Advice for How to Attack Problems

[Heisenberg7]

Hello,

I was lying in bed the other day and an idea came to my mind. How does one actually solve a problem? Now, I know this question is impossible to answer, but I would like to know your strategies when it comes to solving problems.

I would like to make an analogy to this. Let's say that you are a general of an army and your job is to conquer a country. How would you do it? How would an experienced general approach this?

Now, I know that this is probably a funny analogy because none of us are army generals, but the point stands nonetheless. The general would most definitely rely on his experience. This is the same tactic grandmasters use in chess. Magnus Carlsen himself said that most of the time he knows what to play next. He basically doesn't have to think about it. But, what happens when he does? So let's say we give him a board and we tell him to find the best move. How would he do it? I mean sure, his experience would play a huge role. But, what happens when he gets stuck? What happens when he actually has to think about it?

An army general can't just move troops on the field without thinking. He needs a plan that is going to lead him to the victory. At this point, I can see that I am going in a circle, so let's get to the punchline. What problem solving strategies would he use? In other words, what problem solving strategies would a mathematician use?

Mathematicians aren't born, they have to train. But, do the gifted mathematicians have an unfair advantage over the rest of us? Over this summer break, I've probably done more than 500 problems in physics. I'm not sure how much they helped. But they had to have done something, right?

I had this idea in my mind for a while so I'm hoping someone could give me a relaxation. Just to simply get it off my mind.

Thanks in advance

 

[Heisenberg7]

Also, a good book would help. I've realized that I'm not here for the mathematical rigor, but to improve my math skills and get better at problem solving. The sooner one realizes this, the easier their life becomes.

 

[Vanadium 50]

I know this question is impossible to answer

Then why pester people - people whose help you need - for an answer?

You want to solve problems better? Then start solving problems. Don't look for the magic shortcut. Don't look for the perfect book. Don't come running here the instant things get a little sticky. Solve problems.

 

[berkeman]

How does one actually solve a problem?

That is a very general question that does not have one answer. It depends on what type of problem and if it is obvious that it has a solution versus if it could be an unsolvable problem.

Let's say that you are a general of an army and your job is to conquer a country. How would you do it?

That general would have probably spent multiple years in a military training academy (like the US' West Point), and have multiple years of combat experience. Please read through this Wikipedia article for more details about this: https://en.wikipedia.org/wiki/List_of_military_tactics

Magnus Carlsen himself said that most of the time he knows what to play next. He basically doesn't have to think about it. But, what happens when he does? So let's say we give him a board and we tell him to find the best move. How would he do it?

Do you play chess? If so, what resources have you used to learn it and get better at it? https://www.amazon.com/Winning-Chess-Three-Moves-Ahead/dp/0671211145/?tag=pfamazon01-20

But, do the gifted mathematicians have an unfair advantage over the rest of us?

Maybe @fresh_42 can offer some insights...

Over this summer break, I've probably done more than 500 problems in physics.

So, an important emphasis when doing practice problems is not to do a bunch of simpler/similar problems, and rather to do the hardest problems that you can attempt. In many textbooks at the end of each chapter you will find a number of problems, and the hardest problems are generally listed at the end. When I was studying, I would do a few of the first problems at the end of the chapter, and if those went okay I would skip ahead and skim the rest until I got to a different/harder type of problem. There is no reason to do 20 similar problems, IMO; it is much more important to do different kinds of problems, and to get to the harder and more challenging ones quicker. They will likely take some time and re-reading of some parts of that chapter, but that is where the quality learning takes place.

Anyway, in case you were asking a more general question about how to solve problems, I can say from my many years in Engineering that it involves immersing yourself in the problem, and looking at many ways of trying to solve it. I mentioned above that sometimes in the real world we do not know ahead of time if the problem is solvable at all. It could just be hard and solvable with enough work and enough insight (and "aha" moments), or it could be that there is no solution at all (at that price point, or with current technology, etc.).

For hard problems that are solvable with enough work and insight, you may find that it is helpful to step back now and then to "rest" from trying to solve the problem. This employs some brain physiology characteristics, where thinking about other things and then coming back to the problem helps to reset the brain's tracking on the problem, and put you back on a clean path toward a solution. Consider when you sometimes have trouble remembering a person's name, or the name of some other thing, but if you stop thinking about it for a few minutes and come back to it, you get right to what you were trying to remember.

My patents that I'm most proud of came after many months or years of work, where I did not know if the problem was solvable or not, and I kept coming back to the problem with fresh approaches and calculations. That is how you solve hard problems.

 

[Vanadium 50]

500 problems

Do you think this is a lot? It's not.

 

[Heisenberg7]

Then why pester people - people whose help you need - for an answer?

You want to solve problems better? Then start solving problems. Don't look for the magic shortcut. Don't look for the perfect book. Don't come running here the instant things get a little sticky. Solve problems.

I was expecting someone to say this and honestly, it is a dumb question. I just don't feel like I'm doing any better. I feel like I'm stuck. When I am solving a problem, it often gets really really messy and all the ideas just get lost. I guess that's one skill I should work on. I can't say that I'm looking for a shortcut either. I am literally giving up everything I have to get better at my craft, but the progress seems to be stagnating. I guess one must really really give up everything to achieve greatness.

 

[Heisenberg7]

That is a very general question that does not have one answer. It depends on what type of problem and if it is obvious that it has a solution versus if it could be an unsolvable problem.


That general would have probably spent multiple years in a military training academy (like the US' West Point), and have multiple years of combat experience. Please read through this Wikipedia article for more details about this: https://en.wikipedia.org/wiki/List_of_military_tactics


Do you play chess? If so, what resources have you used to learn it and get better at it? https://www.amazon.com/Winning-Chess-Three-Moves-Ahead/dp/0671211145/?tag=pfamazon01-20


Maybe @fresh_42 can offer some insights...


So, an important emphasis when doing practice problems is not to do a bunch of simpler/similar problems, and rather to do the hardest problems that you can attempt. In many textbooks at the end of each chapter you will find a number of problems, and the hardest problems are generally listed at the end. When I was studying, I would do a few of the first problems at the end of the chapter, and if those went okay I would skip ahead and skim the rest until I got to a different/harder type of problem. There is no reason to do 20 similar problems, IMO; it is much more important to do different kinds of problems, and to get to the harder and more challenging ones quicker. They will likely take some time and re-reading of some parts of that chapter, but that is where the quality learning takes place.

Anyway, in case you were asking a more general question about how to solve problems, I can say from my many years in Engineering that it involves immersing yourself in the problem, and looking at many ways of trying to solve it. I mentioned above that sometimes in the real world we do not know ahead of time if the problem is solvable at all. It could just be hard and solvable with enough work and enough insight (and "aha" moments), or it could be that there is no solution at all (at that price point, or with current technology, etc.).

For hard problems that are solvable with enough work and insight, you may find that it is helpful to step back now and then to "rest" from trying to solve the problem. This employs some brain physiology characteristics, where thinking about other things and then coming back to the problem helps to reset the brain's tracking on the problem, and put you back on a clean path toward a solution. Consider when you sometimes have trouble remembering a person's name, or the name of some other thing, but if you stop thinking about it for a few minutes and come back to it, you get right to what you were trying to remember.

My patents that I'm most proud of came after many months or years of work, where I did not know if the problem was solvable or not, and I kept coming back to the problem with fresh approaches and calculations. That is how you solve hard problems.

Thank you for the reply. I appreciate it. The analogies I made about chess players and army generals were just there to explain what I am really trying to say. Anyway, I guess there is only one way to go through this. What this really comes down to is just work I guess. There is no other way to get around this. There is no magic formula. One must invest the hours that Magnus invested to become Magnus. I'm sorry for taking your time to write this reply, I will be more cautious next time. Thanks again.

 

[Heisenberg7]

Do you think this is a lot? It's not.

I am very well aware of that.

 

[Vanadium 50]

I am very well aware of that.

Then don't you have your answer? If you want to get better at solving problems, solve more problems.

There is no royal road to mathematics.

 

[berkeman]

What this really comes down to is just work I guess. There is no other way to get around this.

Yes, work hard, but work smart, modulo my tips in my reply above. Don't work 500 of the same type problems. Don't spend hours all at a time working on the same problem. Comprendes?

 

[fresh_42]

Hello,

I was lying in bed the other day and an idea came to my mind. How does one actually solve a problem?

How many different answers are you prepared to read? Everybody probably has their own strategies.

I once tried to summarize some principles in problem-solving:
https://www.physicsforums.com/insights/how-most-proofs-are-structured-and-how-to-write-them/
... but don't take it too seriously. It's more of a guideline. And, of course, there is still the book of proof
https://www.people.vcu.edu/~rhammack/BookOfProof/

Now, I know this question is impossible to answer, but I would like to know your strategies when it comes to solving problems.

I would like to make an analogy to this. Let's say that you are a general of an army and your job is to conquer a country. How would you do it? How would an experienced general approach this?

The same way you should solve any other problem: firstly, gather all - and I mean all, even the hidden -conditions and given facts. And do not by any means make any additional assumptions. A function is not automatically real, the characteristic of a field is not automatically zero, and a vector space isn't automatically finite-dimensional. If you have to make such assumptions, you need to invoke cases.

Now, I know that this is probably a funny analogy because none of us are army generals, but the point stands nonetheless. The general would most definitely rely on his experience.

First mistake. It is an artificial restriction. Sure, you cannot use tools that are beyond your horizon, but you will possibly have to expand your horizon. I have recently seen documentation about how it was possible for the Germans to conquer France so fast during WWII. It was the experiences of WWI of the French generals and some hidden assumptions about the fortresses at the German border that led to a completely false assessment of the situation.

"No plan survives the first enemy contact." (Moltke, 1890)

This is the same tactic grandmasters use in chess. Magnus Carlsen himself said that most of the time he knows what to play next. He basically doesn't have to think about it. But, what happens when he does? So let's say we give him a board and we tell him to find the best move. How would he do it? I mean sure, his experience would play a huge role. But, what happens when he gets stuck? What happens when he actually has to think about it?

I also heard, not sure if it was Carlsen or someone else, that the art of chess is to decide what to do if there is no tactically obvious move. The strategy is the key, not the tactics.

An army general can't just move troops on the field without thinking. He needs a plan that is going to lead him to the victory. At this point, I can see that I am going in a circle, so let's get to the punchline. What problem solving strategies would he use? In other words, what problem solving strategies would a mathematician use?

I stumbled upon the following problem: Which is the least odd number $m$ such that
$$
n^2=(m+164)(m^2+164^2)
$$
My

Spoiler: Solution

Since it is required that $m$ is odd and positive, we may set $m=2k+1$ and
\begin{align*}
n^2&=(m+164)(m^2+164^2)=(2k+165)(4k^2+4k+164^2+1)\\
&=8k^3+(8+660)k^2+(2\cdot 164^2+660+2)k+(165+165\cdot 164^2)\\
&=8k^3+668k^2+54454k+4438005\\
&=2^3\cdot k^3+2^2\cdot 167\cdot k^2+2\cdot 19 \cdot 1433 \cdot k+ 3\cdot 5 \cdot 11 \cdot 13 \cdot 2069
\end{align*}
which is odd, hence $n^2=(2n_1+1)^2=4n_1^2+4n_1+1.$ Since $n^2\equiv 1\pmod{4}$ we get $n^2\equiv 8k^3+668k^2+54454k+4438005\equiv 2k+1\equiv 1\pmod{4}$ hence $k\equiv 0\pmod{2}$ and we can modify our equation to $k=2k_1$ and
\begin{align*}
n^2&=(m+164)(m^2+164^2)=(4k_1+165)(16k_1^2+8k_1+164^2+1)\\
&=64k_1^3+(32+2640)k_1^2+(4\cdot 164^2+1320+4)k_1+(165+165\cdot 164^2)\\
&=64k_1^3+2672k_1^2+108908k_1+4438005\\
&=2^6\cdot k_1^3+2^4\cdot 167\cdot k_1^2+2^2\cdot 19 \cdot 1433 \cdot k_1+ 3\cdot 5 \cdot 11 \cdot 13 \cdot 2069
\end{align*}
$$
n^2\equiv 4k_1+5 \equiv
\begin{cases}
1\pmod{8}&\text{ if }k_1\equiv 1\pmod{2}\\
5\pmod{8}&\text{ if }k_1\equiv 0\pmod{2}
\end{cases}
$$
\begin{align*}
n^2&\equiv (m-1)(m^2+1)\equiv m^3+m-m^2-1\equiv m^3+2m^2+m+2 \pmod{3}\\
n^2&\in \{0,1\} \pmod{3}\\
&\Longrightarrow m\equiv 1\pmod{3} \wedge n\equiv 0\pmod{3}\\
&n\equiv 1\pmod{4}\wedge n\equiv 0\pmod{3}\Longrightarrow n\equiv 9\pmod{12}\\
&\Longrightarrow n\not\equiv 5\pmod{8}\\
&\Longrightarrow n^2\equiv 1\pmod{8}\wedge k_1=2k_2+1\wedge k=4k_2+2\\
&\Longrightarrow n=24p+a\text{ with }a\in \{9,21\}\\
&\Longrightarrow n^2\equiv 9 \pmod{24}
\end{align*}
\begin{align*}
n^2&=64(2k_2+1)^3+2672(2k_2+1)^2+108908(2k_2+1)+4438005\\
&=512 k_2^3 + 11456 k_2^2 + 228888 k_2 + 4549649\\
&=2^9k_2+2^6\cdot 179k^2+2^3\cdot 3^2\cdot 11\cdot 17^2 k_2+ 13^2\cdot 26921
\end{align*}
\begin{align*}
n^2&\equiv 9\equiv 8k_2^3+8k_2^2+17\pmod{24}\\
k_2&\equiv 1\pmod{3}\wedge k_2=3k_3+1 \wedge k=12k_3+6
\end{align*}
\begin{align*}
n^2&=(m+164)(m^2+164^2)=(2k+165)((2k+1)^2+164^2)\\
&=8k^3+668k^2+54454k+4438005\\
&=8(12k_3+6)^3+668(12k_3+6)^2+54454(12k_3+6)+4438005\\
&=13824 k_3^3 + 116928 k_3^2 + 760008 k_3 + 4790505\\
&=2^9\cdot 3^3\cdot k_3^3+2^6\cdot 3^2\cdot 7\cdot 29\cdot k_3^2+ 2^3 \cdot 3\cdot 31667 \cdot k_3 + 3\cdot 5\cdot 59\cdot 5413\\
\end{align*}
\begin{align*}
0&\equiv 3k_3+3\pmod{9}\Longrightarrow k_3=3k_4+2\wedge k=36k_4+30\\
n^2&=13824 (3k_4+2)^3 + 116928 (3k_4+2)^2 + 760008 (3k_4+2) + 4790505\\
&=373248 k_4^3 + 1798848 k_4^2 + 4180824 k_4 + 6888825\\
&=2^9\cdot 3^6 \cdot k_4^3+2^6\cdot 3^4\cdot 347\cdot k_4^2+2^3\cdot 3^2\cdot 58067 \cdot k_4+3^2\cdot 5^2\cdot 17\cdot 1801\\
n^2&\equiv 72k_4+9\pmod{144}\\
&\equiv (24p+a)^2\equiv 24^2p^2+48pa+a^2\equiv 48pa+a^2\text{ for }a\in \{9,21\}\\
&\Longrightarrow n^2\in\{432p+81,1008p+441\}\equiv \{9\}\pmod{144}\\
&\Longrightarrow k_4=2k_5\wedge k=72k_5+30\\
n^2&=2985984 k_5^3 + 7195392 k_5^2 + 8361648 k_5 + 6888825\\
&=2^{12}\cdot 3^6\cdot k_5^3+2^8\cdot 3^4\cdot 347\cdot k_5^2+2^4\cdot 3^2\cdot 58067\cdot k_5+ 3^2\cdot 5^2\cdot 17\cdot 1801
\end{align*}
For $k_5=0$ we get $n=3^2\cdot 5^2\cdot 17\cdot 1801$ which is no square.
For $k_5=1$ we have
\begin{align*}
\dfrac{n^2}{9}&=2^{12}\cdot 3^4+2^8\cdot 3^2\cdot 347+2^4\cdot 58067 +5^2\cdot 17\cdot 1801=2825761=41^4\\
n&=3\cdot 41^2=3\cdot 1681 =5043
\end{align*}
\begin{align*}
k_5&=1\, , \,n=5043\, , \,k=102\, , \,m=205\\
n^2&=(3\cdot 41^2)^2=3^2\cdot 41^4\\
\end{align*}

Of course, there were at least as many lines that I rejected because they led nowhere. Yet, my answer is not satisfactory. It was a brute-force attempt. It finally led to the correct answer, but it wasn't nice.

If we assume that $41\,|\,m$ then it can be solved in a few lines instead:

Spoiler

$n^2=(m+164)(m^2+164^2)=m^3+164m^2+164^2m+164^3.$ Assume $m=41q.$ Then
\begin{align*}
n^2&=41^3q^3+4\cdot 41^3q^2+16\cdot 41^3q+64\cdot 41^3\\
&=41^3(q^3+4q^2+16q+64)\\
n^2/41^2&=41(q^3+4q^2+16q+64)\\
41\,&|\,q^3+4q^2+16q+64=(q + 4) (q^2 + 16)\\
\end{align*}
If $41\,|\,q^2+16$ then the least solution is $q=5.$ Thus
\begin{align*}
\dfrac{n^2}{41^2}&=41^2\cdot 9\\
n&=3^2\cdot 41^4 \text{ and }m=205
\end{align*}

It remains to show that $41\,|\,m$ or equivalently that $41\,|\,n.$ However, I didn't manage to prove this yet, and the only advice someone on SE gave was to check all $20$ cases if $41\,\nmid\,m.$ That isn't much better than my first solution so I'm still trying to prove why $41\,|\,m$ must be the case. However, this is a luxury problem since it only arise if I want to find a more elegant solution.

We have many problems here
https://www.physicsforums.com/forums/general-math.73/
https://www.physicsforums.com/forums/math-proof-training-and-practice.296/
and pdf where many problems are gathered with solutions here
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
The pdf that contains many of these in one document has 533 pages. It contains all kinds of problems and most of them provide formulas, inequalities, theorems etc. that would be helpful anyway. So it's more than just problems, you can also learn stuff.

If you want to improve your skills, then there is no other way than ...

If you want to get better at solving problems, solve more problems.

There is no royal road to mathematics.

Those 533 pages are a good starting point and the solutions help you to proceed if you are stuck.

Mathematicians aren't born, they have to train. But, do the gifted mathematicians have an unfair advantage over the rest of us?

Sure, that's what gifted means. But you shouldn't compare yourself with the Terrys in the world. However, he has an interesting blog that allows you insights on what the gifted do:
https://terrytao.wordpress.com/

Over this summer break, I've probably done more than 500 problems in physics. I'm not sure how much they helped. But they had to have done something, right?

Right. Here is an interview that also touches the question of time management:
https://www.ams.org/publications/journals/notices/201707/rnoti-p718.pdf

This answer is biased toward math problems, but much of it holds for physics, too. And if you want to find problems, search for "<enter course name here> + exams (+solutions) + pdf."

Edit: The elegant solution for the riddle above for the sake of completion, if anyone got attracted to solve it, and the unlikely event, that the one who posted this on SE and whose post had been closed, finds his way to us where he can get his answer. It is also a real-life example of how problem-solving works. It sometimes takes more than one attempt and time.

Spoiler

What is the least odd, positive integer $m$ such that
$$
n^2=(m+164)(m^2+164^2)=m^3+164\cdot m^2+ 164^2 \cdot m+ 164^3
$$
for some integer $n$?

If $m+164=k^2$ for integer $k$ then
$$
\dfrac{n^2}{k^2}=m^2+164^2 \in \mathbb{Z}
$$
is a Pythagorean triple and we find integers $a,b$ such that
$\dfrac{n}{k}=a^2+b^2\, , \,2ab=164\, , \,m=a^2-b^2$ because $m$ is odd. Thus $ab=41\cdot 2=82$ and $m=1677.$ However, $(1677+164)(1677^2+164^2)=5227013225$ which is divisible by $7$ but not by $49$ and thus no square. Hence there is a prime factor $p>2$ of $m+164$ such that $p^s\,|\,(m+164)\, , \,p^{s+1}\,\nmid\,(m+164)\, , \,s\geq 1$ and $s\equiv 1\pmod{2},$ say $s=2t+1$ for $t\geq 0.$ Set $m+164=p^s\cdot a_0=p^{2t+1}\cdot a_0\,|\,n^2.$ Since $n^2$ is a square, we even have $p^{2t+2}\,|\,n^2.$
\begin{align*}
m^2+164^2&=(m+164)^2-2\cdot164 \cdot m\\
&=(m+164)^2- 2\cdot 164 \cdot (m+164) +2\cdot 164^2\\
n^2&=(m+164)\cdot (m^2+164^2)\\
&=(m+164)^3- 2\cdot 164 \cdot (m+164)^2 +2\cdot 164^2\cdot (m+164)\\
&=(p^{2t+1}\cdot a_0)^3- 2\cdot 164 \cdot (p^{2t+1}\cdot a_0)^2 +2\cdot 164^2\cdot (p^{2t+1}\cdot a_0)\\[6pt]
\dfrac{n^2}{p^{2t+2}}&=p^{4t+1}\cdot a_0^3 -2\cdot 164 \cdot p^{2t}\cdot a_0^2+ \dfrac{2\cdot 164^2\cdot a_0}{p}\in \mathbb{Z}
\end{align*}
If $p\,|\,a_0$ then $m+164=p^s\cdot a_0=p^{s+1}\cdot a_1$ for some integer $a_1$ contradicting our assumption about $p.$ Thus $p\,|\,(2\cdot 164^2)$ and since $p>2,$ we have $p=41$ and $m=41\cdot q$ for some integer $q.$
\begin{align*}
n^2&=41^3q^3+4\cdot 41^3q^2+16\cdot 41^3q+64\cdot 41^3\\
&=41^3(q^3+4q^2+16q+64)\\
n^2/41^2&=41(q^3+4q^2+16q+64)\\
41\,&|\,q^3+4q^2+16q+64=(q + 4) (q^2 + 16)\\
\end{align*}
If $41\,|\,q^2+16$ then the least solution is $q=5.$ Thus
\begin{align*}
\dfrac{n^2}{41^2}&=41^2\cdot 9\\
n&=3^2\cdot 41^4 \text{ and }m=205
\end{align*}

 

[Hornbein]

Polya's How To Solve It.

 

[gleem]

Four general principles have been identified by Alan Schoenfeld in his book "Mathematical Problem Solving" which one could use for any problem.

1. Know the subject well.
2. Learn the heuristics relative to the problem area.
3. Learn how to manage your attempts to solve those problems so as not to waste time with unfruitful approaches
4 Believe that you have the ability to find a solution.

 

[Vanadium 50]

Polya's How To Solve It.

Not recommended.

It's an excellent book, but it is not directed to physics.
We have identified the problem - not working enough problems. There is no substitute.
The magic wand - "do this and you won't have to work so many problems" doesn't exist, and we do the OP a disservice by suggesting there is.

 

[MidgetDwarf]

Read textbook. Work the examples. Attack problems. Struggle. Identify what you think you are missing. Read textbook again and repeat.

You also throw everythijg in the kitchen sink a few times.

That is all

 

[CrysPhys]

Mathematicians aren't born, they have to train. But, do the gifted mathematicians have an unfair advantage over the rest of us?

Why do "gifted" mathematicians have an "unfair" advantage over those who are not? I know it's not politically correct to say so, but not all individuals are the same. Some have specific mental or physical characteristics that give them specific innate advantages with respect to specific mental or physical activities. But just as athletes with specific innate advantages need to train to fulfill their potential, so do mathematicians, physicists, doctors, artists, singers, poets, ... with specific innate advantages also need to train to fulfill their potential. The type and degree of training will vary with the individual, of course. If you wish to stray into a discussion of fairness, it would not revolve around an individual's innate advantages (or lack thereof), but around an individual's opportunities to access the requisite training (or lack thereof).

 

[symbolipoint]

Why do "gifted" mathematicians have an "unfair" advantage over those who are not? I know it's not politically correct to say so, but not all individuals are the same. Some have specific mental or physical characteristics that give them specific innate advantages with respect to specific mental or physical activities. But just as athletes with specific innate advantages need to train to fulfill their potential, so do mathematicians, physicists, doctors, artists, singers, poets, ... with specific innate advantages also need to train to fulfill their potential. The type and degree of training will vary with the individual, of course. If you wish to stray into a discussion of fairness, it would not revolve around an individual's innate advantages (or lack thereof), but around an individual's opportunities to access the requisite training (or lack thereof).

In simpler wording, one needs interest and effort.

 

[CrysPhys]

And ability.

 

[symbolipoint]

I was lying in bed the other day and an idea came to my mind. How does one actually solve a problem? Now, I know this question is impossible to answer, but I would like to know your strategies when it comes to solving problems. .....and then all the rest....

Some courses or classes teach how to "attack" exercise problems. Introductory fundamental physics/mechanics is often one of those kinds of courses. Read the problem description; interpret enough to draw a figure, or diagram to represent the situation; identify formulas (arithmetic, algebraic, including any trigonometry) which may be needed; identify what number needs to be solved for; assign varialbes to every number from or related to the description. Now, write the necessary equations for the description.

Next, solve the problem, or solve for the unknown number but all in variables. Last, substitute the given information values from the description and compute the resulting value for the unknown/s variable.

In GOOD instruction of something like that, the professor will take the students through a few example problems to demonstrate the problem solving process from beginning to end.

 

[hutchphd]

One of the nice things about physics problems is that they come in all shapes and sizes. I would suggest that you find some problems/ questions and choose some that you would really like to know the answer to. Pick one up like a shell on the beach and look at it from all angles. Is it like anything you have seen before? Do this with a few different problems.......

Remember them as you do some research (books or online texts: whatever works best for you). Using similar problems as a starting point formulate an approach. Start with the easiest problem (for you) and ask questions as necessary. If you ask them here be certain that you are complete and clear in you description and ask specific questions (LateX, etc.).

There is no Royal Road but there are Samaritans along the route. Use them only as a last resort.....Samaritans can be testy to the lazy

 

[billtodd]

In simpler wording, one needs interest and effort.

And ability.

And time...

 

[CrysPhys]

And access to resources.
And ....

I initially gave a nuanced response in Reply #16, which included the overarching statement:

The type and degree of training will vary with the individual, of course.

In Reply #17, @symbolipoint simplified the wording of my overall response, but, in so doing, dropped the nuances. Now we have replies adding back in the nuances.

 

[gleem]

Anders Ericsson a Swedish psychologist determined that it usually takes much practice over 10,000^* hours under the supervision of mentors or instructors to become proficient at a given skill. While practice is necessary in general, it is not sufficient, and guidance by teachers or mentors is needed to achieve a goal efficiently and to judge the student's capability. How well do you know how good you are unless someone can challenge and evaluate your ability to provide feedback to give you the confidence to proceed on your own and achieve the success that you are capable of?


* Malcolm Gladwell in his book "Outliers: Stories of Success" popularized the 10,0000 Hour Rule by Ericsson but neglected to note the need for a good teacher.

 

[Vanadium 50]

I don't like the "10,000 hours" rule. How much does it take to earn a BS? 10,000 hours, A MS? 10,000 hours. A PhD? 10,000 hours. This is a gross simplification.

A better estimate is a year in college is, say 35 weeks. A 3 hour course is 100 hours of instruction so probably 300-400 hours total. If an average problem takes 10-15 minutes, that's 1500-2000 problems. The OP is doing this with physics and calculus simultaneously, so double it: he needs to work 3000-4000 problems.

He's unhappy that after 500 he is not Olympiad material (see his posting history). Of course he's not - he's doing 10-15% of what an average student needs, and he wonders why he's not a superstar.

As I said before, there is no royal road to mathematics.

 

[gleem]

The 10000-hour rule is about practice. Don't forget elementary school where you set down foundations for fundamental skills needed but often overlooked like practicing scales and cord progress in music.

School years are about 250 days. Credit 1000 hours in eight years. In HS double the effort in half the time credit 2000 hours since you take twice the number of problem-solving courses.. In college credit 4000 hours for more difficult problems and even more subjects. In graduate school, credit 3000 hours for two years of coursework in even more difficult subjects.

Fifteen minutes per problem in college and grad school, really? For the best 10,000 hours may be high but for most not so. Throw in class attendance which provides some intellectual activity relative to problem-solving and you are way, way over the ten thousand hours of intellectual activity relevant to learning and developing an expertise in physics.

Yes, the OP has a long way to go at his rate, and who is advising him on his progress? Is he making progress?