When I was involved in physics I thought that the range of problems that were solvable were like a continent (linear systems) surrounded by islands (integrable systems and a domain in which perturbation theory works around them.)

I thought that the problems solved by Bethe, Feynman and their cohort were the easy problems and the current generation is struggling with problems that were much harder, possibly impossible to solve.

One example is the problem of chaotic dynamics with six degrees of freedom as opposed to four degrees of freedom. With four degrees of freedom you have unbroken KAM tori that partition the phase space into areas that it will stay in forever. In 6 DOF or more, some tori are still unbroken but it's possible for the system to go around them, so even in a phase space region that has a lot of intact tori (e.g. the motion of the planets around the Sun) you can't prove anything for the long term (e.g. the Earth doesn't get ejected from the solar system.)

Look also at examples such as the halting problem, Godel's theorems, etc. I'm not sure if a "higher intelligence" could actually solve that many problems we can't because the problems are intractable.