Unique Rectangles

Unique Rectangles is probably a misleading name but it is used consistently in the Sudoku world.

This solving strategy is looking to avoid specific instances of deadly patterns. Any deadly pattern is not allowed in a Sudoku puzzle that has a unique solution. There are Sudoku enthusiasts that do not agree with the premise of uniqueness and if you are one of them then this strategy will probably not work for you.

The deadly pattern considered for this strategy is four cells in a rectangle that contain the same two candidates, with the four cells all contained in just two rows, two columns and two boxes.Unique Rectangle - Type 1. If three of the four cells in the rectangle are bi-value cells and contain only the two common candidates then the fourth cell cannot contain either of the two common candidates. If one of the common candidates was used in the fourth corner of the rectangle it would force the same two values into all four corners of the rectangle. This deadly pattern has to be avoided in a puzzle with a unique solution.

The green cells in Figure 1 show an instance where three of the four cells in the rectangle are bi-value cells with the common candidates 3 and 4. If either the value 3 or the value 4 is placed in the blue cell of the rectangle it would force a rectangle where all four cells have either a 3 or a 4 in them. This would mean the values 3 and 4 could be swapped and the puzzle would have two solutions.

This deadly pattern has to be avoided so the values 3 and 4 can be removed from the candidates of the blue cell of the rectangle.Unique Rectangle - Type 2. If there are only two cells in the rectangle that contain additional candidates and those two cells only have one additional candidate and it is the same in both cells, then that candidate will have to be used in one of the two cells to avoid the deadly pattern.

If this additional candidate has to be used in one of the rectangle cells then it can be removed from any other cells that are buddy cells to both of the rectangle cells with the additional candidate.

Figure 2 shows a unique rectangle with candidates 1 and 3. The two blue cells contain one additional candidate that is the same in both cases, the value 6. To avoid the deadly pattern the value 6 will have to be used in one of the blue cells so it can be removed from any other cells in the same unit as both of these cells. In this case column 8.Unique Rectangle - Type 3. For a rectangle that has additional candidates in only two of the corners and those corners are on the same side of the rectangle, one of these addition candidates has to be used to avoid the deadly pattern. This creates a pair of cells that has to use one of the additional candidates, or think of it as a pseudo cell containing all the additional candidates.

If there is an almost locked set in the same container as the pair of cells in the rectangle that includes at least all the additional candidates in these two cells then that almost locked set and the pseudo cell from the rectangle create a locked set. All candidates in the locked set can be removed from any cells in the unit that are not in the almost locked set or the rectangle.

In the example in Figure 3 the blue cells from the rectangle contain the additional candidates of 1 and 2. The red cells are an almost locked set containing the candidates 1 and 2 (and others) so when this is combined with the pseudo cell made up of the blue cells there is a locked set in box 6 containing the candidates 1, 2, 4, 8 and 9.

The values 1, 2, 4, 8 and 9 can be removed from any cell in box 6 that is not part of the rectangle (blue cells) or part of the almost locked set (red cells).

Note - In this example the same result could have also been obtained by finding the almost locked set containing candidates 1 and 2 in column 9.Unique Rectangle - Type 4. For a rectangle that only has additional candidates in two cells and those two cells are on one side of the rectangle, then if one of the common candidates in the rectangle is not present in any of the buddy cells of the two cells in the rectangle with additional candidates, this candidate must exist in one of the two cells on that side of the rectangle.

In this case the two cells with additional candidates cannot contain the other common candidate in the rectangle or it will form a deadly pattern.

Figure 4 shows a rectangle with common candidates 6 and 8. The two blue cells on one side of the rectangle both contain additional candidates. The two blue cells are the only two cells in row 5 that can contain the value 6 and they are the only two cells in box 6 that can contain the value 6. The two blue cells are not allowed to contain both the value 6 and the value 8 as this would form the deadly pattern.

If one of the blue cells has to contain the value 6 then the value 8 can be removed from these two cells.

Hidden Unique Rectangles

Hidden Unique Rectangles are instances where the deadly pattern is more hidden under a clutter of additional candidates. In most cases Hidden Unique Rectangles only have a single bi-value cell and the other three cells contain the two common candidates plus one or more additional candidates.Hidden Unique Rectangle - Type 1 When there are strong links on one of the unique rectangle common candidates between the three cells in the rectangle that contain addition candidates, it removes the possibility of the other common candidate existing in the cell diagonally opposite the bi-value cell.

The example in Figure 5 shows a Hidden Unique Rectangle on candidates 4 and 8. There are strong links between the green and blue cell in column 1 on the value 8 and between the two blue cells in row 3 on the value 8. The value 8 can only be placed into the green or blue cell in column 1 and the blue cells in row 3.

If the value 4 was chosen as the solution to the blue cell in column 1 it would force the value 8 into the green cell in column 1 and the blue cell in column 2. The value 8 cannot go anywhere else in column 1 and row 3. This would force the value 4 into the bi-value cell and create the deadly pattern.

The value 4 can be removed from cell diagonally opposite the bi-value cell, in this case the blue cell in column 1.Hidden Unique Rectangle - Type 2. When there are two bi-value cells in the rectangle and they are in the same row or column and a strong link exists between one of these bi-value cells and one of the other cells in the rectangle on one of the common candidates, the other common candidate can be removed from the non bi-value cells that is not part of the strong link.

Figure 6 shows an example of a Hidden Unique Rectangle on the values 4 and 8. There is a strong link between the green bi-value cell and the blue cell in row 1 on the value 4. The value 4 cannot be placed anywhere else in row 1.

If the value 8 was inserted into the blue cell in row 3, it will force the value 4 to be placed into the green cell in row 3, which forces the value 8 to be placed into the green cell in row 1, which forces the value 4 to be placed into the blue cell in row 1 and creates a deadly pattern.

The value 8 can be removed from the blue cell in row 3.