3. In this question you will count the number of ways to colour the vertices of a triangle
using N colours, up to symmetries of the triangle. Adjacent vertices are allowed to be
the same colour. The group of symmetries of a triangle has order 6; it consists of the
identity, 2 rotations and 3 reflections.
(a) How many ways can you colour the vertices in total (i.e., not up to symmetry)?
(b) How many colourings are fixed by a rotation?
(c) How many colourings are fixed by a reflection?
(d) Use Burnside's lemma to count the number of colourings up to symmetry.
(e) Give representatives of all of the colourings up to symmetry when N = 2.
(f)* How many ways can you colour the vertices of a square up to symmetry?
