I tend to reason through problems on plain old pencil and paper. There's just nothing quite like it; it's completely unstructured.
Otherwise, I mean, there are math problems like Maple, Matlab or Mathematica (those are the popular three) but they of course have a learning curve.
Figuring out that particular problem was mostly just a matter of reasoning; define exactly what you want to accomplish, what variables you need (intersection point, velocity of ball, distance of intersection from center of paddle, minimum and maximum speeds), and then determine how to go from what you have to what you want.
Sometimes maybe it just takes being familiar with the math too, or having solved similar problems or understanding different angles of the problem. For example, I had already seen similar problems like this one in my Calculus classes, so it was pretty easy for me to tackle. Math is important in game development.