My first actual encounter with problem-solving was maybe the word problems of mathematics in junior school.
I think that, because all the other Q&A were information-based i.e. you had to know things to answer things. In those word problems, you don't have to know the answer but how to approach the answer.
Of course n+m=? are also like this but most problems, in reality, are word problems as you have to understand it in a human language and somehow convert it into a model of your choice.
What are "problems"?
Most development happens when you find inefficiencies in different systems. When the model of the world doesn't match our perceived model.
So the problems are basically the inefficiencies.
You want to achieve a goal but there is too much friction and you are thinking about a solution that can reduce if not eliminate that friction.That friction is the problem.
Solutions in disguise
As I was growing up I found that the problems I hated most were not the problems in reality, they were solutions in disguise to other problems and they were the best solution we had currently.
So maybe you have to solve the problem of the problem (solution) you are thinking about.
The average solution
Most solutions are designed for scale and that is for the better of the entire society as it should be but this solution may not work for the outliers.
And the biggest problem I can think of is education and its inefficient solution, schools. ( though the goals of schools are not limited to education but IMHO the other goals are also not fulfilled .)
Thinking is not enough
Most things don't work like you thought in your mind, you have to conduct physical experiments ( not just thought experiments ).
You have to rely on the previous data available to test whether the solution works but that data may be outdated ( due to the dynamic nature of the world ).
Then you turn to surveying people and interviewing them. But often people don't actually know what they want and they can lie to hide away what they may consider inappropriate.
The best is to make an experiment out of it but resources are limited and you may don't have enough power to exercise that solution practically.
Filter the solutions based on your priorities, skills, and society at large to obtain a set of problems that you can work on. The best thing to keep in mind is perceived improvement, 10X is idle. You will not be able to achieve the perceived improvement and you may have miscalculated but bigger the better.
The good thing about friction is, if there is a lot of it ( The problem is hard ), the reward will be higher for you and society at large. ( if true )
The path of least resistance is probably targeting the individuals being concerned directly not through traditional systems and networks.
For eg. Youtube ed, Udemy, Coursera, etc.
If the problem is really hard and you just can't solve it or the perceived problem was wrong, at least you learned something while solving it. ¯_(ツ)_/¯
Framing of the problem
It is physically impossible to go beyond the speed of light but if there are wormholes in space we may be able to travel faster than that. ( the speed hasn't increased, the space is intertwined )
The problem wasn't to cross the speed of light but to travel faster to places.
Refining the problem and pinpointing the stress points may help make the seemed impossible into possible.
Problems you just can't solve?
Intellectual barrier: you may not know enough biology of the brain to make a Brain-Machine Interface (BMI) for eg.
While either learn the required knowledge or team up with the people who have it and do your part that you have expertise in.
Physically impossible: first maybe it isn't, it's just that enough people have not worked on it for enough hours to conclude the impossibility.
Maybe there are knowledge gaps or the previous knowledge is incorrect. The only way is to ask an expert or/and experiment.
And remember what seems magic and science fiction may be the future technology but some are just physically impossible and you can't do anything about it.
Biases and Errors
While conducting experiments you may be having biases that can lead to wrong conclusions and human errors can prevail.
If nothing is broken, don't fix it
It can happen that there were no problems in the first place, just noise in your brain.
Problem-solving at scale.
Often the impact metric is taken as (Improvement * People affected) and that is true in the long run maybe but the improvement is the main metric in the short run.