> (In hindsight, to solve a challenging problem, try to prove why it cannot be solved, since even that is usually hard, and you may prove yourself wrong in the process by solving it!)

This is how I approach almost any hard problem if I get stuck at some point. Attempt to prove that the next step is impossible - a missing link in the attempted proof usually reveals some deeper structure or invariant that you can use for solving the problem.

It's a great way to force yourself into a different perspective and distill the problem down further.