Unique Rectangle Type 1: The Simplest Deadly Pattern
A Unique Rectangle Type 1 is a deadly pattern formed by four cells arranged in two rows, two columns, and exactly two boxes, where three corners contain the same two candidates and the fourth corner contains those two plus extra candidates. This pattern is the most basic form of the [Sudoku unique rectangle](/blog/sudoku-unique-rectangle) strategy. It relies on the logical assumption that a valid Sudoku puzzle must have a single unique solution, allowing you to make a powerful elimination and avoid creating a puzzle with two valid solutions.
What is a Unique Rectangle Type 1?
The Unique Rectangle Type 1 is a specific, forbidden configuration. If it were allowed to remain, it would create a deadly pattern where two solutions become possible, violating the fundamental rule that a proper Sudoku has one unique answer. The pattern involves four cells that form the corners of a rectangle on the grid.
The Spotting Checklist
To correctly identify a Type 1 rectangle, all of the following conditions must be true. First, the four cells must occupy exactly two rows, two columns, and exactly two 3x3 boxes. Second, three of the four corner cells must be bi-value cells containing the same two candidate digits (e.g., {3,8}). Third, the fourth corner cell must contain those same two candidates, plus at least one additional candidate (e.g., {3,8,5}). This fourth cell is the 'victim' or target cell for elimination.
- Focus on pairs of rows and columns that intersect in exactly two boxes.
- Look for common bi-value pairs like {1,2}, {7,9}, or {4,6} that appear in three corners.
The Elimination Rule
Once you have confirmed a Type 1 pattern, the logic is straightforward. You must eliminate the two shared candidate digits from the fourth corner cell. In our example with candidates {3,8,5}, you would remove the 3 and the 8. This leaves only the extra candidate (the 5) in that cell, forcing it to be the solution there and breaking the deadly rectangle before it can form. This is a direct application of Sudoku uniqueness principles.
Worked Example
Consider cells R1C1, R1C4, R3C1, and R3C4. Cells R1C1, R1C4, and R3C1 are all bi-value cells containing only the candidates {3,8}. They occupy two rows (1 & 3), two columns (1 & 4), and two boxes (Box 1 and Box 2). The fourth cell, R3C4, contains the candidates {3,8,5}. This matches the Type 1 pattern perfectly. To prevent a deadly rectangle with two potential solutions, you eliminate the shared pair {3,8} from R3C4. This solves the cell immediately as 5.
How It Differs From Other Strategies
Unique Rectangles are often confused with simpler pair strategies, but they operate on different logic. A Naked Pairs or Hidden Pairs strategy uses basic candidate placement and elimination within a single row, column, or box. The Unique Rectangle strategy is different; it uses the higher-level, global constraint of puzzle uniqueness. It looks at a pattern spanning two boxes and eliminates candidates specifically to preserve the puzzle's single-solution property.
Key Facts
- ▪A Unique Rectangle Type 1 consists of four cells in two rows, two columns, and exactly two 3x3 boxes.
- ▪Three of the rectangle's corners must be bi-value cells holding the same two candidate digits.
- ▪The fourth corner cell must contain those two shared candidates plus at least one extra candidate.
- ▪The elimination rule for Type 1 is to remove the two shared candidates from the fourth corner cell.
- ▪This strategy is based on the logical assumption that a proper Sudoku puzzle has a single unique solution.
- ▪If not eliminated, the pattern creates a 'deadly rectangle' that allows two interchangeable solutions.
- ▪It is one of the most common and easiest-to-spot uniqueness techniques in advanced Sudoku.
- ▪Unlike pairs, this technique considers the puzzle's global uniqueness, not just local candidate counts.
Frequently Asked Questions
Can a Unique Rectangle be in just one box?
No. A true Unique Rectangle Type 1 must span exactly two boxes. Four cells in one box would be a different pattern, like a Naked Quad or a standard pair elimination within that box.
What if the fourth cell has more than three candidates?
The rule still applies. As long as it contains the two shared candidates from the other corners, you eliminate just those two. The cell could have candidates {3,8,5,9} and you would still remove the 3 and 8.
Is the 'uniqueness assumption' always valid?
For published, well-constructed puzzles, yes. It is a standard premise for advanced techniques. In rare, poorly constructed or variant puzzles, it might not hold, but for classic Sudoku from reputable sources, you can safely rely on it.
How is this different from a Naked Pair?
A Naked Pair eliminates candidates from other cells in the same unit. A Unique Rectangle eliminates candidates from one of the pattern's own cells to avoid a two-solution scenario, using the puzzle's uniqueness as logic.
Do all four cells have to be unsolved?
Yes. All four cells in the potential rectangle must be empty, unsolved cells. If any corner is a given number or a solved cell, the deadly pattern cannot form, and the technique does not apply.