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)



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.