Quora Question: What are the best guidelines a person can follow to improve his/her problem solving skills?

In mathematical and scientific circles, a problem is defined as a situation where there is a difference between the desired or goal state and your current state. 

Usually most of us go about in a disorganized way when trying to solve a problem. 

But becoming organized can make you a much better problem solver.

There are certain ways of approaching problems which are followed mainly by mathematicians and scientists but can be applied to any branch.

Here are the guidelines I usually follow.


Bright Idea / Intuition / Inspiration 

High level pattern matching: Use working memory + visualization + organization to hold the whole problem + solution (so far) at a time and rapidly move between its different portions. Change the representation if you can’t make progress. Take a break and then come back if you can’t make progress after thinking for a long time. Zoom out for sketching the solution and zoom in to carry out each part with rigorous arguments. Work on parts of the problem and the problem as a whole.

Understanding the problem

Data? Unknown? Condition? Organize the whole problem with a diagram: invent your own representations (It might be a mathematical structure: graph, network, lattice, matrix, number line, geometrical figure,…). Visualize it and Draw it. Find a structure that holds the problem and solution at once and completely. Get emotionally involved with the problem (Wow! How can this problem be solved? Is it a “to prove” problem? So, this theorem holds true? Wow! Is it a “to find” problem? Guess the answer. (Good for exercising intuition / high level pattern matching.) Extract from memory all the relevant information, theorems, problems (mobilization) and organize/connect/plan with them.

Planning

First make high level plans for solving the problem then carry out the plan with rigorous arguments. Make connections between Data, Unknowns and Conditions. What about working backwards? What could be penultimate step? Related problems? Related Data? Related Unknowns? Related Conditions? Theorem? Structure? Imagine a more accessible related problem and solve it. What makes the given problem hard? Try different strategies, tactics and tools. Don’t get stuck. Change the problem representation and change your perspective. [generalization-specialization: logical quantification]

Carrying out the plan

Rigorously prove and convince yourself that the solution / proof is correct. (The way to convince oneself is to visualize / imagine. Remember, “seeing is believing”. If you can see the arguments in your mind’s eyes, you believe it; in other words, you are convinced.) If you can convince yourself, your mathematical intuition would grow. Otherwise, what is mathematically correct / logically consistent, won’t seem correct at a glance. (This is the problem people have – they know something is scientifically correct, but they get astonished when they see it in action / nature – their subconscious and conscious mind have different ideas – they have read it but haven’t reprogrammed their subconscious beliefs.)

Checking the result

Is it OK? Can you see the whole problem-solution (solution embedded in the problem) at a glance? Is it reusable in other problems? What have you learned that can be reused in developing solutions to other problems? If it’s a “to prove” problem, then the theorem can be reused.

Thinking harder and going to deeper levels of concentration (and mental performance)

Level 1, Level 2, Level 3 and so on. You might find it hard (feel fatigue, etc.) to cross a level but if you push through and succeed, your brain power will expand (with a bigger working memory). Newton used to work on a problem until it was solved. Try other methods to go to deeper level of concentration: try visualizing progressively more vividly; hearing, touching, smelling, tasting progressively more realistically (always visualizing in tandem).

Learning – generalization & organization

“Each problem that I solved became a rule, which served afterwards to solve other problems.” – Rene Descartes.


Generalize the problem, generalize your solution, generalize solutions by others (Say, wow! What an incredible brain!) and find out its applications, generalize strategies, tactics, tools and make more than one representations so that you can hold the strategies, tactics, tools completely in your head at a time. Organize the strategy in your mental problem solving toolbox (hierarchically or in a graph-like structure). Rather than solving a lot of problems without going deeper, concentrate on all the sub-problems, ideas, queries that arise while solving a problem. 

Visualization

Visualize every problem solving strategy, tool, technique, algorithm, algorithmic paradigm, design pattern, computational abstraction as structures and processes.

Learning by organizing Mathematics

Organize all the problem solving strategies, techniques, tools, areas of Mathematics, theorems, identities, structures in your ontology.


References:

1. “How to solve it” – George Polya.

2. The art and craft of problem solving – Paul Zeitz. (many ideas, for example, the idea of working backwards and penultimate step.)

3. Books on Mathematical Problem Solving.

4. Mind Power – Reader’s Digest (Think harder and go to deeper levels of concentration….. from “How to increase energy” – “Usually we make a practice of stopping an occupation as we meet the first layer of fatigue…… But if an unusual necessity forces us onward, a surprising thing occurs. The fatigue gets worse up to a certain point, when, gradually or suddenly, it passes away and we are fresher than before!….We have evidently tapped a new level of energy. There may be layer after layer of this experience, a third and a fourth ‘wind’. We find amounts of ease and power that we never dreamed ourselves to own……habitually we never push through the obstruction of fatigue.”)

5. “Newton used to work on a problem until it was solved.” from উন্নত জীবন – ডাঃ লুতফর রহমান। “নিউটন বলেছেন, আমার আবিষ্কারের কারণ আমার প্রতিভা নয়। বহু বছরের পরিস্রম ও নিরবিচ্ছিন্ন চিন্তার ফলেই আমি আমাকে সার্থক করেছি, যা যখন আমার মনের সামনে এসেছে, শুধু তারই মীমাংসায় আমি বাস্ত থাকতাম। অস্পষ্টতা থেকে ধীরে ধীরে স্পষ্টতার মধ্যে উপস্থিত হয়েছি।”

6. The concepts of working memory, subconscious mind – from books on Psychology.

7. “Each problem that I solved became a rule, which served afterwards to solve other problems.” – Rene Descartes.

8. The Emotion Machine & The Society of Mind – Marvin Minsky. I own hard copies of both the books. I used to consider Marvin Minsky as a “guru” during my college years.

9. Books on Artificial Intelligence (Chapters on Problem Solving, Planning, Machine Learning) inspiration for “hierarchically or in a graph-like structure”.

10. “Get emotionally involved with the problem.” – from “You And Your Research” by Richard Hamming.



Quora Links:

Solving Programming Problems

 

  • Learn Computational Abstractions. “Explanation Based Learning (EBL)” – once you design an abstraction that might prove useful in other problems, generalize and learn it.
    • Control Abstractions
      • Iteration
      • Recursion
      • Recursive Backtracking
    • Data Structures
    • Algorithms
  • Zoom in, zoom out
    • Work on parts, always keeping an eye on the whole. See how changes in part affect the whole.
  • Once a problem is solved, look back and learn new strategies, tools, techniques. Ask questions.
  • Learn common mistakes. Avoid them. Develop Test cases that break the code.
  • Speed up.
    • Typing speed.
    • IDE.

বাংলাদেশে গণিত অলিম্পিয়াডের সংস্কৃতি

বাংলাদেশে গণিত অলিম্পিয়াডের সংস্কৃতির সূচনার পর বেশকিছু ব্যাপার আমরা লক্ষ্য করছি।

আমরা লক্ষ্য করছি, অনেকগুলো ছেলেমেয়ে রাতদিন গণিত করে।

স্কুল কলেজে আমরা গণিত বলতে exercise করি – কিছু নির্দিষ্ট ধাপ বা কম্পিউটার বিজ্ঞানের ভাষায় অ্যালগরিদম মেনে চলি মাত্র। কিন্তু গণিত অলিম্পিয়াডের সমস্যাগুলো সমাধানে ধাপগুলো বা অ্যালগরিদম নিজেকে দাঁড় করাতে হয়। অন্যকথায়, গণিত সৃষ্টি করতে হয়।

একটা উদাহরণ দেই। দুটা সংখ্যাকে গুণ করতে আমরা না বুঝেই কিছু নির্দিষ্ট নিয়ম মেনে চলি – প্রথমে দুটি সংখ্যার সবচেয়ে ডানের অঙ্ক দুটিকে গুণ করি, তারপর হাতে রাখি ইত্যাদি। কিন্তু গণিত অলিম্পিয়াডে এই ধাপ বা নিয়মগুলো – কোন ধাপের পর কোন ধাপ হবে – তা নিজেকে চিন্তা করে বের করতে হয় – অর্থাৎ গণিত সৃষ্টি করতে হয়। আমরা বলি, স্কুল কলেজে তোমরা exercise কর, আর আমরা গণিত অলিম্পিয়াডে problem solve করি। কাজেই এখনও যারা Problem Solving কর না, আশা করি, তোমরাও দ্রুত আমাদের দলে যোগ দেবে!

একটা প্রজন্ম গড়ে উঠছে যাদের গড় IQ আগের প্রজন্ম গুলোর তুলনায় বেশি। অনেকগুলো ছেলেমেয়ে আগের চেয়ে ভালভাবে চিন্তা করতে পারে। আমাদের দেশের সবচেয়ে জনপ্রিয় দৈনিক এ নিজের ছবি দেখা; বিশ্ব প্রতিযোগিতায় নিজ দেশকে represent করা – অনেক বড় inspiration

আমরা লক্ষ্য করেছি, গণিত অলিম্পিয়াডের অনুষ্ঠানগুলোতে অনেক ভাল ভাল কথা হয়। আলোকিত মানুষ হওয়ার, দেশকে ভালবাসার অনুপ্রেরণা পায় ছেলেমেয়েরা। ছোট ছোট ছেলেমেয়েরা দেশের গুণী মানুষদের কাছ থেকে দেখার সুযোগ পায়, প্রশ্ন করতে পারে, কথা বলতে পারে এবং অটোগ্রাফও নিতে পারে!

দুটা চমৎকার ব্যাপারের একটা হল “গণিত শেখো, স্বপ্ন দেখো” – অনেকগুলো ছেলেমেয়ে নিজের জীবন নিয়ে দেশ নিয়ে বড় বড় স্বপ্ন দেখছে। আরেকটা হল একেবারে ক্লাস থ্রি – ফোরের ছেলেমেয়েরা ড. জাফর ইকবালের ভাষায় “পেন্সিল কামড়ে” অঙ্ক করতে আসে!

আমরা লক্ষ্য করেছি, বাংলা মাধ্যমের বেশ কিছু ছেলেমেয়ে বিশ্বের সেরা বিশ্ববিদ্যালয়গুলোতে আন্ডারগ্রাজুয়েট লেভেল এ পড়ার সুযোগ পেয়েছে। মুন পড়ছে Harvard University তে [1], নাজিয়া MIT তে [2] (নাহলে “MIghTy” শব্দটা এভাবে লেখা আমাদের শেখা হত না!), ইশফাক Stanford University তে [3], তানভির Caltech এ [4] (আমাদের শ্রদ্ধেয় ড. মুহম্মদ জাফর ইকবাল এই বিশ্ববিদ্যালয়ে Post-Doctoral Researcher হিসেবে কর্মরত ছিলেন) [5], সামিন Cambridge University তে [6]।
আগে গ্রাজুয়েট স্কুলগুলোতে আমরা এমএস বা পিএইচডি করতে যেতাম। ইংরেজি মাধ্যমের অবস্থাসম্পন্ন ছেলেমেয়েরা পড়তে পারত আন্ডারগ্রাজুয়েট লেভেলে। কিন্তু “বাংলা মাধ্যম” থেকে “স্কলারশিপ নিয়ে” “আন্ডার গ্রাজুয়েট” লেভেলে “বিশ্বের সেরা বিশ্ববিদ্যালয়গুলোতে” পড়তে যাওয়াটা নতুন!

“বাংলা মাধ্যম” থেকে “স্কলারশিপ নিয়ে” “আন্ডার গ্রাজুয়েট” লেভেলে “বিশ্বের সেরা বিশ্ববিদ্যালয়গুলোতে” পড়ার পথ দেখানোর কৃতিত্বের একক দাবিদার বাংলাদেশ গণিত দলের কোচ ড. মাহবুব মজুমদার  [7]; যিনি নিজে MIT থেকে Electrical Engineering এ আন্ডারগ্রাড, Stanford University থেকে Civil Engineering এ মাস্টার্স এবং Cambridge University থেকে Theoretical PhysicsPhD করে Imperial College এ [8] Post Doctoral করছিলেন। ২০০৫ সালে বাংলাদেশ গণিত অলিম্পিয়াডের সাথে সম্পৃক্ত হন এবং স্বপ্নের বাংলাদেশ গড়ে তোলার লক্ষ্যে দেশে থেকে যান। বিদেশী ও ইঞ্জিনিয়ারিং আন্ডারগ্রাড ডিগ্রি থাকা এবং আরও কিছু হাস্যকর কারণ দেখিয়ে তাকে Dhaka UniversityPhysics Department এ যোগ দিতে দেওয়া হয়নি। [9] তিনি স্বপ্ন দেখেন বাংলাদেশে একটা বিশ্বসেরা বিশ্ববিদ্যালয় এবং গবেষণাপ্রতিষ্ঠান গড়ে তোলার। তার মত ভাল মানুষ সচরাচর দেখা যায় না। আমরা তার পাশে থাকবো।

১৯০৫ এ আইনস্টাইনের “Miracle Year” [10] স্মরণে ২০০৫ সালের বাংলাদেশ জাতীয় গণিত অলিম্পিয়াডে আইনস্টাইন এবং পদার্থবিজ্ঞানের উপর একটা প্রশ্ন উত্তর পর্ব ছিল। সেখানে কিছু প্রশ্নের উত্তর দিয়েছিলাম। তাই গণিত ক্যাম্পে ড. মাহবুব মজুমদার আগ্রহের সাথে পদার্থবিজ্ঞান নিয়ে আলোচনা করতেন। মেক্সিকোতে যাওয়ার আগে প্রেস কনফারেন্সে দেখি তিনি স্ট্রিং থিউরি (String Theory) র [11]  একটা জটিল পেপার নিয়ে হাজির!
  
আরেকটা ব্যাপার লক্ষ্য করার মত। আন্তর্জাতিক গণিত অলিম্পিয়াডে আমাদের সাফল্যের মাত্রা দ্রুত বাড়ছে। আমাদের কিশোর – তরুণ গণিতবিদরা ২০০৬ সালে প্রথমবারের মত অনারেবাল মেনশান, ২০০৯ সালে প্রথমবারের মত ব্রোঞ্জ মেডাল, ২০১২ সালে প্রথমবারের মত সিলভার মেডাল জয় করে এনেছে। আমরা আশা করছি, এই ধারা অব্যাহত রেখে বাংলাদেশ গণিত দল ২০১৫ সালে আন্তর্জাতিক গণিত অলিম্পিয়াড থেকে গোল্ড মেডাল নিয়ে ফিরবে! গোল্ড মেডাল জয়ী সেই গণিতবিদ হতে পারো তুমিই!  

গণিত অলিম্পিয়াডের এই সংস্কৃতি সম্ভব হয়েছে কিছু তরুণ – তরুণীর স্বেচ্ছা কর্মোদ্যোগে। আমরা তাদের “মুভারস” (MOVERS – Math Olympiad Volunteers) বলে জানি। একটা শুভ উদ্যোগে দেশের তরুণ তরুণীদের উৎসাহী অংশগ্রহণ আমাদের প্রাণশক্তিতে ভরপুর তরুণ প্রজন্মকে সংজ্ঞায়িত করে।

নাগরিক শক্তি গণিত অলিম্পিয়াডের এই সংস্কৃতিকে দেশে আরও ব্যাপকভাবে ছড়িয়ে দেবে।

Healthcare Reform

The initiative to make sure more people get health-care insurance coverage – the Affordable Care Act, popularly known as the Obama-care, should be seen as positive. 


Ensuring health-care of citizens is a duty of the government. Who else will come forward to take the responsibility of ensuring healthcare of citizens who need it most?

But why do we have so little enrollment? One reason is surely the failure of the website to respond. With the best and brightest Software Engineers in our country, it can’t be a big problem.

Are there other reasons? It’s a new system all together. Is the website too difficult to navigate and use, especially to the people (who are economically lagging and thus technologically lagging) for whom the whole reform is targeted? Why don’t we educate people on how to use the site? We can extend the time of getting insured in the marketplace and let people spend more time to familiarize themselves with the new system.

We can make necessary changes so that those who want to keep their previous insurances can do so.

It’s a new system – we have to solve problems as we face them along the way.

In the meantime, we can come up with creative solutions that solve problems our health-care system face, solutions that minimize costs but are still more effective and reach more people. American Healthcare System is under-performing and a lot more expensive compared to other developed countries.

Why not prioritize disease prevention programs and motivational programs that encourage healthy lifestyle choices so that we have less sick people to start with? Obesity and poor lifestyle choices are surely among causes of diseases. Less sick people translates into less visits to doctors and less visits to doctors translate into less health insurance prices. If health insurance prices go down, more people would have health insurances.

Let’s try to understand it through an example.

Suppose, visits to doctor cost a person say, “A” $100 a month and another person say, “B” $50 a month. Now, the insurance company that sells healthcare insurances to “A” and “B”, sells each of the monthly insurances for $80. So, “A” and “B” together spend $80 + $80 = $160 to buy their insurances and the insurance company has to spend $100 + $50 = $150 to bear the cost of visits to doctor. Now, suppose “A” and “B” learned to take better care of their health. As a result, they have monthly healthcare costs of $60 and $40 respectively. Now, the insurance company has no way but to reduce their insurance fees. Thus, by taking better care of their health, “A” and “B” forced the insurance companies to reduce prices for insurance.


There can be so many different solutions to different inter-related problems if only we are wise enough to “define” the “problem” we want to solve creatively!

Evolution of My Dreams and Realizations

My first ‘Aim in life’, as far as I can remember (It was 1988 / 89; I was 2 or 3), was to become a milkman. I mean, it wasn’t about being a milkman. I wanted to become the honest person appreciated by my parents – a milkman by the name Mubarak. (“I want to become Mubarak”, I used to say). So, what I truly wanted to become was a plain, simple, honest person. 

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Next, I wanted to become a building mechanic. I used to stare at people who built houses in awe. My uncle sent me a toy Mechanical Tool Box.

My next major change in aim occurred when I wanted to join the Military (age: 4-5). Each night, I used to stay awake until the National Anthem with the National Flag was played on BTV and give salute. I watched a Television program depicting Military life. One of my uncles quipped: “The secret: Tahsin wants to become the President!”.

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My mom told me of an incident that took place when I was a baby of few months old (1986). One day, General Ershad was delivering a speech (who was then the President). My mom was studying for her exams. I was lying right beside my maternal Grandfather. My Grandfather suddenly started praying loudly: “God, grant my wish and guide my grandson to become the President and lead the Nation.” My Grandmother called my mom, “Come! Quick! Look how your dad is praying for your son!”     

During my First grade, a serial had an enormous influence on me: “The sword of Tipu Sultan”. Tipu Sultan and Hyder Ali were my childhood heroes. The serial drew me to History. I was deeply influenced by another historical novel during 3rd / 4th Grade – “Khun Ranga Path”. Besides History, books on General Knowledge were among my favorites from an early age. My father bought me my first “General Knowledge” book (Encyclopedia) around 5. Then I discovered “General Knowledge” books (Encyclopedia) in my aunt’s house. Later, I started buying Encyclopedia myself. I used to stare at the Globe of the world and fantasize (
Grade 3 / 4). I fantasized first becoming a King of Ancient Bengal, then King of Myanmar (Burma) and later lifetime President of Kazakhstan. 

I remember playing computer games at one of our relative’s house during Fifth grade. Almost everyone around me wanted to become a Computer Engineer at that time. So I thought I should try to become one myself – a Computer Engineer. 

During my middle school years, I was a voracious reader of novels. Reading novels was the most fun activity I could think of. I could understand different writing techniques employed by novelists. Becoming a novelist, writing great novels was my dream during 7th to 10th grade (1999 – 2002). For living, I would become a Physician or Engineer or Architect. That was my plan.

During 9th / 10th grade, I made up my mind to study Medicine (there was huge encouragement from my parents) and become a Physician besides writing novels.

When I read a book on Psychology (my mom’s book on Educational Psychology from her M.Ed. course), I understood that an intense interest in the workings of the human mind was the chief reason I wanted to become a novelist. Moreover, Literature could only depict subjective human experience, but the objective theories of Psychology applied to all humans.

I thought that I could become a Physician and specialize in Psychiatry or Neurology.

Studying Psychology helped me understand the essence of Science: To understand experimentally provable General Rules that govern everything we see around us.

Studying Psychology books gave me the confidence that: I can come up with original ideas, and that I should question what is written in books.

Trying to understand the theories of Psychology in terms of my own experiences and what I see around me, made me aware of the connection between Real World and the world of Books and Theories.

As I later diversified and ventured into different branches of Science, these realizations and understandings proved invaluable.

One day, as I was preparing for my high school (11th grade) Entrance Exam (later it was decided that Entrance would be based on results of matriculation exam), a Chapter on different forms of Energy from my Physics book grabbed my attention. I thought: maybe I could work on both Psychology / Neurology and Physics. I went through my 9-10th grade Physics book. I bought and read other books (Undergraduate level Physics Textbooks, Stephen Hawkin’s A Brief History of Time and others).

I thought and wrote down my understandings and realizations. I tried to come up with new Theories myself.

Physics taught me to understand “everything” in terms of fundamental constituents and few fundamental laws that govern things we see around us.

Physics made me realize the necessity of learning Higher Mathematics.

Mathematical Olympiad was gaining popularity in Bangladesh at that time (it was 2003). I bought Books and started solving problems.

One of the books published at that time was “নিউরনে অনুরণন” (“Resonance in neurons”). The idea for the name: it’s better to create resonance in your brains’ neurons by solving Mathematical problems rather than leaving the neurons idle!

I found out: the more I worked on problems, the better I could think! My Neurons really were resonating!

My interest in Psychology helped me appreciate brain function improvement and Mathematical Problem Solving. I discovered ways of improving brain function myself.

It was an amazing realization – I could become anyone I wanted if I worked in the right way.

Other Sciences started grabbing my attention.

Psychology drew me to Neuroscience – the Biology of what happens in the mind. Physics led me to Cosmology (the study of the evolution of the Universe) and some of the books described evolution of our planet and Biological evolution. Evolutionary Biology was among my favorites.

At that point, I saw my future as a Scientist: trying to understand the truth and decode the Laws of Nature.

I became interested in Computer Science and Engineering as I read an article portraying the field of Artificial Intelligence. The article was written by Dr. Ali Asgar included in one of his popular science books (Grade 11). I bought Undergrad Texts on Artificial Intelligence and started reading.

Psychology and Neuroscience always grabbed my attention. So when I found out that there is a subfield in CS that tries to emulate intelligence on computers, I got hooked instantly. 

Later, I participated in International Mathematical Olympiad, and met people who were serious participants in programming contests and I felt that I really liked contests and competitions. Besides, computation seem to be everywhere – required in almost every branch. I could do Physics and Biology on Computers. I read an inspirational book (“Medhabi Manusher Golpo” – Prof. Dr Kaykobad) which depicted lives of eminent Computer Scientists and students of Computer Science. The choice was either Physics or Computer Science and Engineering, but my parents wouldn’t let me study Physics. Choosing Computer Science and Engineering also made sense when I considered practical aspects. I thought: I could still pursue my multi-disciplinary interests besides studying CSE at college. 

The Majors I considered at that time included: Computer Science and Engineering, Physics, Mathematics, Neuroscience, Nanotechnology / Nanoengineering & Bioengineering / Biomedical Engineering.

[If you find my life and my understandings interesting you might like Looking back and connecting the dots.]

Lets move a few years forward … During March / April 2013, I thought, I should analyze and understand and learn from and codify everything I see around me – just as I did with the sciences and engineering. I started with the political situation in Bangladesh. I wanted to figure out what would happen if I start my own Political Party. Next, I applied my analysis to other domains: Mechanical Engineering, Economics, Computer Science.

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I come across new understandings and realizations almost on a daily basis. I look forward to share my newer understandings at sometime in not too distant future: “Living to tell the tale”, truly!
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