When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute.
On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It was the sort of thing that he really wanted to be true; he liked things smooth and beautiful. I, on the other hand, hoped it was false as I like the weird and exceptional in mathematics. It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.
I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
My single (quite small) contribution to mathematical research was a counterexample because it was all I could do. The story does illustrate that it can be helpful to have people with different tools, hopes, and motivations working on a problem, though. I was not, and will never be, even a shadow of that great mathematiciam I studied under, but on that occasion, I had reason to look in a different direction than he did.