Julia Robinson [1], who played a crucial role in resolving Hilbert's Tenth Problem, was once asked by the personnel department at her university to submit a description of what she did. She gave them a description of her typical work week:
Monday: Try to prove theorem
Tuesday: Try to prove theorem
Wednesday: Try to prove theorem
Thursday: Try to prove theorem
Friday: Theorem false
A good start is to try easier problems that will hopefully provide some insight into the hard problem. I.e. if you have some theorem you want to prove about all matrices of a certain form then do it for 2x2 and 3x3 matrices first or something.
Straight up numerical examples help me a lot as well, so a first approach is to write some code that generates specific cases I can play around with.
Obviously the usefulness of this varies a lot depending on the field and the problem.
Personally I really like coming up with counterexamples to stuff, i.e. if someone has an idea for something they think is true I'm really good at coming up with random examples that break it, so if I'm trying to prove something occasionally I take a break and try to work out what a counterexample would look like and this often provides insight into why counterexamples can't exist.
> A good start is to try easier problems that will hopefully provide some insight into the hard problem. I.e. if you have some theorem you want to prove about all matrices of a certain form then do it for 2x2 and 3x3 matrices first or something.
Other times, it helps to go the other way. If you have a load of numbers to work with and can't see a pattern, replace them with variables; there will probably be more patterns in your derivation if you do that, and you'll be able to simplify more easily.
I don't think I've worked on any problems that took more than a couple days to solve. And yeah that feels intuitive. But I suppose I would have thought these long-unsolved general case math problems were mired in emergent properties at large numbers that made these kinds of approaches impractical.
Well, "these kinds of approaches" isn't really a binary thing. When you're looking at a problem, you try a big mix of lots of different approaches - maybe you try making the problem smaller, and you get something that you could possibly crack if you made this extra assumption, and then your friend talks about their own unrelated work and you think "hold on, there's a bit of an analogy there", and you go back to your books and discover that a handy missing piece is actually Lemma 3.6 of some famous text, and you take a lot of showers, and eventually maybe the walnut shell has softened enough that you can peel a bit of it off.
My record for one I've actually solved is like 6 months working on a problem (on and off). I'm actually more of a physicist than a mathematician (although my work is all about proving theorems and lemmas), so the objects I work with tend to be at least possible to play around numerically with for special cases.
a huge part of it is trying to solve / formulate easier versions of the problem, or problems that are similar or related to the original problem. Or making some stronger assumptions to get rid of the clutter / all of the moving variables and distill it down to the smallest form a human brain can handle! For many of the "huge" unsolved problems, there tends to be a program of "dominos" or "ledges" you hope to work on in some order that will make the original problem fall.
that way you don't just meander idolly from day to day, but instead gain some intuition for the central problem (and of course have publishable work to appease the grant gods / the university).
trying to code up some of the work to experiment is also useful, but that can be a research problem of its own :-)
i read polya's book as an undergrad to my detriment. 10 years later and having "solved" several problems, i can authoritatively say that it is not how difficult problems are solved. the real "how to solve it" for problems that aren't exercises is closer to what feynman said in one of his books: a good physicist [or mathematician] has 5-10 problems on their mind at all times and when they learn of a new technique they apply it to their problem. such learning happens by reading monographs, papers, and going to conferences. that is to say that math (or theoretical physics) is socially produced/developed.
