This problem gives a visual and engaging framework in which to practice mathematical thinking, conjecturing and start to grapple with issues of topology.

This is initially an entirely practical activity. Suggest students try to copy and colour in the images. Ask about the minimal colourings and ask that students create a well-reasoned argument that colouring with fewer colours is not possible.

Then ask students to experiment with other patterns and try to conjecture concerning either the minimal or maximum number of colours required to colour each type of picture. This phase of the problem will be longer and should involve experimentation coupled with reasoning.

A key aspect is to understand the underlying mathematical aspects of the images, and students will eventually realise that the key aspects concern the sort of mathematics underlying Euler's formula.

The second part involves an almost combinatorical categorisations of regions. Working as a group, students can individually invent images with 2, 3 or 4 regions and bring those to the board when ready. With each new image the class could collectively decide whether or not the image is genuinely 'new and different' or simply a distortion of an existing image. Eventually, no new images will be
found. The focus can then switch to the question of proving that there are no new topologically different images.

What are the most important mathematical aspects of the two images?

How might you represent the images mathematically?

If you pick a region to colour first, how far can you colour the image using only two colours?

Why are the first pair of images topologically the same, whereas the second pair are not?

More able students should place their focus on providing the cleanest, simplest possible explanations. They could also look at the follow on problem Torus Patterns.

The reasoning required in this problem could be developed via the game Colouring Curves Game.