Say you're making hex tiles with roads (like in Civ 5), but some of the tile variations are just rotations/reflections of others. How many unique tiles do you need? Why, obviously (2⁶ + 2⋅2 + 2⋅2² + 2³ + 3⋅2³ + 3⋅2⁴) / 12 = 13.

Let me tell you about Burnside's lemma. (1/n) The lemma is a beautiful result from group theory that talks specifically about enumerating things that are specified up to some transformations. But do we really need some cryptic lemma for that? On the image above I've easily managed to figure out all 13 cases. (2/n)