How can I improve my ability to solve hard problems?

Start thinking it's an easy problem and its half done.

What does it take to solve hard problems:There are 3 things essential to problem solving. The right paradigm Pattern recognition Insight 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 compare problem 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. 3rd Principle. 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. Make sure you read this excellent answer by . Suggested reading: http://en.wikipedia.org/wiki/How_to_Solve_It http://www.amazon.com/Seeing-What-Others-Dont-Remarkable/dp/1610392515 http://www.amazon.com/Art-Craft-Problem-Solving/dp/0471789011/ref=sr_1_1?s=books&ie=UTF8&qid=1374558366&sr=1-1&keywords=paul+zeitz&tag=rnwap-20 http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X http://www.amazon.com/Asking-Right-Questions-Critical-Thinking/dp/0205111165

I feel the same way you feel every time there is jump in the difficulty of the work I am doing; it is completely expected and it still terrifies me. The reason you do so well on things you consider easier is that you have a toolbox in your brain of problem solving techniques. Instead of solving every problem you have from the basics, you apply a pattern recognition algorithm-like-thing to the problem, and then hit it with the matching thing from your toolbox. The reason you are having so much trouble is that the toolbox for your current level is not full. Give it time, you'll get there. Additionally, since you don't have any shortcuts for problem solving at this level, you might get frustrated or overwhelmed. It is perfectly natural. Just step back a bit, write down what you know, what you need, and what you know how to do, and then see where that goes.

There is actually no essential difference between learning to solve difficult problems or easy ones. If you're certain that you can't solve a particular problem yourself, then you should to find somebody who can and learn from them. This will allow you to gain knowledge about new approaches and techniques for solving. If your asking about problems that have yet to be solved, then you're going to need genius and a bit of luck. If this is the case, then all I can do for you is wish you the best of luck!

Many people find it difficult to solve some problems because they believe you have to be smart to do so, and they also believe that intelligence is fixed.  For someone like this initial failures cause anxiety that tends to focus their thoughts on only things they've done before.  If you are like this, the first thing to do is to understand that intelligence is not fixed and that you can get better at solving problems by keeping at them and using a variety of approaches.  At some point you will then start seeing challenges as exciting and rewarding. This doesn't mean that you will suddenly find all problems easy.  I once assigned an advanced physics problem to high school students who couldn't solve it.  Then I found that I couldn't either and stopped assigning that problem.  Every year I tried to solve that problem again with my usually successful approaches and failed.  Then one year I decided that I would stop thinking in algebraic terms and try something else.  Once I visualized the problem (made it geometrical), I solved it in minutes without writing anything down!  Basically don't keep trying something that isn't working.  There are many ways to examine the same thing.  Come at it from different angles and in different ways. While persistence is a good thing, working on something to exhaustion is not so good.  If you can't solve a problem at one point, and you've made a good effort, think about what you've done and where it's working and where it's not.  Then do something else for a while or even take a nap or go to sleep.  When you see things as interesting challenges, your brain keeps working on them during sleep or when you're doing other things. I find that students often want to be taught only one way to solve any given problem type.  When you think this way, you're setting yourself up for failure.  Try to own the math or the science your working with.  That means that you play with it and go beyond assignments to understand it in many different ways.  Read about it widely.  Search engines can help with this.  You don't need to read entire books.  Think of things that you're really good at.  What do you do with this that you're not doing with the difficult problems?  Apply the approaches that already work for you in other contexts.

Well, it's because they're harder.  You need to develop a larger foundation of mathematical knowledge before you can proceed with a difficult problem.  If you know the subject you can be good at solving the problem.  For instance in my calc 2 class we covered work problems.  This class puts an emphasis on science and engineering.  So one of the problems may go a little like this. "If a water tank is shaped like the paraboloid z = 7 - x^2 - y^2 and has a height of 6 meters, what is the work done in moving the water out of the top?" Obviously you need the background knowledge necessary to solve this.  I had attempted to learn some of the mathematics behind quantum mechanics in 10th grade after learning calculus in 9th grade.  Little did I realize I needed to learn several more semester's worth of mathematics to even begin to understand most of the topics in that book.  I had bought a book on Lagrangian dynamics, linear algebra, and I wanted to get a book on multivariable calculus too.  I know a little linear and multivariable calculus, but I never got around to studying them in depth to the point that I wanted to. Not only does it take the necessary background knowledge, but it takes creativity and a good intuition and understanding regarding the underlying principles.  My 9th grade algebra 1 teacher would always put two extra credit problems on our tests.  The last ones were really difficult.  I managed to get one of them correct which was better than pretty much everyone else in the class.  I think I had more time with that specific one as I'm generally a slower test taker and you had to finish the main part of the test first of course, but if you give me time with a difficult problem with necessary information, I can usually find a way to solve it, but I'm not extraordinary at it.  Still need a little practice and to learn some things. So, it takes knowledge and creativity to solve difficult problems.

There are two aspects of your question. 1. I would say it doesn't matter that you have to be really really good in solving hard problem. You may be able to solve some other kind of problem, means your brain structure could be designed in such a way that you are fantastic at something else. Like holistic brains are less analytical and vice-versa also. So don't bother too much about it. Find the best in you and accept it and be proud of it, no need to be best at everything. May be you are best at quick response, communications, explaining from top-level etc., there must be something special, enjoy that. 2. If you really want to learn how to solve the hard problems, I think Calvin has already explained good things, I would just want to add few more things. 1. Understand the concept from the core 2. Do the analysis by yourself about multiple possibility of concept, think about How an author would write such problem? Can you do that? This will help you in finding the application of a problem, and this essentially makes your learn reverse part also, means if there is a problem, you'll be able to apply some concept/theory and proceed to solve. 3. Sometime take a break, sometime you'll be able to solve your problem using sub-conscious mind. Keep the problem in your mind and relax and go to sleep, many times it happens as you find the solution in next 2-3 days, because mind is having capacity than we think. It's a practice of meditation also. Wish you all the best. Nishal

I’ve answered similar questions here on Quora, so will add to that.First a response to bad advice.[Calvin] 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.Of course, if you are a college student, time management is important and the above rule might be prudent. But if you want to become better at solving hard problems, do not get into a mindset of always giving up based on the clock. Pick a hard problem and give it your best attempts with all your spare time. Even if you eventually give up, you will have gained skills.Now for some new hints: In addition to trying an easier problem, consider solving a harder problem! Sometimes this gets you out of a rut and onto the right path. I’m in Don Knuth’s The Art of Computer Programming Vol 3 for solving an Open Problem he posed. Another person also solved it, but I “got the nod” for solving a stronger, more difficult version of it. Play tournament chess. A) It is a continuous sequence of hard problems; B) provides rewards for solving those problems; C) takes all the fear out of time management. In the final exam for Advanced Calculus (epsilon and delta proofs!) a proof was asked for a theorem we students had never seen before. With the clock ticking, I’m sure many of my fellow students had a variety of negative attitudes. My positive excitement level went up! This was as much fun as a chess tournament! I applied knowledge and problem solving skills at full efficiency and won! Side note - Calvin’s comments on chess were also quite inaccurate. Widen your perspective and toolbox. If the problem is in subject X, don’t limit yourself to skills learned in class X, apply everything that might be at all relevant. Also do not assume that you are not worthy of inventing new tools to solve the problem. It’s OK if you never do find something new, but don’t reject a new idea as doomed before even trying it out.

P - Practice and perseverance C - confidence and positive attitude M - continue motivating yourself and keep your eye on your goal Apart from very good techniques that above answers already included one should also possess these qualities to excel in problem solving