This is quite a general question... apologies if it is too general; I'm just wondering if anyone has any useful tips or advice.
As part my final year university project I've made an algebriac/combinatorial conjecture (I won't say exactly what as I'm not asking for homework help) about something that looks like it might be true... but I also wouldn't be entirely surprised if it was false. I'd like to disprove or prove it, so I've split my time between trying very hard on paper to come up with a proof, while simultaneously computationally searching for counterexamples (the search space is quite large and awkward).
On both fronts I seem to be going nowhere... I haven't identified a single counterexample yet (I have lots that do satisfy the conjecture though), but equally a proof always seems to be just out of reach (there always seems to be one logical step in the way).
Basically this is the first time I've really had to do my "own" maths, so to speak, so I don't really have enough experience or gut instinct to know inuitively whether this is worth pursuing. I fear it's very possible that one of
- the counterexample is computationally out of reach;
- the proof is beyond my mathematical abilities; or
- one them is actually within my reach and I'm about to give up prematurely.
So I'm wondering if people with more experience might be able to share any strategies they use when they find themselves in this situation. Are there indicators/tell-tale-signs I should be looking for to help decide what's more likely to be the case? And does anyone have any advice on how "urn the problem around and start looking at in a different light?
Many thanks.
Congratulations. You're learning how real mathematical research works.
You alternate between believing and doubting. When you are stuck on one you suspect the other and switch modes.
Over time you sense how to choose problems that are goldilocks for you: hard enough to be interesting, not so hard as to be impossible. But it takes time to develop that intuition.
One suggestion for now: talk to people - fellow students, faculty you know. Sometimes explaining your conjecture to a new listener will trigger an insight. Perhaps try explaining to your rubber duck. Or ask about a fragment of your puzzle here on stackexchange.
Finally: you might be able to write an excellent final year project in which you present evidence showing that your conjecture is interesting and difficult.