Actually, I'd love to know of any cases where an outsider with a nontraditional background like this was able to solve a famous unsolved problem that had withstood significant academic and economic scrutiny. Is this a collective myth, or does it ever actually happen?
The only example that comes to mind was Ramanujan, who made many new contributions in number theory, but it's not quite the same thing -- though his genius was unparalleled, he was also working on domains that were at the time not as widely studied as nuclear fusion is today.
Einstein: physics-trained, in Switzerland, married a physics classmate, learned electromagnetism from his father and uncle who were in the power generation business, taught physics, worked in the patent office, certainly a good place to be exposed to the froth of new ideas. Read the Isaacson biography.
> There is a story that Tartaglia learned only half the alphabet from a private tutor before funds ran out, and he had to learn the rest for himself. Be that as it may, he was essentially self-taught. He and his contemporaries, working outside the academies, were responsible for the spread of classic works in modern languages among the educated middle class.
After Tartaglia's solution for Cubic equations and Ferrari's solution for Quartic equations were published in 1545, no doubt that finding a solution for 5th degree polynomials became a hot topic. http://en.wikipedia.org/wiki/Quintic_function
> Finding the roots of a given polynomial has been a prominent mathematical problem.
But even though it was a hot topic, it took 300 years until Galois came around with a method to determine which Quintic equations can and which cannot be factored to "radicals".