Waistcoat and Handcuffs
If you are wearing a waistcoat (as many stylish mathematicians often do), it is a relatively easy task to take the waistcoat off, turn it inside out and put it back on so that the lining is showing. Easy, that is, provided your arms are not handcuffed together! However, using a bit of ingenuity and some mathematics, this is still possible.
Equipment needed: a waistcoat (preferably one which is a different colour on the inside, so you can see the difference) and a pair of handcuffs, or some other way of attaching your hands together at the wrists.
First you must put the waistcoat on normally, and handcuff yourself. The first challenge in turning the waistcoat inside out is to get it off - if you are struggling, the easiest way to do this is by pulling it over your head so that it hangs in front of you on top of your hands.
The waistcoat now needs to be turned inside out so that when it is put back on, it is the other way round. The way to achieve this is to notice that the two shoulders of the waistcoat, which would normally sit on your left and right shoulders when it is the right way in, will sit on the opposite shoulders when the waistcoat is inside out! This means that turning it inside out is easily achieved by passing one half of the waistcoat through the other - by pushing one armhole through the other and pulling the rest of the waistcoat through after it. This all happens in front of you, with your hands still passing through the centre. Then, once the waistcoat is inside out, you can flip it back over your head, to the sound of applause from anyone watching.
How does this work?
There is a branch of maths called Topology. In this link you can read about uses of Topology in Genetics. Topology is concerned with what shape things are - but you are allowed to squash things a little bit, as long as you don't tear or glue. So, a person with their arms handcuffed, if you ignore the head and body parts, is essentially just a circle. The waistcoat is effectively a pair of armholes joined together - the back of the waistcoat doesn't matter, and the waistcoat can be seen as a figure of eight shape. If you picture a circle, passing through the two holes of a figure of eight, then swapping the two loops of the figure eight is a very simple task, because you can pass one through the other. When you consider the problem as simple topology, it does not seem so impossible after all!
If you like this trick you may enjoy the book: Martin Gardner, Knotted Doughnuts, Freeman, 1986