Problem-solving improvement

[Jakelaker]

One of the most important skills required to become a successful quant (or anything, in general), is (mathematical) problem-solving.

What strategy do/would you use to improve your problem-solving in both the short- and long-run?

Really interested to read what everyone has to provide!

 

[Daniel Duffy]

Georg Polya is the man

How to Solve It - Wikipedia, the free encyclopedia

And hard work.

 

[bigbadwolf]

Polya is passé -- he was a contemporary of Fred Flintstone. If you must read Polya, go for his 2-volume "Mathematics and Plausible Reasoning."

Try Engel's "Problem-Solving Strategies."

 

[Daniel Duffy]

Polya is passé -- he was a contemporary of Fred Flintstone. If you must read Polya, go for his 2-volume "Mathematics and Plausible Reasoning."

Try Engel's "Problem-Solving Strategies."

Well, that's your opinion. You are becoming emotive. Can you be a bit more precise? Take one of his topics, tell why it is stone-age and we can then talk about it.
His ideas encompass many thought processes needed in not just maths but also software development.

I bought Plausible based on your recommendation some time ago. It's really outdated.

 

[bigbadwolf]

Well, that's your opinion. You are becoming emotive. Can you be a bit more precise? Take one of his topics, tell why it is stone-age and we can then talk about it.
His ideas encompass many thought processes needed in not just maths but also software development.

I bought Plausible based on your recommendation some time ago. It's really outdated.

You do know that "How to Solve It" is an abridged version of M & PR? There is nothing wrong with Polya but there seem to be better books in the market today -- Engel being one of them.

 

[Jakelaker]

Thanks for the recommendations, I'll definitely check them out.

How about for logical problem-solving in general? Are the skills developed from studying George Polya's and Engel's books, and applying them to problems, transferrable to problem-solving in general? If not, what would you suggest?

By this I mean, you can boil everything (within context) down to a problem and then solve it. Whether it's minimising distance travelled while mowing the lawn, fixing sleeping patterns, etc.

 

[Daniel Duffy]

Thanks for the recommendations, I'll definitely check them out.

How about for logical problem-solving in general? Are the skills developed from studying George Polya's and Engel's books, and applying them to problems, transferrable to problem-solving in general? If not, what would you suggest?

By this I mean, you can boil everything (within context) down to a problem and then solve it. Whether it's minimising distance travelled while mowing the lawn, fixing sleeping patterns, etc.

Aristotle's decomposition?

 

[Jakelaker]

Aristotle's decomposition?

Sorry, what's that? I can't find anything on google.

 

[Daniel Duffy]

Sorry, what's that? I can't find anything on google.

Maybe analysis and synthesis.
Analysis > Ancient Conceptions of Analysis (Stanford Encyclopedia of Philosophy)

In software: Functional Decomposition

 

[Jakelaker]

Maybe analysis and synthesis.
Analysis > Ancient Conceptions of Analysis (Stanford Encyclopedia of Philosophy)

In software: Functional Decomposition

Thanks, I'll give that a look.

 

[Daniel Duffy]

You're welcome. I use the technique all the time to break a software problem into independent components and use Polya's questions. Here is a test case for the Monte Carlo method that I wrote.

 

[Jakelaker]

You're welcome. I use the technique all the time to break a software problem into independent components and use Polya's questions. Here is a test case for the Monte Carlo method that I wrote.

Cool! Going to be a while till I understand that document since I have no knowledge of MC method, SDEs, geometric BM, etc. My knowledge is most of Sheldon Ross' Introduction to Probability Models.

 

[Daniel Duffy]

No problem; the specifics of MC are not critical. The message is that Figure 2 is a baseline blueprint to which specialization, generalisation etc. etc. can be applied.

And developers: don't forget the magical number 7!

The Magical Number Seven, Plus or Minus Two - Wikipedia, the free encyclopedia

 

[bigbadwolf]

How about for logical problem-solving in general? Are the skills developed from studying George Polya's and Engel's books, and applying them to problems, transferrable to problem-solving in general? If not, what would you suggest?

By this I mean, you can boil everything (within context) down to a problem and then solve it. Whether it's minimising distance travelled while mowing the lawn, fixing sleeping patterns, etc.

Ah, you're looking for the holy grail. There is no such approach or method. A genuine problem requires something new by way of perception and insight. Otherwise it wouldn't be a problem but just a variant of something already solved previously.

 

[Daniel Duffy]

Ah, you're looking for the holy grail. There is no such approach or method. A genuine problem requires something new by way of perception and insight. Otherwise it wouldn't be a problem but just a variant of something already solved previously.

Discovery is mostly random. What about Popper?

https://www.amazon.com/Logic-Scientific-Discovery-Routledge-Classics/dp/0415278449

 

[bigbadwolf]

Discovery is mostly random. What about Popper?

https://www.amazon.com/Logic-Scientific-Discovery-Routledge-Classics/dp/0415278449

For some reason I don't care for Popper. My favorite philosopher of science is Feyerabend. For math I like Lakatos ("Proofs and Refutations") and Hadamard ("On the Psychology of Invention in the Mathematical Field"). And maybe some Gardner ("Aha! Insight").

 

[Jakelaker]

Ah, you're looking for the holy grail. There is no such approach or method. A genuine problem requires something new by way of perception and insight. Otherwise it wouldn't be a problem but just a variant of something already solved previously.

I'm looking for what others would do to achieve that goal.

My current plan is to do daily puzzles/brain teasers, baby steps at a time, from sources like Heard on the Street, online, etc and try to use every opportunity I have to find the flaw in my thinking (supplemented with How to Solve It/ Engel's while studying math at university).

There is a particular puzzle nurtured my critical thinking: "You have 9 balls, equally big, equally heavy - except for one, which is a little heavier. How would you identify the heavier ball if you could use a pair of balance scales only twice?". After testing the obvious cases, I just kept thinking about solving the puzzle (it was like waiting for the 'Aha moment') but in retrospect, nothing was going on in my mind. Then I asked why I can't solve it and that was when I realised that I really trusted my false intuition and should have verified earlier that the (highlight the following to reveal the text that solves the puzzle)3-3 pair configuration would work or not.

This discovery is similar to the bats and balls puzzle: "A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?" which is a question used in a Cognitive Reflection Test (Cognitive Reflection Test - Wikipedia, the free encyclopedia), for those who never seen it.

 

[Daniel Duffy]

This discovery is similar to the bats and balls puzzle: "A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?" which is a question used in a Cognitive Reflection Test (Cognitive Reflection Test - Wikipedia, the free encyclopedia), for those who never seen it.

1. Impulse answer (wrong)
2. Use variables x and y, x = .., y = .. QED

Daniel Kahneman in his book calls these System 1 and System 2, respectively.

Then I asked why I can't solve it and that was when I realised that I really trusted my false intuition and should have verified earlier that the (highlight the following to reveal the text that solves the puzzle)3-3

Exactly what DK states!

https://www.amazon.com/Thinking-Fast-Slow-Daniel-Kahneman/dp/0374533555

 

[Jakelaker]

1. Impulse answer (wrong)
2. Use variables x and y, x = .., y = .. QED

Daniel Kahneman in his book calls these System 1 and System 2, respectively.

Then I asked why I can't solve it and that was when I realised that I really trusted my false intuition and should have verified earlier that the (highlight the following to reveal the text that solves the puzzle)3-3

Exactly what DK states!

https://www.amazon.com/Thinking-Fast-Slow-Daniel-Kahneman/dp/0374533555

I started this book a few months ago and was hooked when he introduced scientific identities those two schools of thought. I need to resume reading and develop some discipline, haha. Thanks for reminding me about DK.

 

[Daniel Duffy]

Looking at research and research degrees in mathematics, it is clear that 90% is based on existing results. Most PhD students generalise and specialize existing research, yes? (unless you are John Von Neumann).