This activity came from watching this Numberphile video . A worksheet of the activities below is available here.
Suppose that we want to colour the whole plane so that any two points at distance 1 from each other would have different colors.
You can imagine this as walking on a tiled floor in jumps of 1 and each jump has to be onto a different colour.
What is the minimum number of colours we would need to use?
Finding an upper bound to the number of colours needed
Task 1
It doesn’t have to be a regular tiling, but would 4 colours work if we coloured the plane like above? How big would the squares have to be?
Task 2
Find a number of colours that would work for a square tiling. How big would the squares have to be?
Task 3
Show that we would need at most 7 colours using a different shape tile (isometric paper will be helpful!).
Finding an lower bound to the number of colours needed
Task 4
Use this diagram to show that we would need at least 3 colours to colour the plane.
Task 5
Create a graph of points to show that we need at least 4 colours.
Further reading
This problem is called the Hadwiger–Nelson problem and was stated in 1950. It was recently (2018!) proved that you need at least 5 colours by an amaetur mathematician who was working on it as a break from his job as a maverick biologist intent on extending the human lifespan! The middle dot in the picture below has to be white rather than red, green, blue or yellow like the rest. It is still an open problem to find the smallest number of colors. It could now be 5, 6 or 7 and nobody knows.
Links:
https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/
https://www.theguardian.com/science/2018/may/04/60-year-old-maths-problem-partly-solved-by-amateur
https://mathworld.wolfram.com/Hadwiger-NelsonProblem.html