You have to consider all possible symmetries and spins using Burnside's lemma and MUST use all colors. For now I figured that for a simple spin around an axis you get: 0 deg = 6! 90 deg = 270 deg = 6!/4 180 deg = 6!/2

But I'm not sure if it's correct. What's the right way to do this? Thank you!

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    $\begingroup$ Must you use all six colors? Or may you repeat colors? If you must use all six colors, it is easier to approach via direct counting. One of the sides must be color $1$, orient that to the top. Choose the color for the opposite side. Then from the remaining colors, take the "smallest" and orient that to be the front face. Choose colors for the remaining. This gives final total of $5\cdot 3\cdot 2\cdot 1=30$ arrangements. $\endgroup$ – JMoravitz May 29 '17 at 5:12
  • $\begingroup$ In the case that you are allowed to repeat colors, then you should not be using factorials at all in your calculations. There are $6^6$ possible ways to color a cube ignoring symmetries. For rotations, you need to keep an eye on the axis of rotation, and don't forget that you can spin along an axis on opposite corners as well. E.g. spinning across axis going through centers of top and bottom faces by $90^\circ$ will give $6\cdot 6\cdot 6=6^3$ arrangements (choose top, choose bottom, and then choose front face colors: all remaining colors must same as front face). $\endgroup$ – JMoravitz May 29 '17 at 5:17
  • $\begingroup$ Does "color a cube" mean that you assign a color to each of the $12$ edges? Or does it mean that you color the vertices? Faces? Diagonals? $\endgroup$ – bof May 29 '17 at 6:00
  • $\begingroup$ You must paint a face color 1 so paint that first and you might as well orient that to the top. You have 5 choices for the opposite/bottom face. Call that color 2. You must have a face color 2 so just paint one. Any face can be the basis of orientation. Once done though you have fixed the cube. There are three more colors so 3!= 6 choices left. A total of 30 ways. $\endgroup$ – fleablood May 29 '17 at 6:23
  • $\begingroup$ This type of question has appeared frequently for example at this MSE link. $\endgroup$ – Marko Riedel May 29 '17 at 20:55

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