Intersection Removal
This method may look hard, but is actually easy to spot and very common. We look at the intersection
of a square with either a row or a column. The intersection consists of 3 cells in a square which are aligned
either in a row or a column. If a given candidate appears only in the intersection with regards
to either the square or the row/column, we can eliminate it from the other house.
Since we look at a square versus row or column and the eliminations can appear in both, we can
differentiate between four cases (each is illustrated with an example).
- Square x Row - elimination in the square
- Square x Row - elimination in the row
- Square x Column - elimination in the square
- Square x Column - elimination in the column
Intersection Removal - Example 1
Consider the intersection of row A with the top-middle square. The intersecting cells are highlighted in dark blue. Looking at candidate 9, we see that it appears only in the intersecting cells in row A. Hence, the 9 will be the value of either A4 or A5, but we do not know where exactly. What we do know is that both of these are in the top-middle square and so no other 9 can be in the square. We can eliminate the candidate 9 from C5. To see why the method works, let us consider placing the 9 on C5. This forces its elimination from A4 and A5, and there will be no place for the 9 in row A.
Intersection Removal - Example 2
In the next example, we are faced with a similar situation - an intersection between a row and a square, but this time the elimination appears in the row. Consider the intersection of row H with the bottom-middle square. In this square the candidate 2 appears only in H5 and H6. Hence the value 2 will be in one of these, but cannot be anywhere else in row H. Again, if we think of placing 2 in, say, H7, this would leave the square without a cell for the 2.
Intersection Removal - Example 3
The previous two examples were looking at an intersection between a row and a square. Symmetrically, we can search for an intersection between a column and a square. Considering row 1 and the bottom-left square, we see that candidate 2 appears in the column only at the intersection. Hence, we can remove all other 2s in the square.
Intersection Removal - Example 4
Finally, when the intersection is between a column and a square, we can eliminate from the column. The candidate 5 is eliminated from column 6, by using the same logic.
