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
For the question of toplogy, ask yourself why the first of these
two images are topologically the same whereas the second pair are