# How can I improve my problem solving abilities so that I stop missing the obvious?

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?

• 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. – cr001 Nov 5 '15 at 4:31

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.