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How to Solve It (1945)

math.utah.edu

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Re: How to Solve It (1945)

#11

Post by jvanderbot »

rahimnathwani wrote: post #9
ergocoder wrote: post #8

I read it and it doesn't help much. What helps with solving problems like math and algorithmic problems is to go through a lot of problems to see different patterns and strategies of solving problems. I'm talking about going through thousands of problems. That is very effective.

This is consistent with a sentence on page 1 of the book: 'The student should acquire as much experience of independent work as possible.'

Not only that, my take on this book is it's meant to help you classify those patterns and better recall them by going through consistent triage and process.

Re: How to Solve It (1945)

#12

Post by bee_rider »

> What is the unknown? What are the data? What is the condition?

…

> Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

What sort of problems are they solving, that they can somehow identify the relevant data to the point that they know once they’ve chomped their way through the data, the problem is done? It seems oddly constructed (I can only imagine that I know I’ve only been given relevant data if I’m working a textbook problem or playing a video game; somebody has set the problem up for me, but clearly this was written by somebody prestigious, so that isn’t it).

Re: How to Solve It (1945)

#13

Post by RcouF1uZ4gsC »

Or you can just be like von Neumann

From: https://www.tennessean.com/story/opinion/columnists/teachabl...

>George Pólya, one of his university teachers, said, “I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him, he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.”

Re: How to Solve It (1945)

#14

Post by morkalork »

js2 wrote: post #6

Ah yes, just four steps: 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back. --- Compare with the Fenyman Algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution. https://wiki.c2.com/?FeynmanAlgorithm (The discussion on FeynmanAlgorithm links back to Polya's book since not everyone is Feynman.)

>Write down the problem

Sometimes this step is enough to get you 80% of the way to solving a problem.

Re: How to Solve It (1945)

#15

Post by UltimateEdge »

bee_rider wrote: post #12

> What is the unknown? What are the data? What is the condition? … > Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? What sort of problems are they solving, that they can somehow identify the relevant data to the point that they know once they’ve chomped their way through the data, the problem is done? It seems oddly constructed (I…

I think the author is talking about maths problems, or proofs of theorems/propositions.

A problem might give you one or more mathematical objects, and ask you to show that some further condition holds true. To get started, you might consider how the given properties of those objects will help you to achieve your goal (and typically you would need to use every given property).

Re: How to Solve It (1945)

#16

Post by bonoboTP »

bee_rider wrote: post #12

> What is the unknown? What are the data? What is the condition? … > Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? What sort of problems are they solving, that they can somehow identify the relevant data to the point that they know once they’ve chomped their way through the data, the problem is done? It seems oddly constructed (I…

It's about advanced school exercises or math contest questions that are designed by someone, not real-world/research problems, where you don't even know how to best frame the issue or whether it's even solvable or the appropriate thing to tackle at the time.

Re: How to Solve It (1945)

#17

Post by bee_rider »

bonoboTP wrote: post #16
bee_rider wrote: post #12

> What is the unknown? What are the data? What is the condition? … > Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? What sort of problems are they solving, that they can somehow identify the relevant data to the point that they know once they’ve chomped their way through the data, the problem is done? It seems oddly constructed (I…

It's about advanced school exercises or math contest questions that are designed by someone, not real-world/research problems, where you don't even know how to best frame the issue or whether it's even solvable or the appropriate thing to tackle at the time.

Oh! That’s a huge difference, haha.

Re: How to Solve It (1945)

#18

Post by bonoboTP »

js2 wrote: post #6

Ah yes, just four steps: 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back. --- Compare with the Fenyman Algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution. https://wiki.c2.com/?FeynmanAlgorithm (The discussion on FeynmanAlgorithm links back to Polya's book since not everyone is Feynman.)

Or his steps to do science:

1. Guess 2. Compute consequences 3. Compare with experiment - if they disagree, the guess was wrong.

https://www.youtube.com/watch?v=OL6-x0modwY

Re: How to Solve It (1945)

#19

Post by jxramos »

bee_rider wrote: post #12

> What is the unknown? What are the data? What is the condition? … > Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? What sort of problems are they solving, that they can somehow identify the relevant data to the point that they know once they’ve chomped their way through the data, the problem is done? It seems oddly constructed (I…

> Did you use all the data...

It's a good articulation that informs one while working on complex stuff. Here's a recent example of this where the above advice came to mind while reading over this advice the other day (I believe someone linked it on HN)

> A good way to stress-test this sort of false argument is to try to run the same argument without the initial assumption that X is false. If one can easily modify the argument to again lead to a contradiction, it shows the problem wasn’t with X – it was with the argument. https://terrytao.wordpress.com/career-advice/be-sceptical-of....

Re: How to Solve It (1945)

#20

Post by mathisfun123 »

ergocoder wrote: post #8

I read it and it doesn't help much. What helps with solving problems like math and algorithmic problems is to go through a lot of problems to see different patterns and strategies of solving problems. I'm talking about going through thousands of problems. That is very effective.

> I'm talking about going through thousands of problems.

you don't need thousands of problems. you don't even need hundreds, unless, no offense, your medium-term memory is very poor.

personal anecdote 1: in between undergrad and grad school i decided i was gonna try this "solve all of the problems" approach, as opposed to my usual "sit there and ponder approach", in order to prepare for eventual quals in grad school. i started with calculus, using apostol's calculus (famous for its rigor and difficulty right?). some sections have double digits (maybe even 100? i don't remember) problems and invariably (no pun intended) by the time i got about a quarter of the way through they got trivially easy. i did finish and do all the problems in both volumes. i didn't feel i learned any of it better than the first time i took calc (wherein i didn't solve many at all beyond assignments). i did not keep on with this kind of slavish dedication and just skimmed the rest of the books. i didn't end up doing a phd in math but i did take math and cs theory classes and i did well.

personal anecdote 2: after my MS i did hundreds of leetcode problems. it was roughly the same phenomenon: in every category it only took about a dozen to be able to solve the remainder trivially (yes even hard DP problems).

and i'm willing to bet (if you're on this board) your memory is better than mine (i smoked incredible amounts of pot in high school...).

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