Painting by Numbers
How many different colours of paint would be needed to paint these
pictures by numbers?
Age
16 to 18
Challenge level
Problem
With painting by numbers, a line drawing is split into regions which are to be painted according to the rules that:
1) No two regions of the same colour can share a border
2) Two regions of the same colour are allowed to meet at a point.
Consider these two images, one made from intersecting ellipses and one made from overlapping rectangles (include the outer square boundaries in the image)
Image
Image
Part 1
What is the smallest number of colours needed to colour these pictures according to the colouring rules? Prove your result clearly.
Consider the more general question of colouring other pictures created from intersecting ellipses or overlapping squares. Can you form any well-considered conjectures, prove any results or find any interesting examples?
Part 2
Imagine that painting by numbers pictures are draw on a sheet of rubber which can be distorted - stretched or shrunk at any point. How many 'topologically different' sorts of pictures would there be with two, three or four regions? How could these be painted according to the rules?
Extension: Try the Torus Patterns problem.
NOTES AND BACKGROUND
The issues raised in this problem concern a famous mathematical theorem the proof of which was surrounded by difficulty and controversy. You can read about this in our fascinating mathematical history article 'The Four Colour Theorem'. The issues raised by the theorem are accessible at school level, but full understanding takes you right up to university level pure mathematics.
Mathematics and art are in many ways closely related. You can read more about the mathematics of art on the Plus website pages
Art + Math = x by Carla Farsi with Marianne Freiberger
Teacher package: Maths and art contains many fascinating links and articles
Getting Started
To get started, simply try colouring in some pictures. This is the best way to get a feel for the structure of the problem.
Once you have made a conjecture, you will be able to test it out on other pictures to gain confidence in its validity, or to reject it if you create a counter example.
To prove a conjecture ask yourself the question: what could go wrong to make it false? This is a good way to focus on the key underlying mathematical structure.
To begin with, you can certainly pick a region and give it a colour. The colour of certain other regions will necessarily follow. Try this approach and see where you and up with a choice.
For the question of toplogy, ask yourself why the first of these two images are topologically the same whereas the second pair are not:
Image
Teachers' Resources
Why do this problem?
This problem gives a visual and engaging framework in which to practice mathematical thinking, conjecturing and start to grapple with issues of topology.
Possible approach
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.
Key questions
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?
Image
Possible extension
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.
Possible support
The reasoning required in this problem could be developed via the game Colouring Curves Game.