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...).
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 someone else or write a document that could teach someone else. It's an ineffective use of time to do that more than once, for later review of the material just skim what you wrote on your first review.
3. 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. Treat it like any other life story: learn the lessons they did, but don't let it become baggage that you irrationally follow.
Back on topic of Polya's How to Solve It, I took a no-credit uni course where the professor taught in Socratic style that expected us to (loosely) follow Polya's process. It was a great experience, and I've always been fond of Polya's steps because of it.
> 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. Treat it like any other life story: learn the lessons they did, but don't let it become baggage that you irrationally follow.
I don't know how old you are but bravo - it took me a very very very long time to realize this. The converse to this is to always be very transparent and very careful when giving novices advice - when junior PhD students ask me about some kind success I've had I always make sure to emphasize the ways in which I got lucky first and only after talk about how I applied myself.
Edit: and in full transparency, I believe i had this epiphany after at some point listening to the funny poem/song/thing (from the 90s):
Everybody’s Free (to Wear SUNSCREEN): THE SUNSCREEN SONG (Class of ’99)
> 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.
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.
Yes you'd need to go through a lot of problems because every problem is slightly different
Realistically, it is rare that you would encounter hundreds of identical problems.
Take realistic examples: topcoder.com, Facebook's Hacker cup, and Google codejam. Or any university exam. Or SAT. Or Olympiam math. Their problems are unique because they can't just create the same problems over and over again.
There are also other aspects like solving the problems faster and being able to write up a solution correctly. Coding it correctly at the first try is really a magical moment.
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.
is hard for a calculus book, i.e. is an introductory real analysis book. If you can master analysis at the level of baby Rudin without doing hundreds of problems, you're very talented. If you still don't need to do hundreds of problems by the time you finish papa Rudin, you're a mathematical genius.
Yes but I was talking about calculus as an anecdotal example. Not analysis.
> baby Rudin
This is definitely starting to look like "no true Scotsman". Anyway I did the regular amount (5-10) per section and got an A.
> you're very talented.
Trust me not the case. Just average person doing average number of pset problems.
> papa Rudin
I didn't read this, I went on to read Schilling's Measures, Integrals, and Martingales instead because I already knew enough functional analysis and wasn't interested in complex analysis.
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.
<snark>
Yes, yes. I get it. We should all invest in your "LLM, but for Ride-Sharing" startup.
</snark>
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.
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.