What are some secret things geniuses do to solve problems—techniques anyone can apply?

These are my observations as a mathematician.

Let their dreams work for them : Often reported by stalwarts like Madam Curie, Tesla,Newton,Einstein etc. that often difficult problems are solved in the dreams. They keep problem alive in their mind while going to sleep and let their subconscious mind work for them.Can Our Dreams Solve Problems While We Sleep?
Utilize the power of subconscious or back ground thinking : This needs no explanation.It happens to all of us that if [pullquote]All geniuses (most of them) are known to be loners and introverts.Some of them were absent minded. Why ? Because they tend to live in the world of their own thoughts and theories.[/pullquote]we want to remember some detail for a while and can’t get it, some time later all unannounced it pops up in mind out of nowhere!. This phenomenon is also accounted in more detailed form as Eureka effect.
Cutting themselves from reality : All geniuses (most of them) are known to be loners and introverts.Some of them were absent minded. Why ? Because they tend to live in the world of their own thoughts and theories.16 Super Successful Introverts. Introverts here mean that people immersed in their own thoughts and NOT antisocial. Absent mindedness can also be attributed to the fact that these people are so immersed in their primary goal that they lack focus anywhere else.The term for it is ‘Absent-minded Professor’ syndrome.Are you often absent-minded?. Absent-minded professor

There is a story doing rounds.(I copied it from the link given).

Einstein was once traveling from Princeton on a train when the conductor came down the aisle, punching the tickets of every passenger. When he came to Einstein, Einstein reached in his vest pocket. He couldn’t find his ticket, so he reached in his trouser pockets.

It wasn’t there, so he looked in his briefcase but couldn’t find it. Then he looked in the seat beside him. He still couldn’t find it.

The conductor said, ‘Dr. Einstein, I know who you are. We all know who you are. I’m sure you bought a ticket. Don’t worry about it.’ Einstein nodded appreciatively.

The conductor continued down the aisle punching tickets. As he was ready to move to the next car, he turned around and saw the great physicist down on his hands and knees looking under his seat for his ticket.

The conductor rushed back and said, ‘Dr. Einstein, Dr. Einstein, don’t worry, I know who you are. No problem. You don’t need a ticket. I’m sure you bought one.’

Einstein looked at him and said, ‘Young man, I too, know who I am. What I don’t know is where I’m going.” (copied) A Wonderful Story About Albert Einstein And The Reverend Billy Graham.

Ten Facts You Should Know about Albert Einstein

Ability to restart from scratch always : Genuis are often famously patient.They have the ability to start everything again from scratch.
Believing in some sacred source of knowledge : Many great visionaries have reported throughout history that there is some invisible super consciousness which works to get them the desired results.many discoveries have ‘eureka moments’.

Many scientists cannot really explain the process of their inventions.They will say that the final idea just struck them.[pullquote]They often have larger canvas in their mind, a larger goal to follow.They rather have a mission and NOT goals.[/pullquote]

Indifferent to results : Edition made 10000 mistakes before inventing the bulb. Had he bothered about the results every time , he must have stopped long before. Geniuses believe in the journey and NOT the destination.They just go on working hard.
Gazing at the larger picture : They often have larger canvas in their mind, a larger goal to follow.They rather have a mission and NOT goals.
Some times starting with no clear goal : Well began is half done. They just believe in starting and not finishing always.
Challenging conventional wisdom : All genius people ‘think different’ and it is no longer a secret. They leave the conventional path often to get break through.
Humble in taking help or suggestions from earlier works and colleagues and ruth less in rejecting ideas if they do not fit.