How much of mathematics is just dumb luck?

And so i've began working for PS, part of which requires me to help out in SEASME on wednesday night, the other part is thinking of math questions and solutions. Oh, and apparantly Borsuk conjecture is to be a long term goal lol. No progress made in that area at all.

Well, so the point is that, i was working on a problem, thinking of how to solve it. Apparantly the solutions out there use some kind of convergence, which PS doesn't like. Like any new problem, i've been spending quite a bit of time thinking about it, which would make my hourly wage quite low ... To cut a long story short, i chanced upon a solution in the morning :) It used a high-power theorem, of which i have no idea why it is true. And the only reason why i know it exists, is cause i chanced upon it. Yupz, its called the Borsuk-Ulam theorem, which i saw when i was searching up info on Borsuk conjecture.

Hence the question, how much of mathematics is just dumb luck? Just chancing upon the correct theorem, the correct idea, the correct method?