What are proven ways to solve hard problems?

- huge amounts of time, preferably multiples of the amounts of time that other people also working on the same problem spend on it

- taking good care of your subconscious thought processes: adequate sleep, traversal of real space (i.e. walks/runs), evaluation of physical & emotional state and the identification of dismissed intra/inter-personal issues, and amounts of time each day /not/ spent on the problem

- working on other challenging problems also, as there may be some cross-pollination of ideas

- exposure to extremely creative ideas in completely different (& often consumptive) domains; novels, movies, tv shows, art

- time devoted to learning from peers, and creating with them, also. finding those with parallel and antiparallel value systems may widen your horizon of modes of attack

- knowing when it's best to pause your work and resume it in the future, or when to drop your current approach

- great documentation skills, as your brain will forget the assumptions you relied upon, or the method you used to reach some milestone/write some code once you re-read it in a compressed format

Try to break the problem down into smaller parts. That may help if some of those parts are already being used with other projects, allowing you to concentrate on others. Beware of Murphy!

Soviet problem-solving method TRIZ might be exactly what you are referring to:

https://en.wikipedia.org/wiki/TRIZ

Also, have a look at Polya's "How to solve it":

https://en.wikipedia.org/wiki/How_to_Solve_It

https://www.amazon.com/Progress-Debunked-Creation-Destructio...

My framework is:

1) Identify hard problems — they are pivotal in unlocking the next level of progress. If unresolved, they all but ensure the status quo.

2) Their solution requires System 2 thinking, autopilot interruption, lack of distraction and imposition of both physical and time constraints.