Unique Rectangless

Index
Unique Rectangles 1
Unique Rectangless use the information that a puzzle only has one solution to exclude candidate values. This technique can only be used on puzzles which have only one solution. When used on a puzzle with multiple solutions, it is possible that you only find one of them.

What we are looking for is the a pattern called the "Deadly Pattern". The "Deadle Pattern" is a pattern of four cells with the same 2 candidates that share 2 rows, columns and blocks. In the puzzle below the deadly pattern is formed by the candidates 8 and 9 in the yellow marked cells.
  • When the top left cell is 8, the top right cell is 9, the bottom left cell is 9 and the bottom right cell is 8.
  • When the top left cell is 9, the top right cell is 8, the bottom left cell is 8 and the bottom right cell is 9.
These values can be exchanged without having any invluence at the rest of the puzzle. This means that a "Deadly Pattern" is formed. Because we know that this puzzle only has one solution, we can break this pattern. There are several ways to do this, categorized in four Unique Rectangless. The example below shows Unique Rectangles 1. In this case, one of the cells forming the "Deadly Pattern" has one or more extra candidate values. In this example, the bottom right cell. The candidates that form the "Deadly Pattern" can be removed from this cell. The candidates 8 and 9 can be safely removed from the bottom right cell.
Unique Rectangles 1
Unique Rectangles 2
Unique Rectangles 2 is formed when two cells of the "Deadly Pattern" have the same one extra candidate. This candidate can be removed from all cell that share the same group with both of the cells with the extra candidate.

In the example below the "Deadly Pattern" is formed by the candidates 3 and 6 in the yellow marked cells. The top left and the rop right cell both have one extra candidate 4. We cannot break the pattern because we do not know wich cell has the value 6. We do know that one of these cells gets the value 6, so this candidate can be safely removed from the blue cells.
Unique Rectangles 2
Unique Rectangles 3
Unique Rectangles 3 is more difficult to spot. You need an extra cell to establish this rectangle. Unique Rectangles 3 is formed when two cells of the "Deadly Pattern" together have two extra candidate values. These two cells, together with a cell that has only these candidates, can be used to exclude candidate values from all other cells sharing the same group with these three cells.

In the example below the "Deadly Pattern" is formed by cell A, B, C and D. Cells A and C contain the extra candidates 1 and 7. Cell E is used to create a Naked Pair together with the other to cells. It is a Naked Pair because we cannot use both cells A an and C, but we need both of them because we don't know wich one. We can now exclude candidate value 1 from the blue marked cell.
Unique Rectangles 3
The example below also illustrates Unique Rectangles 3. The "Deadly Pattern" is formed by the cells A, B, C and D. The candidates 1 and 3 can be exluded from all the blue marked cells.
Unique Rectangles 3
Unique Rectangles 4
In the example below a Unique Rectangles 4 is highlighted yellow. The "Deadly Pattern" is formed by the cells A, B, C and D. In the sixth row the only that have a candidate value 6 are cell C and D. Cell C or cell D must be 6. Because of this, neither cell C or D can be a 9, since this would create the "Deadly Pattern". In the example below, we can savely exclude the candidate value 9 from cell C and D.
Unique Rectangles 4