Teaching the fundamental group via everyday examples This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning about the fundamental group and/or covering spaces?  

To be more precise, I am teaching a short course on the fundamental group and covering spaces, from chapter one of Hatcher's book.  I want to motivate the material with everyday objects or experiences.
Here are some examples and then some non-examples, to explain what I am after.  First the examples:
$\newcommand{\RR}{\mathbb{R}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\ZZ}{\mathbb{Z}}$


*

*The plate (or belt) trick; this is a fancy move that a waiter can make with your plate, but it is more likely to appear in a juggling show.  It is "explained" by the fact $\pi_1(\SO(3)) = \ZZ/2\ZZ$.

*Tavern puzzles: before trying to solve a tavern puzzle, one should check that the two pieces are topologically unlinked.  You can decide this by computing $\pi_1$ of the complement of one of the pieces, and then checking the other piece is trivial. 

*The game of skill, the endless chain (also called fast-and-loose), is explained by computing winding number, ie computing in $\pi_1(\RR^2 - 0)$. 

*In the woodprint Möbius Strip II the ants illustrate the orientation double-cover (an annulus) of the strip.  One could also perform the usual game of cutting the Möbius strip along its core curve to demonstrate a double cover of the circle by the circle. 


Noticeably missing are any real life toys/puzzles/games that rely on the idea of homotopy.
Now for the non-examples:


*

*Impossible objects such as the Penrose tribar that exist locally, but not globally.  These can be explained via non-trivial cohomology classes.  But homology and cohomology are not discussed in this course. So - no cohomology!  You can find many real-life examples of cohomology discussed here.

*Winding number (in the form of linking number) also arises in discussions of DNA replication; see discussions of topoisomerase.  However DNA is not an everyday object, so it is not a good example. 

*There are no draws in the board game Hex.  This is equivalent to the Brouwer fixed-point theorem.  This example is not very good, because most people don't know the game. 

 A: I recently heard this puzzle from Dror Bar-Natan, and there's a nice solution using the fundamental group.

There are $n$ nails arranged in a line on a wall. Find a way of hanging a picture from these nails so that if any 1 nail is removed, then the picture will fall.

To solve it, you can first reformulate it as follows: nails correspond to punctures in the plane, and removing a nail corresponds to filling in a puncture. The fundamental group of such a space is freely generated by loops around each puncture, and filling in a puncture corresponds to quotienting by one generator. We'd like a loop that is killed in each of these quotients, and it's easy to write one down inductively using iterated commutators.
A: My favorite example is this. You want to hang a picture (to which a piece of rope is attached in the usual way) on a wall. 
There are two nails drawn in the wall, close to each other.
One has to hang it with the following conditions:
a) The picture must be hanging (does not fall).
b) When one nail is removed (any one), the picture falls.
When you solve this, generalize to n nails.
I asked this question many people from 10 years olds to participants of a conference
in low dimensional topology. The average time required to find a solution is about the same, and it is strictly greater than $0$.
Rope and nails or some substitute were always present when this question was asked).
I use this example in my teaching in two ways. When I explain the correct statement of Cauchy's theorem in Complex Variable, and the difference between the fundamental group and
the first homology group, and when I teach what is a non-Abelian group.
My second favorite example is the belt trick which is associated to the name of Dirac, and which is mentioned in the question. Sometimes I do it without a belt, using just my hand
holding a pencil.
EDIT. Sorry, it is a duplicate:-) But I have a reference:
http://www.math.purdue.edu/~eremenko/problems.html
I did not invent this problem myself:-(
It was one German student who asked me this question when we were hiking in the mountains
about 15 years ago. I don't remember his name:-( 
