Problems are solved by sleeping

I find that a notebook (or in my case a smartphone) by the bed is a great tool. Write down any and every idea you get while falling asleep/waking up.

I figured out the orbits in an inverse r^5 force field during a particularly productive nap in the winter of 1993. (IIRC, the orbits are closed, like those in an inverse r^2 field, but---bang!---they intersect with the center of force.)

"Solve world hunger: Xerox donuts."