Two volunteers are attached together, using two pieces of rope with a slip knot at each end, so that each person has their own wrists joined together but the two ropes are crossed around each other. The challenge is to see if the two people can separate themselves, without taking the ropes off their hands and without cutting the ropes or untying the knots.
This may seem impossible, and a little thought will discover that if the two people are thought of as complete loops along their hands and the ropes, this task would be completely impossible. In fact, this is not quite the case and there is a little bit of extra space which you many not initially notice.
The ropes are tied around the two peoples' wrists - but they are not completely tight! The gap between the loop of rope and your arm may be exploited. If we take a loop of one person's rope, and push the middle of the loop through the wrist hole on the other person, we find that the loop may be (watching out for things getting twisted) passed over their hand, and then the rope will easily slide out from the loop around their wrist, leaving the two people separated!
How does this work?
There is a branch of maths called Topology, which is concerned with the shapes things are fundamentally - if something can be stretched or squashed to form a different shape, topology would consider the two things to be equivalent.
If the rope was directly glued to your hands, the shape you make would be equivalent to a circle (ignoring your body and head). Two circles linked together cannot be unlinked without breaking the circle. However, this is not the case we are dealing with in this trick, and having a rope tied around your wrists means you are not quite a completely closed circle. The gap between you and the rope, which is big enough to fit a piece of rope through, is the key to getting yourself unlinked.
This trick was demonstrated by Professor Christopher Zeeman at the Royal Institution Christmas Lectures, which he gave on the subject of mathematics in 1978.
If you like this trick you may enjoy the book: Martin Gardner, Mathematical magic show, Penguin Books, 1985