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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!

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  • $\begingroup$ 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. $\endgroup$ – Michael Lugo Dec 4 '20 at 20:41
  • $\begingroup$ Do you have 6 faces to color? R rotations of a coloring allowed (what you mean by keeping in mind orientation)? $\endgroup$ – Moti Dec 5 '20 at 5:33
  • $\begingroup$ For all sides black you count 3 or 4? $\endgroup$ – Moti Dec 5 '20 at 5:35
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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 $$\frac{2^6+2×2^3+2^4+2×2^4+2×2^3}8=18$$

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