I chanced upon Alon Amit's amazing answer on Quora to the question:
I am not amazingly good at maths, so it is obvious to me whether I can solve a problem or not after 5 minutes. How do people spend hours on problems?He paints a beautiful problem, which is easy to describe, and which anyone can have fun with. You should check it out.
One of the bad misconceptions people have about math problem solving is that it’s a mechanical process: either you know which algorithm to follow, in which case just fucking follow it and be done with it, or you don’t, in which case there’s nothing you can do.