Direct Hidden Single

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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, focus on digit $1$. For every cell that already contains a $1$, whether it is a given or an answer, we cross out all empty cells in the same row, column, and box. That tells us that only the cells not crossed out can still take digit $1$.

In the example above, if we look at row $4$, only cell $R4C2$ remains available for digit $1$. Therefore, $R4C2$ must be $1$.

This pattern of crossed-out cells resembles the cross-hatched shading used in drawing, so it is called cross-hatching.

If we inspect the peer cells of $R4C2$, we find that only digits $3$, $5$, $6$, $7$, $8$, and $9$ have appeared. In other words, $1$, $2$, and $4$ could all still go in $R4C2$; 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 $1$ 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. For example, in the grid below, if we focus on digit $2$, we find that in column $3$, only $R7C3$ is not crossed out. Therefore, $R7C3$ must be $2$.

Likewise, if we focus on digit $2$ in the grid below, we find that in box $3$, only $R3C8$ is not crossed out. Therefore, $R3C8$ must be $2$.