What does it take to solve hard problems: There are 3 things essential to problem solving.
- The right paradigm
- Pattern recognition
The right paradigm: This is the most important part about problem solving. You need to approach the problems the right way. Even if you are intrinsically motivated and want to study because you're interested make sure you don't despise problem solving. Many smart people do this. When people sit to solve problems, they want to just get through because they want to improve in their domain. Make sure that you enjoy the process, that's the most effortless and the most effective way to get better at problem solving. Problem solving is about persistence. Also, never feel frustrated when you are stuck. Easy problems suck. Whilst some people may feel happy solving 'em, they just suck. It is the tough problems, the ones that spin your head, they stretch your mind. Once a problem stretches your mind, you never see things the way you did before. Solving hard problems changes the way you think and look at the world.
Pattern recognition: Make the gears mesh. I'll take an analogy and compareproblem solving to a system. In order to understand a system perfectly, you need to know how things fit in with each other (meshing of gears). The other important thing is you need to see the big picture. The one thing that separates a genius from his peer is that he has the ability to see things in problems that others don't. He forms the not so obvious connections in problems in order to crack them. He does not do it because he has some innate power to solve problems.
Here's what geniuses do: They ask questions. It's not that whenever you see a problem your brain starts popping up everything useful that is essential to solve a problem. Questions are like cues in a scavenger hunt, they head you in the right direction. There is no other way to do it whatsoever. The genius does this over and over again till it becomes subconscious. It's only a matter of a couple of seconds before he solves a problem that takes a day or two for an average person, it's these abilities that make it so. I will elaborate more on the asking questions and methods to solve questions later in the answer.
Insight: Insight is just the next level of pattern recognition. It is pattern recognition done subconsciously. There is no other way to get insight on a subject other than deliberately practicing hard stuff and stretching your mind. Let's take the case of chess. I suck at chess, just like any other amateur I think in terms of pieces. If the knight is vulnerable, I find ways to attack it. I try my best at killing all the valuable pieces of my opponents. I think in terms of pieces. Grandmasters do it different. They think in term of the big picture, once you are proficient with the details you just need the big picture to win. Once you see the big picture your skills sky rocket. Grandmasters don't think in terms of pieces or think about the next 3-4 steps. They think about the game as opening, middle game and end game. They are predicting what their opponent would do at any point of time and how to exploit his positions. The moment the opponent misses on something, the grandmaster then goes ahead with his attacking strategy that he has thought about a while back. They just stop seeing pieces or do the nitty gritty details. They think about when to do the Sicilian defense or when to go for the Queen gambit and about what to do when the gambit is accepted. After practice even these processes go subconscious and the problem solving increases.
Feynman on problem solving:
"Right. I don't believe in the idea that there are a few peculiar people capable of understanding math, and the rest of the world is normal. Math is a human discovery, and it's no more complicated than humans can understand. I had a calculus book once that said, 'What one fool can do, another can.' What we've been able to work out about nature may look abstract and threatening to someone who hasn't studied it, but it was fools who did it, and in the next generation, all the fools will understand it. There's a tendency to pomposity in all this, to make it deep and profound." -- Feynman, Omni 1979
The pragmatic approach to problem solving: I'll discuss 3 methods here and certain things that you should keep in mind when solving problems.
Method no. 1: Subconscious method: In this method, the subconscious mind is used to solve hard problems. This is my go to method to solve the really hard problems. This method involves priming your mind about the problem, think hard for like 15 minutes. Then leave the problem aside, later when you're taking a walk in the park the solution strikes you. Once you prime on a problem (i.e. think about the problem in a holistic manner, about the details and different approaches to solve the problem) the subconscious starts solving it without you even knowing. You can do the toughest problems in the world using this method provided you have all the tools to solve the problem. Having the right tools/skill set to solve a problem is easy, the problem lies in how to use them. It's easy to know every rule about calculus but when you're confronted with a hairy problem the thing is you just don't know how to apply your tools.
Method no. 2: George Pólya's method (source: wiki):
Pólya's 4 principles:
1st Principle. Understand the problem: Ask yourself these questions.
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Can you think of a picture or a diagram that might help you understand the problem?
- Is there enough information to enable you to find a solution?
- Do you understand all the words used in stating the problem?
- Do you need to ask a question to get the answer?
2nd Principle. Devise a plan:
- Guess and check
- Make an orderly list
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
- Use a formula
- Be creative
- Use your head/noggin
Next is easy.
Carry out the plan 4th Principle. Understand how you solved the problem and what approach can be used to solve similar problems you'll encounter in the future.
Method no. 3: The analogy method: Whenever you're solving an abstract problem try to make a real world analogy and now analyze the analogy. I think this method is under rated. It's so much easier to apply reason and logic to an analogy than to operate on an abstraction symbolically.
Other things to keep in mind:
- "Every now and then a man's mind is stretched by a new idea or sensation, and never shrinks back to its former dimensions." - Oliver Holmes
- Make sure your fundamentals are clear.
- Ensure that you have the right prerequisites and are proficient in the skills required to solve the problem.
- When learning from textbooks, do the exercises and keep a solution manual handy. Do not spend more than 30 minutes on a problem, after that open the manual.
- If you are simply stuck, try solving problems/concepts of (n -1) difficulty level.
- Use additional learning resources. Read reviews on amazon, find recommendations on stackoverflow/Quora.
- Sometimes every effort you make is futile, no matter how hard you work. That's because you're headed in the wrong direction and you don't know, take feedback from professors and peers to prevent this.
- You may sometime doubt yourself that you do not have the mental capability to solve high level problems. Don't think that. Everyone has to practice the right way. No child prodigy ever beat a grandmaster in his first game in chess. Grandmasters play a bazillion games before they get to that level. If in doubt, read the Feynman quote mentioned in the answer.
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