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I'm a generally good math problem solver. I get decent scores on contests, top of my class in math courses, and have a pretty wide array of knowledge from which to relate concepts in order to solve problems in unique ways. But I have a defect; I never see the "best" way to solve a problem.

I know best is a vague term, so I'll try to define what I mean. Most problems can be solved an infinite number of ways, but a lot of problems on contests and in school have a desired solution. There's almost always one really quick and effective way of solving a problem that gets right to the answer and avoids clutter whilst also being clever and informative. I always miss this solution.

I'm not being self-degrading either. I mean, of course simple questions I'll see the intended solution, but anything that takes a little more than the zero amount of brainpower to solve is out of the question.

Here's an example problem. What is $\int_0^{\infty}x^5e^{-x^3}\text{d}x$? The intended solution is to use integration by parts to get that the indefinite integral is $-\frac13e^{-x^3}(x^3+1)+\text{constant}$. I did it this way:

I know that $\Gamma(n)=\int_0^\infty x^{n-1}e^{-x}\text{d}x$. Now this sort of looks like the integral in question, just with $-x^3$ as the exponent on $e$. So, substituting $u^k=x^{n-1}$, we get $\Gamma(n)=\frac{k}{n-1}\int_0^\infty u^{k-1+k/(n-1)}e^{-u^{k/(n-1)}}\text{d}u$. So we just have to find $k$ and $n$ such that $k - 1 + k/(n - 1) = 5$ and $k/(n - 1) = 3$. Doing this gives us $n = 2$ and $k = 3$. Thus we have $$ \Gamma(2)=3\int_0^\infty x^5e^{-x^3}\text{d}x $$ and since $\Gamma(2) = 1$, our integral is equal to $\frac13$.

This is seriously overcomplicating things, and this is mild compared to what I do sometimes.

Point is, I always miss the intended solution or the easiest solution, and I always take some roundabout path to the answer. So, my question is:

What can I do to rid myself of this habit? How can I strengthen my problem solving skills so that I see the obvious more quickly, instead of "bashing" away at problems until I get the right answer.

Addendum: I think about this a lot, and I think one of the reasons I'm bad at seeing the obvious solution is because I read a lot of math, rather than do a lot of math. Instead of spending my time solving problems "at my level" (whatever that means), I go off and read a book on something much higher level. I have no problem understanding and absorbing the material, but I'm not actively doing any problem solving, so maybe that's the cause?

Another thing is just that I rarely practice contest math or do optional homework questions. I spend time solving problems that aren't meant to have solutions, things that I just think of off the top of my head that I'm curious about. For example, is there a closed form for $\sum_{k=1}^n k^r,\ r\in\Bbb{C}$? This was something I toyed with for about a week, and got some really interesting and nontrivial results, but would never see on a contest. I also have a feeling that my inability to notice obvious things in problems can effect my ability to solve real world problems like this.

So, what can I do to keep this from happening over and over again?

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  • $\begingroup$ This is going to be tough. In general everyone has a preferred way of solving problem and you must have done some problems which you managed to solve in the "desired way" as well but you are just focusing on problems you did not manage to. In theory doing a lot of practices using methods you are not used to will do the trick but that's going to be pretty painful. $\endgroup$ – cr001 Nov 5 '15 at 4:31
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At first attempt to answer this question, my impulse is to say it's not so bad to miss the obvious answer. New advances in mathematics are typically not made by doing things the obvious way. Being able to think laterally is a good skill, and it seems to come naturally to you. Your gamma solution is pretty clever, to be honest, and if you keep being that clever then eventually you may be clever in a way that nobody has ever been clever before. So I'm not sure that the premise that it is a defect that you should rid yourself is a good one.

But you want to finish your exams with good marks before time runs out, so here at least is my method of dealing with this problem.

We'll use your case of integrals. As you go through the lower division calculus classes, you're essentially given a set of tools, one at a time. First you're shown how to integrate by simple antiderivative. Later you get u-sub, parts, trig sub, partial fractions, etc. Each tool I learned went in a mental toolbox, and when I saw a question on an exam, I would go through my tools mentally. Is this the antiderivative of something? No. Is it a good u-sub? I can't see any part of it that's cleanly a derivative of another part, so no. There's no trig, so it's not a trig sub. Is it a parts problem? Etc etc. If you can organize your tools in your mind then you can draw on them at the appropriate time.

Your mind is already doing this, really. The $\Gamma$ function is a tool, it's just probably not the tool your instructor intends for you to use. I think that if you spend mental effort organizing the toolbox (by way of e.g. a mental list), you're more likely to bring forth the right tool. A good way to figure out what tool is intended to use is to just look at the top of the page. Your homework assignment or textbook will probably list the section that the problem comes from. If it says "$\S$ 5.3 Integration by Parts" then chances are you should be looking for ways to apply parts.

Lower division exams typically don't require much lateral thinking. In those classes you learn techniques and then apply them. Sometimes instructors will throw in twists but usually it's pretty straightforward - if you can follow directions you can do it. If you become (or are) a math major and you get to the upper division classes, you will benefit greatly from lateral thinking while you learn abstract algebra and analysis. Those classes aren't about applying techniques, they're about learning the ideas of math and when to apply them, and it's not always so straightforward.

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