<p>Like - if your solution is appearing to be too long in math, you clearly realize that you're doing something wrong - as the solutions that come in the solutions manual are all short and elegant (especially so for the hard problems). We're also taught that "short and elegant" is always best. And indeed, most of the solutions I've seen at all levels are all very short, including the solutions I've seen for the International Math Olympiad and solution manuals I've seen for grad level textbooks.</p>
<p>Which then begs the question - which fields have problems with solutions that MUST be pages long? Engineering?</p>
<p>I remember that once, when my friends and I were doing a problem set question, we spent more than an hour on one problem, and were working on our sixth page when we decided to start over. We approached the problem from a completely different angle, and realized that it was actually incredibly simple if one thought about the question in a less obvious way. We finished it in ~5 lines.</p>
<p>Uh... I'm going through Neural Networks: A Comprehensive Foundation, and a lot of the derivations are often 10-15 pages long, and that's only after they are highly polished and specific.</p>
<p>Oh and Andrew Wiles proof of Fermats Last Theorem anyone?</p>
<p>Well if you're looking at textbooks designed for the average joe, then they won't be 10+ pages long, because the average joe can't concentrate on that long a proof/solution.</p>
<p>And the long solutions are often more helpful than the small ones, because the highly polished small ones don't teach you how to think, whereas the longer solutions require much more understanding and concentration.</p>
<p>It's not just textbooks for the average Joe - it's also the AoPS books, answers to AMC/AIME problems, and graduate level textbooks (at least for one real analysis book I encountered at that level). There's a lot of question fine-tuning in such textbooks (including the ones used for MIT OCW - though admittedly I don't look at engineering problems :p) such that they all have neat solutions and neat integral value answers (no numerical analysis)</p>
<p>Yeah - my concern is - such competitions are designed to be predictive of talent. The issue is - is talent in closed problems where you can use all sorts of "tricks" that you can't use in "real problems" desirable? Are students better off learning those tricks or in doing something useful for research? I know quite a few students who like math research but who disdain competition math.</p>
<p>most of my higher level math and physics classes actually had SHORTER answers bc my prof's used shorthand/"etc" to solve them.</p>
<p>also my physics teacher would love to use logic to solve problems that normally would take pages to solve. they end up being a half a page.</p>
<p>my multi var/diff eq teacher would do three steps and then be like 'blah blah blah' and draw a line down the board and then write the answer and say, 'you know how to get there.'</p>
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The issue is - is talent in closed problems where you can use all sorts of "tricks" that you can't use in "real problems" desirable? Are students better off learning those tricks or in doing something useful for research? I know quite a few students who like math research but who disdain competition math.
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<p>I don't think you really need to know that many "tricks" to do good in contest math. It's just that you have control over what the problems and the solutions in contest math look like since you already know the answer, but you can't really do that with research.</p>
<p>Well there is still a large overlap between competitions and research. Research problems usually take the form of: I want to go from A to G, and I have to proceed probably by going from A->B->C->D->E->F->G. Of course you need some insight and such to figure out B might be. But to get from A->B, is often times similar to solving a competition problem, perhaps a little more computationally nasty.</p>
<p>Good points. Competition math ability certainly does correlate with research ability - it's just that it's hard to pinpoint the strength of such correlation. Most able people don't seem to have too much of a problem with transition from competition/textbook problems to research. Of course a few people will end up disappointed.</p>
<p>Sometimes I do get annoyed with the pervasiveness of "perfect problems/answers" in textbook/competition problems.</p>
<p>Typical problems are written by someone. That someone is interested in creating a problem with a given solution - usually an elegant one. If the writer can't craft a problem with a 'nice' solution, he won't publish the problem.</p>
<p>Research problems, on the other hand, aren't made in the same way. They arise from research, and so can have extremely convoluted, long, complex solutions. Read any math research paper - the theorems provided usually take up a few pages each at least, even for relatively shallow results.</p>
<p>For deep results, a paper the size of a book isn't uncommon.</p>