## Sudoku: Hidden relationships

### Hidden singles

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

Figure 1 shows an example in which the symbol 7 remains in only one cell in the box, and hence all other symbols in the cell can be removed.

Each 9mer is examined to see if any symbol occurs in only one of its cells. Or putting it the other way round: cells are examined to see if they contain a symbol which is unique within any of its three 9mers. If such a symbol/cell combination is found, because the symbol has to occur somewhere in the corresponding 9mer, it must be in this cell, and the cell can be set accordingly.

### Hidden pairs

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

Figure 2 shows a box in which the symbols 2 and 9 occur only two cells and so the other symbols in these cells can be removed.

Each 9mer is examined to see if any pair of symbols occur only in the same pair of cells. If so, then these symbols must be the solutions for these two cells: they are ruled out everywhere else in this 9mer, yet they have to occur somewhere, so it can only be here. We do not know which of the two symbols is the solution to which cell, but having discovered this pattern, we do know that any other symbols in the two cells involved can be removed.

### Hidden triples

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

Figure 3 shows a row in which the symbols 5,7 and 8 occur in only three cells and so the other symbols in these cells can be removed [click for larger image].

Each 9mer is examined to see if any trio of symbols occur only in the same trio of cells. If so, then these symbols must be the solutions for these three cells: they are ruled out everywhere else in this 9mer, yet they have to occur somewhere, so it can only be here. 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 symbols in the three cells involved can be removed.

### Locked candidates 1

This algorithm uses the Rule 1, i.e. the property that a symbol can only occur once in each 9mer.

Figure 4 shows an example where, for the bottom box, the symbol 7 can only occur in the left hand column. Hence it can be removed from the rest of that column.

If a symbol can only occur in one row or one column of a box it cannot occur in those rows or columns outside of the box.

Boxes are examined to see if any of their symbols are restricted to a single row or column.

If any such symbols are found they can be removed from the segments of those rows or columns outside of the boxes.

### Locked candidates 2

Figure 5 shows an example where, for the middle column, the symbol 9 can only occur in the middle box. Hence it can be removed from the rest of that box.

This algorithm uses Rule 1, the property that a symbol can only occur once in each 9mer.

If, for a particular row or column, a symbol can occur in only one box, then it cannot occur elsewhere in that box.

Rows and columns are examined to see if any of their symbols are restricted to a single box.

If any such symbols are found they can be removed from the rest of the cells in the box.