Sudoku HowTo

# Remote Pairs

Remote Pairs uses the properties of bi-value locked sets extended along a chain of cells to identify candidate removal.

A bi-value locked set in a unit, called a naked pair, means that the two candidates contained in the locked set cannot be used in any other cell in the unit. Remote Pairs is a way to extend the reach of this logic beyond the bounds of a unit.

If a number of bi-value cells containing the same two candidates exist in a puzzle and these cells can be connected using pairs of cells in the same unit, a pair of cells with an odd number of connections between them behaves like a naked pairs but only affects common buddy cells.
The Remote Pairs example in Figure 1 shows a number of bi-value cells with candidates 2 and 5 chained together using the pairs of cells contained in a single unit.

In this case the chain was started at the cell tagged with an 'A'.

Consider the two green cells tagged with an 'A and a 'B' which have an odd number of connections between them, in this case three. If the value 2 is placed in cell 'A' and the naked pair logic followed along the chain of connections then cell 'B' would have to be a 5. If the value 5 is placed in cell 'A' and the naked pair logic followed along the chain of connections then cell 'B' has to be a 2.

Either way, this shows that the cells tagged 'A' and 'B' will eventually have to contain a 2 and a 5. There is no way to tell which cell will contain which candidate at this point, but the 2 and the 5 will get used in these two cells. So far this is just like a naked set, but these two cells are no longer in the same unit. They do however have some common buddy cells and these common buddy cells are of interest.

The yellow cells are buddy cells to both the green cells. If the two green cells have to eventually end up containing the candidates 2 and 5, the yellow cells cannot contain either the 2 or the 5. The candidates 2 and 5 can be removed from all of the yellow cells.

This process will work on longer chains as long as there is an odd number of connections between the two cells used to identify candidate removal.

If two cells with an even number of connections between them are considered, the logic of naked pairs followed along the chain of connections will show both cells will end up containing the same candidate, but there is no way to identify which one it will be. There is not enough information here to identify possible candidate removal.
Figure 2 shows a Remote Pairs example with seven connections between the two cells used to identify candidate removal.

The cells tagged with an 'A' and a 'B' contain the candidates 4 and 6. These candidate cannot be used in any cells that are buddy cells to both the green cells.

The yellow cells can have the candidate 4 removed.

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