Sudoku: Naked relationships

Naked pairs

This algorithm uses Rule 2, the property that each of the symbols 1-9 must occur in every 9mer. If, for any 9mer, a pair of symbols are the only candidates for a pair of cells, then they must be the solutions for those two cells. Between them they are the only possible solutions for the two cells: if either one is set, the solution to the other cell must be the remaining candidate. We don't know which candidate belongs in which cell, but that does not matter

Figure 1: a naked pair hint. In this box, the pair 4 and 9 occur alone in two cells so all other 4s and 9s can be removed.

Each 9mer is examined to see if any two cells share the same two candidates, and those candidates only. If this is the case, then these symbols must be the solutions for these two cells. We do not know which of the symbols is the solution to which cell, but having discovered this pattern, we do know that any other occurences of these symbols in the 9mer can be removed.

Naked triples

This algorithm uses Rule 2, the property that each of the symbols 1-9 must occur in every 9mer.

Figure 2 shows a naked triple example in which the symbols 1,2,4 are the only candidates for three cells in this column. This means that these symbols can be removed from all other cells in this 9mer.

The application of Rule 2 to triples is more subtle than when applied to pairs. In the case of pairs, both symbols have to occur in the same two cells. For triples, three cells and three symbols are involved, but the candidates do not need to occur in all three cells. The symbols simply have to relate to one another so that between them they are the only possible solutions for the three cells: if any one is set, the solution to the other cells must be one of the remaining two symbols, and if one of those is set, the final one is fixed. This is possible when the candidates in three cells overlap by at least one.

Each 9mer is examined to see if any three cells share the same three symbols, and those symbols only. Each of the three cells must contain at least two of the three symbols and no others. If this is the case, then these symbols must be the solutions for these three cells. We do not know which of the three symbols is the solution to which cell, but having discovered this pattern, we do know that any other occurences of these symbols in the 9mer can be removed.