Suppose there is a hexagon in the plane. Consider two colorings of the edges of the hexagon equivalent if you can rotate the hexagon so that edges of the same color map to each other. Suppose you color the sides black and white such that three edges are black and three are white. Describe the possible equivalence relations and the sizes of their equivalence classes.
Workings:
For simplicity's sake I'm going to illustrate this as a line:
Let $B$ denote a black edge.
Let $W$ denote a white edge.
Some possible arrangements are:
$B-B-B-W-W-W$ (Three Black edges "in a row")
$B-W-B-W-B-W$ (Black and White edges interchanging)
$B-B-W-W-B-W$ (Two Black lines in a row)
I believe these are the equivalence classes though I am not sure.
Any help will be appreciated.
Whether or not these are your equivalence classes depends heavily on what your equivalence relation is. For example if the relation is that two hexagons are equivalent iff there exists a way to make their colors match up exactly by performing a sequence of rotations and reflections about an axis of symmetry, then the classes would be the ones you listed above.
If however you remove the reflections part of that relation you would have one more equivalence class than you have listed above, namely $\;W-W-B-B-W-B$.
(I realize that this is worded somewhat awkwardly, so please leave comments if something is unclear.)