The harmonic (series) beetle: live illustrations of mathematical theorems In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely elastic tape with constant velocity.
Every hour the tape is lengthened out by 1 meter, and the beetle remains at the
same rate of the tape it has already reached to the moment. Will the beetle ever reach
the end of the tape?
This is not a paradox but a calculation of a mathematically idealised model
"from life". Do you have some other, probably nicer examples which illustrate
some standard but deep results in analysis, algebra, probability, geometry,
and so on, and so on, and so on. Please keep in mind an average undergraduate
student as the audience for your example and allow others to use it in his/her
teaching.
Thank you!
 A: How do you hold a pizza (with your hands) so that the toppings don't slide off? That is, how do you keep the pizza "straight"? You bend it through the middle and use one of your fingers to support the bend. Why does it work? Because a pizza is inherently "flat." You would expect any reasonable definition of curvature to give a (flat) pizza slice a 0. It turns out this expression (Gaussian curvature) is the product of two numbers, $\kappa_1$ and $\kappa_2$, corresponding to the two orthogonal curvatures. Thus, one of the two curvatures must be 0 -- so if you're bending one direction, the other must necessarily be "straight" (no curvature). 
A: There are all these examples surrounding fixed point theorems. 
The following is somehow a cliche. Take a sheet of paper, crunch it, and put it on top of its original position. Then there is a point that lies on the vertical of its previous position. This illustrates the fixed point theorem for contractions in Banach spaces. 
There are also a lot of examples in probability theory.
Here is one related to the harmonic series. In your youth, you may have collected cards depicting soccer players, martians, whatever. There is a finite number of cards to collect, say n. Each packet of corn flakes comes with one of them, at random. And of course you want your mom to buy this precise brand of flakes so as to get the whole collection. May be you have wondered what is the average number of packets she should buy so that you can complete the collection. The answer is 
$$
n \sum\limits_{k=1}^n {1 \over k}
$$
This is asymptotic to $n\log n + n\gamma+ 1/2$, where $\gamma$ is the Euler constant. This is the simplest mundane example I know involving that constant. So for example, if there are $n=150$ cards to collect, you need to buy an average of 519 cards.
A: I once thought of the following variant of your beetle problem.  Suppose a kid has a piggy bank with a dollar in it and gets an allowance of a dollar a week (so, this story is implausible).  Then, of course, his savings grow without bound.  If he has $n$ dollars, then the next week he receives a fraction $1/n$ of his savings, and so $1 + 1/2 + 1/3 + \dots$ must also grow without bound.  Tell this to your calculus students but don't analyze it -- it requires infinite products!
