<p>Are there any especially interesting cases of when you solved a problem?</p>
<p>===</p>
<h1>I’ll post a couple of strategies and thoughts of my own:</h1>
<h1>anyways we met today and we talked a bit about problem solving strategies. He mentioned a prof that said that he often had to find the right solution only after 100 different tries. my model of problem solving is divided into 2: (a) generate variance in strategies and (b) select the best strategies from such variance. Generally most people falter on step (a). =P </h1>
<p>finish math problems, hardest assignment I ever got (from my perspective) since it involved proofs. Apparently all it needed was an application of some rule but I did a full-blown proof with lemmas from combinatorial proofs. That was fun. I barely managed to finish it in time for turn-in.</p>
<h1>I started the proof the night before. I stared at it for a while, attempted to consult other books and websites, came at it fruitless, so decided to wait for a flash of insight. That didn’t need to come, actually, the best way to start is just to start working out special cases and seeing if the mind would make an “inductive insight” from special cases, so to speak. Then I found the “inductive insight” and found that I assumed a few combinatorial rules (I got the insight of them from noticing that when you’re analyzing a row of Pascal’s Triangle with an even number of entries - the sum of the odd entries of each row is half the sum of the row per se). So I had to prove those combinatorial rules. The proof came with an inspiration from an exercise in Scheinerman’s “Mathematics - a Discrete Introduction”. I like this a lot.</h1>
<p>=> One of the key questions I always had: Is it better for you to solve challenging problems that are accessible for yourself? Or better for you to solve difficult problems that are so difficult that you need to work with others to solve it? (aka Caltech psets). Hm. This led to a second question.</p>
<p>=> I asked Nate if he knew of professors who go to the offices of other profs to ask for help with a math question. After all, many professors probably can’t do Caltech psets. =P He said yes, but gave no examples. The only example I know was that Einstein wasn’t that great in math so he needed help from some people in order to get at his theories. </p>
<p>(this happens when you find a pattern in say, Pascal’s Triangle, but when you are mathematically immature enough as to not be able to prove that pattern). </p>
<p>==
My other question is this: what of math textbooks that give hints to problems? As compared to textbooks like Rudin?</p>
<p>There are several elements here.
(a) what is better in inspiring your interest?
(b) are you a reliable judge of comparing between the two methods of inspiring your interest?
(c) what results in fewer attention lapses, in more consistent motivation, in a clear estimate of progress that you can make? And progress as measured by how well you understand the problem, the surrounding theory behind it, and related problems?</p>
<p>Inspiration is a matter of taking out a piece of paper and a pen and scribbling down random stuff, random equations, and random patterns, until you come up with something correct yet so completely random that it’s brilliant. </p>
<p>Inquiline, I’m teaching myself Partial Differential Equations and Multiple Integrals at the moment, and they are surprisingly interesting. I’m using Elements of the Differential and Integral Calculus by Granville, Smith, and Longley. I found it at a library book sale amongst a pile of dusty, old books. It’s a gem.</p>
<p>Indeed, Fourier made two HUGE assumptions in his analysis. later mathematicians, doubtful of the truth of such assumptions, had to prove them right (were they? I can’t find any good commentary on his work). Point notwithstanding - you assume some facts are true and </p>
<p>Ah yes CT - why don’t you go read Fourier’s “The Analytical Theory of Heat” - which is highly relevant to PDEs?</p>
<p>Excellent, IK. Thank you. I have downloaded it. I want to finish this current book before I start another one. Multiple Integrals and Applications of Partial Derivatives are the last two chapters. After that, I want to take a look at the reduction formulas. Then I’ll start reading Theory of Heat. </p>
<p>That website has some good books. The philosophy books seem very interesting.</p>
<p>eh so I’m just curious, how do you use online books to learn Math? is it just theory, or are you still able to work problems through from looking at it online?</p>
<p>Well said.
Though I get a lot of random ideas when I’m doing something boring, like walking to the supermarket or something, and I need to have something to think about. Not the best way to solve a math problem, but certainly a good way to come up with problems to solve. Like a couple days ago, I was walking to dinner, and suddenly got the idea to model how people influenced each other…</p>
<p>Wanting to solve the problem is the inspiration itself, I think. It’s more fun when you have to try out a lot of different methods before you find the right one. It also makes you look more closely at the question, which is extremely important for even the seemingly easiest questions (looking at linear algebra problems now about vector spaces, for example, I’ll answer questions too quickly and forget certain little nuances).</p>
<p>And wait, ChaosTheory, are you teaching yourself Partial Differential Equations or Partial Derivatives? I thought PDE’s were supposed to be verrrrrrrrrrrrrry hard (I don’t think math majors are supposed to encounter them until Graduate level). Good luck with that!</p>
<p>um if theres a hard problem first I try for about 4 minutes. Frustrated, I fluctuate between screaming and whimpering for maybe 40 seconds, then i start crying+weakly hit textbook/floor/nearest object for about 90 seconds. Exhausted, I fall asleep, Then in the morning, there’s a 50/50 chance I’ll finish it very quickly because for some weird reason i suddenly know how to do it, or I just give up.</p>
<p>Wow, ChaosTheory, that’s dedication. I liked Partial Derivatives, but I think I’m going to at least wait until after I take an ODE course before I try to study any part of PDE’s. </p>
<p>And commenting on something fizix2 said, since I miss lots of things reading through the first time, I agree that sudden flashes of insight come at some of the most random moments. But I love that. Just as long as I’m near pen and paper, or else I’ll be keeping the problem in my head until I find some haha.</p>
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