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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

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.

wizzwizz4 5y ago
> 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.
spywaregorilla 5y ago
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.
- Smaug123 5y ago
  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.
- eigenket 5y ago
  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.

wizzwizz4 5y ago

> 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.

spywaregorilla 5y ago

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.

Smaug123 5y ago
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.
eigenket 5y ago
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.