Direct Hidden Single

By the rules of Sudoku, once a digit is placed in a cell, that same digit cannot appear again in any other cell in the same row, column, or box. This gives us a method of elimination: for a given digit, we can look at every cell where that digit is already placed and rule out all empty cells that share a row, column, or box with it. After that, only the remaining uneliminated cells can still contain that digit.

For example, in the grid below, consider digit $3$. For every cell that already contains a $3$, whether it is a given or a answer, we eliminate all empty cells in the same row, column, and box. That tells us that only the cells left uneliminated can still take digit $3$.

In the example above, if we look at row $6$, only cell $R6C4$ remains available for digit $3$. Therefore, $R6C4$ must be $3$.

This elimination pattern resembles the cross-hatched shading used in drawing, so it is called cross-hatching.

If we inspect cell $R6C4$ the way we would for a Direct Naked Single, we find that its peer cells contain only digits $1$, $4$, $6$, $7$, and $9$. In other words, $2$, $3$, $5$, and $8$ could all still go in $R6C4$; these are its candidates. That observation alone is not enough to determine the digit. Only by using cross-hatching can we see that, in this house, digit $3$ has exactly one possible location. That is why this technique is called a Hidden Single.

Direct Hidden Singles can also appear in a column or a box. Using the same grid, if we consider digit $4$, we find that in column $6$, only $R7C6$ remains uneliminated. Therefore, $R7C6$ must be $4$.

Likewise, if we consider digit $2$, we find that in box $7$, only $R7C3$ remains uneliminated. Therefore, $R7C3$ must be $2$.