Generalities

"Those who believe that, in the study of nature, great things spring from great opportunities are deceiving themselves.
The germ of an idea, like that of human beings, remains invisible until it finds its own ground, and like them, starts out frail, weak and hidden ."

J.B. Dumas
Academic eulogies 1879

Let's philosophize

I have often noticed that the quest for solutions to a problem resides in the way one looks at it.
In the incredible balance of the universe, all natural phenomena leave traces of their existence before, during and …after.
The pursuit of the keys or laws which determine a phenomenon consists in knowing how to observe, because nature or the creator, in kindness or spite, has put both the problem and its solution in the same place. It all depends on the care given to observing and the light projected on the object.
If, when coming from dazzling light, we enter a dimly lit room, we first distinguish only some of the objects there. However, objects that were initially invisible gradually begin to appear and eventually, the entire room becomes visible.
It is thus the same in the search for hidden laws of the universe.
Expressed differently, the problem and its solution form a whole. One bears within it the imprint of the other; seeing one is like seeing the other. He who asks the question bears the answer within him, even if he is not conscious of it or if he never discovers it.

A method

Rather than turning a problem over in your head without coming up with a solution, it is often judicious to not only visualize it intellectually, but also to attempt to materialize it physically. Try to perceive the problem concretely with your eyes and hands, or in a more general way, with all your senses. Let's say 50% in your head and 50% in your hands.
It's by our hands that instinct can assist the intellect; our fingers find the answer when our mind is reluctant to.