Trace the Surface: Indicate that you are going to launch a challenge: Ask to trace on one side with one colour and on the other side with another colour. Emphasise that no matter where they start, they will end up back at the starting point without lifting the pen or crossing an edge. The audience will be amazed that they can't do it and end up where they started having used just one colour and not two as stated in the challenge.

Cutting the Möbius Strip: Ask the audience what will happen if you cut the Möbius strip down the middle. Wait for the answers.

Now, explain the surprising property of the Möbius strip when you cut it down the middle. Unlike an ordinary loop, which would yield two separate loops when cut, the Möbius strip produces a single, larger loop. This time, have a physical Möbius strip ready to demonstrate this by making a single cut along the strip.

Have some strips prepared in advance:

A Möbius strip is a one-sided surface that can be constructed by affixing the ends of a rectangular strip after first having given one of the ends a one-half twist. This space exhibits interesting properties, such as having only one side and remaining in one piece when split down the middle. The properties of the strip were discovered independently and almost simultaneously by two German mathematicians, August Ferdinand Möbius and Johann Benedict Listing, in 1858.

If you cut it straight in half, (that would mean two cuts) you just get two strips half the original size. However, try to cut it lengthwise and you fail to separate the strip. You still get a ring with a couple twists that's not a mobius strip. Cut that lengthwise again and you get that same shape with a mobius strip half the length interlocked.

An animation of how this works can be found here:

Animation - How this works - Click 

A nice explanation of how to do the effect can be found here:

Animation - Click 

Applications: Möbius strips are found, such as in conveyor belts or certain types of industrial machines. This shows that even though the Möbius strip may seem like a curious mathematical concept, it has practical uses.