Locked Candidates

An intersection is the set of cells shared by a row or column and a box.

Consider a candidate. If, within a house (row, column, or box), that candidate can appear only inside a particular intersection, then the candidate is said to be locked in that intersection. No matter which cell in the intersection finally contains the digit, the digit must also belong to the other house that forms the same intersection. Therefore, in that other house, the candidate cannot appear outside the intersection, and those candidates can be eliminated.

In other words, for the two houses that form an intersection, if a candidate appears only inside the intersection in one of the houses, then the same candidate can be removed from cells outside the intersection in the other house.

Depending on where the locking is found and where the elimination is made, Locked Candidates can be divided into four cases:

• Case 1: A candidate is locked in the intersection of a box and a row, discovered from the box. Therefore, the candidate can be eliminated from other cells in the row.
• Case 2: A candidate is locked in the intersection of a box and a column, discovered from the box. Therefore, the candidate can be eliminated from other cells in the column.
• Case 3: A candidate is locked in the intersection of a row and a box, discovered from the row. Therefore, the candidate can be eliminated from other cells in the box.
• Case 4: A candidate is locked in the intersection of a column and a box, discovered from the column. Therefore, the candidate can be eliminated from other cells in the box.

For example:

Case 1 (Box $\Rightarrow$ Row): In the grid below, candidate $5$ in box $3$ can appear only in the intersection of box $3$ and row $3$. Therefore, no other cell in row $3$ can contain $5$; if any do, those candidates can be eliminated.

Case 2 (Box $\Rightarrow$ Column): In the grid below, candidate $7$ in box $4$ can appear only in the intersection of box $4$ and column $3$. Therefore, no other cell in column $3$ can contain $7$; if any do, those candidates can be eliminated.

Case 3 (Row $\Rightarrow$ Box): In the grid below, candidate $8$ in row $9$ can appear only in the intersection of row $9$ and box $9$. Therefore, no other cell in box $9$ can contain $8$; if any do, those candidates can be eliminated.

Case 4 (Column $\Rightarrow$ Box): In the grid below, candidate $5$ in column $8$ can appear only in the intersection of column $8$ and box $9$. Therefore, no other cell in box $9$ can contain $5$; if any do, those candidates can be eliminated.