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Intersection Reduction

The intersection reduction technique is the first candidate elimination technique that we've come across so far. What this means is that it allows you to eliminate possibilities from cells (i.e. to tell us where a digit cannot go), but not always to place a digit in any particular cell. It is nevertheless a very powerful and important technique, which is necessary for solving most non-trivial puzzles.

Intersection reduction can be split arbitrarily into two different techniques, "Pointing Pairs" and "Box-Line reduction". Because it is easier to introduce the concept this way, I will examine both of these in order (although they are in fact the same technique).

Pointing Pairs

The pointing pair rule is best stated as follows: "If a digit can only occur on one row or column within a box, that digit cannot go anywhere else in that row or column". An example of this in practice is shown below:

Before I explain this example, let me point out that the above puzzle has been fully annotated. This means that all possibilities for where numbers can go in empty cells have been filled in. It is, in theory, never necessary to annotate a puzzle (because it is theoretically possible to remember where certain numbers can go and where they can't). For the purposes of explaining candidate reduction techniques, however, it is far easier to see how they work if you can see which numbers could go where. Thus, the small numbers in the empty cells represent the places where that number could potentially go.

In the above example, the pointing pair technique allows us to eliminate "4" from the two cells highlighted in red. In other words, it tells us that "4" cannot possibly go in either of those cells. To understand why, look at the two cells highlighted in orange, in the top left box. These are the only places that the digit "4" can go in that box. So, we know that one of those two orange cells must be a "4". But these cells are on the same row as the ones marked in red. So neither of the cells marked in red can possibly be a "4", because that would mean there would be two "4"s in that row.

Box-Line Reduction

Box-Line reduction is another way of looking at intersection reduction. There is no example of pointing pairs that can't be seen as a Box-Line Reduction and vice-versa, provided that hidden singles have been spotted before using intersection reduction. Simply stated the rule of box-line reduction is: "if a digit can only occur on a row or column within one box, then it can be eliminated from everywhere else in that box". To illustrate this, here is the previous example, as a box-line reduction.

In this example, the digit "4" can be eliminated from the cells highlighted in red. The reason for this is that, in the top row, the cell highlighted in dark blue cannot contain a "4", so it must be in one of the orange cells. Because both the orange cells are in the same box, there can't be a "4" in any of the other cells in the box, so we can eliminate both of the "4" candidates in the red cells.

As I explained earlier, there is no reason for using pointing pairs over box-line reduction, since they give exactly the same result. So you can just use which ever one you spot most easily.

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