2 Colouring a Squiggle
Draw a squiggle on you page without taking your pen losing contact and join it up so both ends make a loop. Here's one example:
Now, unlike colouring maps which requires 4 colours, you can colour your squiggle in with only 2 colours so that no two regions are in the same colour. Give it a go, it will always work. Here's my example coloured in:
Why is this always true?
Well if you started with just a single loop with no crossings then it can obviously just be coloured in with only 1 colour. As we move it around like a piece of string there are only two ways in which you can create more regions. The first looks like this:
Where by "twisting" the string you add another region, which you could always just do as the same colour to the original region. The second way of making another region is to bring two pieces of string over one another to create an opening in their intersection. However these pieces of string always had the region between them as the same colour, so the regions on the other sides must have been different colours as well, so we can colour the new region in to match. Once again there is no problem:
So the only moves which introduce new regions both are fine without introducing more colours and so the whole thing must be fine. Done.