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How to Solve It (1945)

math.utah.edu

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Re: How to Solve It (1945)

#41

Post by bollu »

paulpauper wrote: post #33
OldGuyInTheClub wrote: post #28

Earlier quoted context omitted.

"How To Solve It" does not account for actual ability. It was recommended in high school to those of us interested in math contests. My big lesson was that if you need this book, you're going to be roadkill in math contests. If you have the chops to be a successful mathlete, you don't need this book. And so it came to pass. There are some people who have the ability to see a problem well enough to analyze it and make…

beyond the book, what contributions did he make? it would seem like the techniques should have helped him solve problems but Wikipedia does not show anything else

Surely you’re joking. Polya pioneered so much combinatorics I don’t even know where to begin. Heck, a lot of counting problems reduces to polya’s enumeration theorem. The page of “references” to stuff named after him should be a clue to you.

Re: How to Solve It (1945)

#42

Post by dang »

Related. Surely there have been others?

How to solve it - https://news.ycombinator.com/item?id=17726102 - Aug 2018 (1 comment)

George Pólya: How to Solve It (1945) - https://news.ycombinator.com/item?id=13255739 - Dec 2016 (1 comment)

George Pólya – How to Solve It (1945) - https://news.ycombinator.com/item?id=12823608 - Oct 2016 (1 comment)

G. Polya: How to Solve It - https://news.ycombinator.com/item?id=153425 - April 2008 (1 comment)

Re: How to Solve It (1945)

#44

Post by mathisfun123 »

hgsgm wrote: post #39
mathisfun123 wrote: post #20

Earlier quoted context omitted.

> 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.…

Of course if you solve the same problem with trivial variations, they become trivial exercises. Try harder problems.

> Try harder problems.

What part of "apostol's calculus" and "hard DP" did you not understand?

I love when people are so confident in their reflexive dismissal that they're literally blinded by it. So confident that they don't even stop to consider whether their dogma might be wrong.

Re: How to Solve It (1945)

#45

Post by lcuff »

On a totally orthogonal axis: My biggest problem with tackling math problems early in life was psychological. My inner critic would say nasty things to me when I didn't get the answer right away, making it impossible to stick with exploring the problem. Once I overcame that, I got lots better at math.

And working on yet another axis, the most useful tip I've gotten on how to think about problems is to "think in extremes". Perhaps more applicable to physics than math, but a classic example is the puzzle where a ship in a lock (closed system/giant bathtub) has an engine break down. As the engine is being hauled out for repairs, the chain breaks and the engine drops into the lock beside the ship and falls to the bottom of the lock. Does the water in the lock rise, fall or stay the same? The only person I ever told the problem to that solved it described his technique: Imagine the engine is the size of a pea and weighs 20 tons. Shazaam: The water in the lock falls because it is now displacing its volume, instead of its weight in water.

Re: How to Solve It (1945)

#46

Post by brightlancer »

CBLT wrote: post #24
mathisfun123 wrote: post #20

Earlier quoted context omitted.

> 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.…

100% agree. I went through a similar phase of being infatuated with work ethic, and all I got were the following three lessons learned: 1. Trying to "put more effort" into artifacts (e.g. note taking) while learning something the first time is counterproductive. Write something lightweight and put your focus on paying attention. 2. Do one intensive review session while you still remember the content. Either teach som…

> Everyone who offers advice is just telling you their life story, dressed up as a simple directive. "Work harder" often means they wished they worked harder in some part of their life.

I think both parts of these are wrong.

There are a lot of times that folks who will advise others to do the thing(s) they wished they'd done themselves, but it's definitely not everyone all the time, and I'm not even sure if it's most people most of the time.

As often, probably more often, folks advise others to do the thing(s) that they _did_ do.

Occasionally, folks are able to give advice that is specific to the recipient. It's definitely not the norm, but I've run into it often enough.

tl;dr Don't dismiss someone's advice because you think it's all about them, particularly because you think it's all about their mistakes.

Re: How to Solve It (1945)

#47

Post by mathisfun123 »

brightlancer wrote: post #46
CBLT wrote: post #24

Earlier quoted context omitted.

100% agree. I went through a similar phase of being infatuated with work ethic, and all I got were the following three lessons learned: 1. Trying to "put more effort" into artifacts (e.g. note taking) while learning something the first time is counterproductive. Write something lightweight and put your focus on paying attention. 2. Do one intensive review session while you still remember the content. Either teach som…

> Everyone who offers advice is just telling you their life story, dressed up as a simple directive. "Work harder" often means they wished they worked harder in some part of their life. I think both parts of these are wrong. There are a lot of times that folks who will advise others to do the thing(s) they wished they'd done themselves, but it's definitely not everyone all the time, and I'm not even sure if it's most…

> it's most people most of the time

it absolutely is most people most of the time. just ponder how many advice columns and platitudinous commencement speeches and gurus and old cranky uncles and pretentious 8th grade teachers (and etc etc etc) there are. hell think of the likelihood of getting XYed in our very own industry.

> folks advise others to do the thing(s) that they _did_ do.

the meat of the point isn't that people exaggerate. even giving people the benefit of the doubt, that they don't exaggerate, they're giving you advice that's 99% of the time non-transferrable, i.e., completely useless for you and serving only the purpose of enabling them to take a nice trip down memory lane. if i'm asking someone for advice, that's an expert, i'm not asking them what worked for them in the instance of the problem they face; i'm asking them about my instance. and it is a very rare person that knows you well enough and takes the time and care and effort to advise someone individually and/or germainely.

Re: How to Solve It (1945)

#48

Post by OldGuyInTheClub »

bollu wrote: post #41
paulpauper wrote: post #33

Earlier quoted context omitted.

beyond the book, what contributions did he make? it would seem like the techniques should have helped him solve problems but Wikipedia does not show anything else

Surely you’re joking. Polya pioneered so much combinatorics I don’t even know where to begin. Heck, a lot of counting problems reduces to polya’s enumeration theorem. The page of “references” to stuff named after him should be a clue to you.

Definitely. Whatever one may think of "How to Solve It," Polya was a renowned mathematician. The book is like the Richard Hamming lectures that often come up on HN: Great thoughts from a great mind but not easily applied by the average or even above average Joe.

Re: How to Solve It (1945)

#49

Post by retrocryptid »

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.

Yes, yes. I get it. We should all invest in your "LLM, but for Ride-Sharing" startup.

Seriously thought... the pattern matching vs. rational directed approaches to general problem seem to have analogs to CNNs vs. Constraint Programming approaches to GAI. Maybe, in the same way that neural-networking solves certain types of problems brilliantly but not so much with other problems, rational directed problem solving and "work a lot of problems" / pattern matching have different problem domains in which they excel.

Re: How to Solve It (1945)

#50

Post by boppo1 »

OldGuyInTheClub wrote: post #48
bollu wrote: post #41

Earlier quoted context omitted.

Surely you’re joking. Polya pioneered so much combinatorics I don’t even know where to begin. Heck, a lot of counting problems reduces to polya’s enumeration theorem. The page of “references” to stuff named after him should be a clue to you.

Definitely. Whatever one may think of "How to Solve It," Polya was a renowned mathematician. The book is like the Richard Hamming lectures that often come up on HN: Great thoughts from a great mind but not easily applied by the average or even above average Joe.

Richard hamming lectures? I'd google, but I wouldn't know if I'd found the ones you meant.
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