Casework

When to use Casework

I like to say that casework can be applied to anything in combinatorics, because when it comes down to it casework is just a type of counting, and combinatorics is fundamentally counting. That being said, casework is technically brute forcing, which we want to avoid unless it is nescessary or optimal. There will sometimes be better solutions to casework, or combinations of when to use casework, etc.

Casework is best used when there are distinct and manageable cases (small case sizes, not many). Of course, this hardly simplifies it, so we'll go into a few cases where casework tends to be likely:

In number theory/combinatorics questions where one digit is set, either by divisibility or by a clear constraint (e.g. "divisible by 5" would indicate what the last digit is, and you can solve for the remaining digits by cases for if it is \(0\) or \(5\))

When there are boundaries ("at least", "at most"), where you can solve for all cases that are under or over the constraint

When there are already clearly defined cases and they are easy to solve (e.g. how many combinations are there of 3 items taken 2 at a time?)

When finding the opposite would take too long. If finding the "complement" or opposite would be easier, that is indicative of complementary counting.

Casework will work in a lot of situations, if not all, but it helps to know when it is best.

How to Divide Into Cases

Cases must fit two situations: they must be mutually exlcusive and collectively exhaustive. In a more simple vernacular, they must not overlap, and alltogether they must cover the entire data set. For instance, if we said that we had knights that were either red or blue and then were also either magic or non-magic, we can't say our cases are red and magic, because that could overlap and would discount blue-non-magics. We can't do red-blue-magic-non-magic because those would overlap. Thus, we would have to do red-blue or magic-non-magic.