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As an undergraduate (I don't study maths primarily, but employ these in the study of econometrics/economics), I've come to realise that solving difficult problems in fact hinge on one's ability to recognise relationships between two things, and then (though not always) expressing this relationship in some equation which moves one towards a solution. This relationship to be recognised will be one which was previously not recognised, or at least not explicitly covered in class.

My question therefore, is how can I get better at this skill at recognising relationships? I'm glad that I've pinpointed the skill on which solving difficult problems hinge (or at least have a working theory), but I still fall short when actually solving difficult problems in problem sets and exams.

For example, suppose in an exam I am asked to show that the expectation of a non-negative random variable is equivalent to the integral of its complementary CDF from $0$ to $\infty$, that is, $E[X]$ = $\int_0^\infty P(X >x)dx$, where $X$ is a non-negative random variable. I had previously only learnt the basics of probability distribution function, cumulative distribution function, expectation and variance etc., but had never had to connect the concepts of CDF and expectation. Mike Spivey, in the accepted/best answer to this post, illustrated why this proof holds true, and, to use the language I employed previously, I note that his explanation hinged on recognising the relationship between column addition and row addition. This is seen from his sentence, "we could also add up these numbers row-by-row, though, and get the same result."

This is of course but one example - my question is how do I train myself to be able to recognise such relationships, especially under the time constraint of examinations, so that I may solve difficult questions? Is there a systematic way to go about doing this?

Charlz97
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    They did that because they had seen it before. Almost all proofs that people ever do consist in applying ideas they have seen before. Occasionally, an idea gets applied in new places, or applied with a small variation, as a result of trying previously known ideas one after another to see which one fits. So, solve many problems and see many solutions by other people. – MoonLightSyzygy Dec 26 '19 at 14:07
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    It's way less intellectual than you might think. Just do hundreds of problems and soon enough you'll be trained like a monkey, that you don't even have to think anymore (I am emphatically not joking). Like, right when I saw you asking to prove $E[X] = \int_0^\infty P(X > x)dx$, I immediately thought "prove for indicator functions, then do usual generalization trick to get all non-negative random variables". I guarantee you thousands of others have the same reflex. – mathworker21 Dec 29 '19 at 12:58
  • Have you seen Polya's How to Solve It before? –  Dec 29 '19 at 13:50
  • It's the punch line of a joke, but "practice, practice, practice" is a very good idea. – Somos Dec 29 '19 at 19:51
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    99.9% of all solutions come from composition of tricks seen before. Many of these tricks are not explicitly taught, so people figure them out in the process of doing. Would you think of solving a geometry problem by parameterizing all objects on a plane, thus converting it into a purely algebraic problem? How about converting a real-valued integral to a complex-valued integral in order to compute it? Or the Feynman technique of differentiating an integral with respect to a constant, that we temporarily assume to be a free variable? The is no rule to this, experience can't be generalized – Aleksejs Fomins Dec 31 '19 at 17:25
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    There is an urban legend, that examiners in some universities in soviet union had a special folder with mean math problems. These problems were carefully selected for two properties: 1) The problem would look trivial to anybody who sees the answer and 2) Even for an expert it is nearly impossible to come up with an answer in an hour's time. It is told that these problems were used to deny university access to some extremely bright students for some political reasons. The poor souls actually thought they were stupid when they saw the answers. – Aleksejs Fomins Dec 31 '19 at 17:30

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