Let's assume you've studied the well-known computer science tricks like divide and conquer, greedy algorithms, and the various tree searches, and that's insuffi

Let's assume you've studied the well-known computer science tricks like divide and conquer, greedy algorithms, and the various tree searches, and that's insufficient, even after specializing for your reduced domain. (These will get you 90% of the way there 90% of the time, and usually that's good enough. Sometimes it's not.) For integer tricks like this, you'll want to look at number theory. Modular equivalence is particularly useful, but lots of things in that space are useful. However, there are a lot of other tricks that rely on geometric sequences, or geometry themselves, and other higher maths -- a lot of advances like this are simply connecting random theorum a to situation b, if you can envision the situation in the correct space.