 logic involves a hidden pair or subset within the pattern's cells. If the only placements for two pattern digits are in two cells within the pattern, those two cells become a locked pair for those digits, allowing for eliminations in their shared house (row, column, or box). These concepts also apply to more complex patterns like Hidden Unique Rectangles.
How is this different from a standard Unique Rectangle?
A standard Unique Rectangle is always a 2x2 pattern involving four cells and two digits. Extended Unique Rectangles involve more cells (six or nine) and typically three digits. The core uniqueness principle is identical, but the geometry and candidate interactions are more complex. The 'deadly pattern' in an EUR is not a simple rectangle but a larger, often interlinked, set of bi-value cells. The elimination types (1, 2, 4) are named analogously, but the deductions apply to the larger cell set. It's also distinct from an Avoidable Rectangle, which deals with a potential deadly pattern already partially resolved.
When should I look for an Extended Unique Rectangle?
Search for EURs in the mid-to-late stages of solving tough puzzles, especially when you have many bi-value cells remaining. They are rare but powerful. Focus on areas where you see clusters of bi-value cells sharing rows, columns, and boxes. The 2x3 pattern is the most practical. If you find a potential 3x3 pattern, check if a 2x3 subset within it already forms the deadly core; you rarely need to consider the full nine cells. Our hint engine finds that most players spot these patterns after mastering standard Unique Rectangles and other intermediate techniques.
- Scan for three digits that appear frequently together in bi-value cells within a confined area.
- Remember, the deadly set usually involves exactly three digits spread across five bi-value cells in a 2x3 grid.
Key Facts
- ▪Extended Unique Rectangles are deadly patterns larger than the classic 2x2 rectangle, applying to 2x3 and 3x3 arrangements.
- ▪The most common and useful form is the 2x3 pattern, involving six cells and typically three candidate digits.
- ▪An Extended UR Triple consists of three bi-value cells whose candidates together total exactly three digits (e.g., [1,2], [1,3], [2,3]).
- ▪In a Type 1 EUR, a cell with extra candidates beyond the deadly pattern must use one, so the pattern digits can be removed from that cell.
- ▪Type 2 EUR logic eliminates a 'guardian' digit from a row or column if that digit appears in all pattern cells across that house.
- ▪Type 4 EUR logic identifies a hidden pair within the pattern's cells, locking digits and allowing eliminations in the shared house.
- ▪A full 3x3 pattern must occupy exactly 3 rows, 3 columns, and 3 boxes, but a deadly 2x3 subset is the minimal needed for eliminations.
- ▪Extended UR patterns can be oriented vertically (2 rows x 3 columns) or horizontally (3 rows x 2 columns).
Frequently Asked Questions
Is the uniqueness assumption always valid for Extended Unique Rectangles?
Yes, it applies if you assume the puzzle has one solution. Like standard techniques, EURs are valid for proper, well-formed puzzles. They are not guessing.
Can an Extended UR have more than three digits?
Theoretically yes, but patterns with three digits (a deadly triple) are the most common and practical for solvers. Larger sets are extremely rare.
How is an Extended UR different from an Avoidable Rectangle?
An Avoidable Rectangle is a potential deadly pattern already broken by a solved cell. An Extended UR is a larger unbroken pattern you must prevent from forming. Learn more about Avoidable Rectangles.
Do I need to find a full 3x3 pattern?
No. A 2x3 window is sufficient. Removing any column from a 3x3 leaves a smaller multi-solution pattern, so only the 2x3 core is needed for logic.
Are Type 3 Extended Unique Rectangles used?
Type 3 logic (conjugate pairs) exists but is complex and rare. Most solving guides and solvers focus on the more applicable Type 1, 2, and 4 eliminations.