"Seeing" the solution I often hear people say that so-and-so [some famous physicist] can see the solution without calculation. What exactly does this mean? Is it an intuition? But how does one have intuition when it comes to say QM? How might one develop such an intuition?
 A: I will share with you a story from a famous physicist, Feynman,  related by him at a workshop back in the 80s.

During the, I think it was called the Manhattan project, when they were scrambling to create the A bomb, all physicists were employed in doing the laborious cross section calculations, integrals within integrals, that took the best of them ( I think Schwinger and Bethe were there at the time) at least a week to calculate. It was group work, everybody checking the calculations against the others.
Feynman said that he clearly remembered when the idea of the diagrams came to him whole, ( I suppose like Athena from the head of Zeus). He said he remembered his position on his bed, his blue jean legs up the wall.
He started going to the meetings and reproducing the calculations the others had taken a week to do, overnight. When he got confidence that his method worked he started teasing them, by working on the current problems and coming up with the solutions the next day. He had great fun before he told them his method.

At the time Feynman was a trained physicist; in addition he had eidetic memory, books he read he could reread mentally.
Building up an intuition in physics does need a lot of perspiration, but the inspiration part should also be there, and surely genetics must play a role in an eidetic memory.
A: It doesn't need to be such complicated stuff – "seeing the solution" can also apply to perfectly simple differential equations. For instance, in
$$
  \tfrac{\partial^2}{\partial t^2}y = -\eta\cdot y
$$
every physicist would immediately tell you the solution is $y(t)=\sin(\sqrt{\eta}\cdot t)$, or the corresponding cosine or complex exponential / linear combinations. Even many high school students would say this is pretty obvious.
But then, this very equation once turned up in a maths course I was in (functional analysis, not really advanced but not a beginner's lecture either), and when I told them this everybody was rather baffled. They said you can't just do it this way, you rather have to


*

*classify this a 2nd order ordinary linear differential equation with constant coefficients

*make the general ansatz of $y(t)=A\cdot e^{\lambda t}$

*put it in the equation

*get $\lambda^2=-\eta$, therefore $\lambda\in\{\pm i\sqrt{\eta}\}$

*write out these two complex solutions
$$
     y_1(t) = A_1\cdot e^{- i\sqrt{\eta}\cdot t},\qquad
     y_2(t) = A_2\cdot e^{ i\sqrt{\eta}\cdot t}
   $$

*try to find, since we're looking for real solutions rather than complex ones, linear combinations of $y_1$ and $y_2$ that are real. And this way eventually find out that the solution is a sine.


Physicists don't have the time to go through all of these steps every time they encounter this equation (it just comes up far too often), but they just know the solution. Now extend this to more complicated problems, then there will often still be clever guys who have seen the equation before (or, more often, a sufficiently similar one) so they can immidiately tell you the solution others will take hours to calculate.
A: There is nothing spectacular about "seeing" the solution, everyone does it. The reason they tell you this is to try to get you to do it. The famous physicists become famous because they see it for a new problem, where no-one else saw it before, and an important problem. Please, do it too. The purpose is to learn something about nature. Never in the history of the world, has anyone seen something internally that not everyone else could also see. They just saw it first. It does require learning the appropriate background, a short explanation from the seer, and questions/answers, but nowadays pretty much every good physicist can see everything that Newton saw, and Einstein, and Feynman, etc.
There is no magic to it, anyone can do it, but it does require a lot of mental work, and a tremendous amount of internal intellectual honesty, so you can catch when you did something wrong. Feynman was very good at it, but equally so was Bethe, and Landau, and Gribov, and many others. Feynman was easily the best at explaining to others how to do it, through his terrific popular works.
The thing that makes Feynman so much more famous than all the others is nothing to do with his physics (which is very great, don't get me wrong). It is his essentially American character, like a physics Walt Disney or Ronald Reagan, he fought alone against the big-shots and nay-sayers. By his own design, he fits into the American media story of the individual genius, standing out from the crowd. American capitalism uses such fables to justify awarding large amounts of wealth to a small number of people, and they need examples of people who actually do something exceptional who people will think are somehow exceptional people, people who deserve to be rich. This is not really Feynman's fault, he doesn't think this way. But I am sure that he understood that the media thinks this way. The bias for capitalist-reinforcing is strong, so much so that communists like Tullio Regge and Shoichi Sakata have been generally excluded from history books. 
As for "seeing the solution", the best examples are cases where you can't do it by symbol manipulation. For instance, suppose I give you beads on a line of parallel circular tracks, all connected by springs at tension. I pull the first particle. What happens? Did you set up differential equations, or did you see the waves run down the spring?
