I am an undergraduate student studying some elementary calculus and statistics. In my honor calculus class, my professor gave one of final exam problem:
$$\lim_{n \to \infty} \int_{[0,1]^{n}} \left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^{2013.1214} d \mathbf{x} = ?$$
At first sight I've brutally tried to calculate it directly: changing variables with $x_{1}=y_{1}$, $x_{1}+x_{2}=y_{2}$, $\cdots$, $x_{1}+\cdots+x_{n}=y_{n}$. Of course it has failed and the time was ran out.
After the exam, I asked this problem to my friend (who did not took this exam). Just after 10 second he saw this problem, he said: "each $x_{i}$ is picked from $[0,1]$ independently, so as $n \to \infty$, its average will go to $\frac{1}{2}$, so the value of limit would be $(\frac{1}{2})^{2013.1214}$. Try to justify this by using the central limit theorem"
I was really shocked, that just after first glance he got how to solve this and realized the meaning of this problem, and felt I became an almost fool and complete idiot, becuase even though I already took the elementary statistics class to learn the central limit theorem, when the time to apply to solve problem was came, I couldn't think to apply that for this calculus problem - but my friend, just after the first glimpse, realized that the essential meaning of this problem is asking where the certain power expectation of average of uniform distribution goes as the size of sample gets bigger.
How can I raise my intuition to catch the essential meaning contained in the mathematical problem and move on from that? Just think over and over again? Or are these kinds of intuition just a gift from the god?
I don't know. Was it my ‘feminine intuition’ that helped me realize that $\displaystyle\sum_{n=0}^\infty\left[\frac{(2n-3)!!}{(2n)!!}\right]^2=\frac4\pi$, or merely plucking $n=\frac12$ into Vandermonde's formula $\displaystyle\sum_{k=0}^n{n\choose k}^2={2n\choose n}$, and using the fact that $\Gamma\left(\frac12\right)=\sqrt\pi$ ? :-)