To start really getting somewhere with attacking hard problems in the wild, I would guess we need to have a cursory understanding of a wide variety of math topics, and how they link together.
How much do we really need to know to get somewhere, and in what order do we need to learn it? Where can we learn how the different topics in Maths link together?
For example, graph theory, number theory, calculus, trig, complex, ... We learn each topic but rarely discuss how they all weave together.