**Context:** The solution I gave here used the theory of tempered distributions; also there were various bits of handwaving that needed to be filled in. I finally found a completely elementary proof of the "Magical Property" - the answer to the question below was the last piece in the puzzle. I got a major chuckle out of the solution - passing it on because I didn't get you guys anything for Christmas...
$\newcommand{\sinc}{\text{sinc}}$

Define $\sinc(t)=\sin(t)/t$ as usual.

**Question:** How can one give an elementary proof that $$\int_{-\infty}^\infty\sinc(t)\sinc(t-n\pi)\,dt=0$$for $n\in\mathbb Z$, $n\ne0$?

**Comment:** If you note that $\sinc(t)=\frac12\int_{-1}^1e^{ixt}\,dx$ then this is more or less obvious from the Plancherel Theorem. We want a solution much more elementary than that - the answer uses nothing but calculus, and no "hard" calculus either...