Examples of useful, insightful, and interesting hand-waving It seems to me that some hand-waving (by which I mean some arguments that aim at giving some form of intuition on the problem even at expenses of complete rigour [and not mnemonics for high-schoolers or totally bogus oversimplistic smoke curtains]) may be really useful at times to get some insight on a problem. 
For example, in Levi's Mathematical Mechanic there are many intelligent examples of problems where some physical intuition (while not perfectly rigorous) may help yield some result and even converted into a formal argument.
So, I would like to collect a "big list" of "useful" (and possibly somewhat sophisticated), insightful and interesting heuristics and hand-waving arguments (which may also include some reference to physical principles e.g. see this) 
 A: Answers by Christian Blatter often contain A little bit of Physics.
This one is a wonderful example:

How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

A parabola is the trajectory $\,\vec{r}\,$ of a particle with initial position
$\vec{s}$ , initial velocity $\,\vec{v}\,$ and constant acceleration $\,\vec{a}$ .
This leads to the representation
$\,\vec{r}(t) = \vec{s} + \vec{v}\, t + \vec{a}\, t^2$ , as has been employed in:


Proving that for each two parabolas, there exists a transformation taking one to the other

What is wrong with this method for a rotated and shifted parabola?

The following answer is inspired by the physical - what is "slimness" - and the physics / mechanics of solid bodies
- Moments of inertia .
With respect to the latter, any - "slim" or "fat" - 2-D body can be thought as an ellipse (of inertia). Then there is a wonderful relationship between the physical, the physics, and the mathematics of 
Steiner ellipses :


How fat is a triangle?

Couldn't really distinguish between physics intuition and a mathematical argument at this place:


Proof of the extrinsic to intrinsic rotation transform

A: I use the following in my calculus class to find limits of various quotients when the variable approach zero. They are:
$\begin{eqnarray}(BIG + small)^r &=& BIG^r + r\ BIG^{r-1} small + \cdots\\
\sin(small) &=& small + \cdots\\
\cos(small) &=& 1 - {1 \over 2}small^2 + \cdots\\
\tan(small) &=& small + \cdots\\
e^{small} &=& 1 + small + \cdots\\
\ln(1+small) = &=& small + \cdots
\end{eqnarray}$
