Thanks for asking this question!
I went to one of India's topmost colleges, and secured a gold medal. Thereafter I worked on my PhD in chemical engineering from the University of Wisconsin Madison. I happened to work with very senior people in my company, and I can tell you how some of them think. I also follow publications in Harvard Business Review, and on some of the financial blogs. Today I felt like contributing a few bits to this topic that interested me.
I would like to take the “thinking in and out of the box” approach, that I somewhat used to develop my own imaginative thinking. You may have heard, 'thinking out of the box' approach at some or the other places. I would get to it in just a bit.
1. Think about an overall picture first
Let me give you a very simple example of house moving. What comes to your mind when I say "You have to move your house". You would start thinking about packing various things, and usually the biggest and most voluminous things would come to your mind. But often, you would realize that the smallest of the things take up an enormous time to be packed and moved. What do you do then? You simply throw them into boxes according to some organization, and you go to your new home, and unpack the boxes and organize the things according to some order. In this case, the overall picture is that you divide your house into various rooms first and think about them individually. Then think about individual items in the respective rooms one after the other.
2. Think about every problem as if it can be divided into pieces and put into boxes.
Just like when you pack things in your home and put them into various boxes, you should make sure, the boxes must not mix. Would you throw spices with clothes? No! Just like that, most engineering problems are heavily convoluted, but if you break them into many pieces, the smaller problems are just like those boxes. The boxes must not mix, and they should be handled with care one after the other. Each box in itself is small enough, and can be handled easily if done with patience. All the boxes put together make the entire house. The problem of moving the whole house sounds big, but that of moving tiny boxes doesn't.
3. Organize everything very well.
There is already too much chaos in the world, around you, and in the little things you do. If you have 250 papers to take care of, wouldn't you put them into various files? The same way, organize your computer program, organize your projects, organize your meetings, organize your personal activities and so on.
4. Think with analogies, but never forget the basics.
Thinking with analogies is simpler, but often one may be mislead into incorrect direction. If you were to pack your kitchen, and all you were packing initially were utensils, you would roughly know the size of the boxes, but if you were to pack your books, and put them into boxes, the boxes would be too heavy to handle. Would the analogy work here? It would, but you must use smaller boxes. If you pack the biggest possible box with books, you would not be able to lift it. The basics tell us that the density of books is about 250 kg per/m^3. In the same way, always think with analogies first, but never forget the basics.
5. Always multiply numbers
Often by multiplying just a few numbers, your overall answer may totally change. To your mind, it may strike that your bed would be the hardest to pack. If you think about packing your kitchen, and it takes a minute to pack every utensil, and if you have ~ 250 utensils, that would be simply ~ 250 mins. Whereas packing your bed might take just an hour. Because here you multiplied the numbers, the time it would take to pack and unpack the kitchen may be more than packing your bed.
Computer programmers spend a fair amount of time just to create algorithms that are O(n), O(n log n) etc, just because they know that they are going to be multiplied by big numbers.
6. Plan very well
When you begin implementing solutions, it is often best to write on a paper what your overall plan looks like. Then you should begin execution. Many times, people do not plan well, and it takes them longer to solve problems.
7. Think about how a robot will do something
Put yourself in a robot's shoes and follow actions just like they have been told to you. If the actions are wrong, perform them, but this way you would figure out that if a robot was to perform them, they would have been wrong. Then when you have found out what is not correct, fix that problem. Continue to make fixes until a robot can perform those actions. When this is possible, you have made a task simpler and automatic. This is pretty much how computer programmers must think!
8. Ask questions to others
Many of your friends, or colleagues will give you their best ideas. You should consult them how they solved a problem. Then think about it, and then isolate yourself to solve them yourself. Then look back some of the solutions, and try to solve them yourself. You would one day get very close to building your own imagination towards solving problems.
Anything and everything in life has to be practiced. This is what makes a man perfect. Elon Musk practices engineering, and Usain Bolt practices running. They cannot give up their practice and hope to be best.
10. Do not hesitate to work hard. And by hard, I mean super hard.
People like Elon Musk work ~ 16 hours or more a day and they constantly put their thoughts to actions.
11. Read as much as you can.
I can tell you that there have been several evenings in my life that I spent only reading different things. Just get your hands on the books, research publications, financial blogs, and so on. You would just know too much about the world.
When you have practiced some of these concepts in different aspects ~ such as physics, statistics, computer programming, engineering, management, and real life world, everything in the world will begin looking simpler and you would be able to see the world in terms of mental images. It is then you can begin to walk the pathway to innovation and see the problem solving in your mind without taking notes, and be able to do more with your imagination.
Last but not the least, your friends are no better than you. According to Steve Jobs, the world that you see around you, was created by people no better than you. So if you believed you could change it, you would be the one changing it. Never ever give up, just keep going, and keep building stuff. Success will be yours.
Stay blessed, and stay inspired!