Pairs: two cells, two digits

Sometimes singles run dry. Every cell has two or more candidates and every digit has two or more places. The next step is to reason about two cells at once.

Suppose two cells in the same row can each only be 1 or 2. One of them is 1 and the other is 2. We don’t know which way round, but either way the row’s 1 and 2 are in those two cells.

So no other cell in that row can be 1 or 2. Cross them off. That is a naked pair. It is case analysis: two cases, and the same fact is true in both.

Write the two cells as X and Y. We know: X is 1 or 2, Y is 1 or 2, and X ≠ Y because they share a row. So either X = 1 and Y = 2, or X = 2 and Y = 1.

Take any other cell Z in the row. In the first case, Z can’t be 1 (X has it) or 2 (Y has it). In the second case, the same. Every case agrees, so Z is not 1 or 2.

Three cells whose candidates, together, are only three digits work the same way. A cell in a triple can have just two of the three.

There is also a hidden version. If two digits can only go in the same two cells of a row, those cells must be those digits, and their other candidates can go.