# How many ways are there to color a square prism with two colors?

How many ways are there to color a square prism with two colors (black and white) such that repetition is allowed and all faces can be the same color keeping in mind orientation?

I counted out the cases and got 18 but I wanted to know if there is an easier way to answer this.

Thanks!

• This is a job for Burnside's lemma (en.wikipedia.org/wiki/Burnside%27s_lemma) but it's been a while since I've used it so I don't want to do it myself. – Michael Lugo Dec 4 '20 at 20:41
• Do you have 6 faces to color? R rotations of a coloring allowed (what you mean by keeping in mind orientation)? – Moti Dec 5 '20 at 5:33
• For all sides black you count 3 or 4? – Moti Dec 5 '20 at 5:35

## 1 Answer

A square prism (not a cube) has $$8$$ rotational symmetries:

• identity leaves $$2^6$$ colourings fixed
• two $$90^\circ$$ rotations about the unique axis leave $$2^3$$ colourings fixed each
• $$180^\circ$$ rotation about unique axis leaves $$2^4$$ colourings fixed
• two $$180^\circ$$ rotations about axes perpendicular to the side faces leave $$2^4$$ colourings fixed each
• two $$180^\circ$$ rotations about axes parallel to diagonals of the square top and bottom leave $$2^3$$ colourings fixed each

Thus, by Burnside's lemma, the number of colourings is $$\frac{2^6+2×2^3+2^4+2×2^4+2×2^3}8=18$$