(Orono, Maine, USA)
Suppose you have solved three cells in a row (or column) within a region. (Rows with letters, from top; columns with numbers from left.) For instance, you have solved
If there is already a solved cell that is not in row C, and whose value is not equal to C1, C2, or C3, then that value must go in C7, C8, or C9. (That is, in the row you have partially solved, and in the other neighboring region in the same direction.)
Similarly, in region A7 – C9, if there is a solved cell not in row C and value not equal to C1, C2, or C3, that value must go in C4, C5, or C6.
You now have a new value in row C in a neighboring region, and not in row C in the other neighboring region. This, in turn, fixes the row in A1 – C3 where that same value must be placed.