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I can find everywhere (e.g. wikipedia, ruwix.com and MIT) the information that the standard $3\times3$ Rubik's cube can be scrambled in $4.3 \times 10^{19}$ different configurations. These are calculated keeping the center cubies in a defined position (say, white on top and green in front), so that rotations of the cube are already ruled out. But I cannot find anywhere the number of "essentially different" arrangements, in the sense I'm going to (try to) explain.

Suppose that from the solved state I perform just one $90°$ clockwise rotation of the face on the right (the move called "R"). Let me call the new configuration A. Then, starting again from the solved state, I perform a single $90°$ clockwise rotation on the face on the left (move "L"), obtaining configuration B. Now, among the $43$ quintillions, A and B are counted separately as different permutations; but I don't consider them to be essentially different: after a $180°$ rotation about the vertical axis and a reassignment of the colors (swapping green $\leftrightarrow$ blue and red $\leftrightarrow$ orange, if the cube respects the standard color scheme) the two configurations become the same. I'd call this arrangment "cube with one face turned $90°$ clockwise". In fact another way to see the equivalence is that, after a proper rotation of the cube in my hands, I can reach the solved state from either A or B with the same sequence of moves (in this simple case, one single counterclockwise rotation).

There are four other configurations that I consider to be equivalent to A and B, being essentially the same arrangement, for a total of six (one for each of the faces that can undergo the initial rotation). Yet I cannot just divide by $6$ the $43$ quintillion figure hoping to get the number of arrangements, since not all of them has the same multiplicity. For example if, from the starting position, I turn clockwise both the right and the left face I find an arrangement (if needed, let's call this C) that has only two other equivalents; therefore three configurations among the $43$ quintillions (one for each of the main axes of the cube) that should count as one arrangement (= "two opposite faces turned $90°$ clockwise").

It reminds me the caution needed while counting the possible polyominoes, depending on whether one considers "free", "one-sided" or "fixed" ones (see here). I don't know shortcuts to avoid checking each element's geometrical symmetries to understand its multiplicity. Also related is the assessment of the possible latin squares (or their widespread counterpart, sudokus): it's not the actual numbers/symbols (that can always be relabelled, see this article and the cyted paper) that matter, but their "pattern". It seems to me that my Rubik's cube question mixes both concerns, and the number of possible configurations prevents any hope to brute-force an answer. Are their smart arguments to tackle the problem? Is there someone who has already asked (and answered?) my question?

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This message on the Cube Lover's Archive calculates this number as 901,083,404,981,813,616 but note that they factor out not only rotations (which would be almost a factor 24) but also reflections (for a factor of almost 48).

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  • $\begingroup$ Thank you, indeed that is a quite challenging reading for me! Meanwhile, I believe that factoring out reflections too would not imply an exact extra factor of $2$: there are arrangements that are symmetric under reflection (e.g. the easy checkerboard, the solved state itself...), so they are counted once in both cases. Anyway I don't know, they might be a negligible fraction of the possible arrangements. $\endgroup$ – lesath82 Nov 4 '20 at 11:15
  • $\begingroup$ @lesath82 Yes, it is not quite an extra factor of 2, and it is indeed only a very tiny fraction of the arrangements that cause the difference in any case since so few arrangements have symmetry. The number I quoted is 1/47.9999998 times the normal number of arrangements. $\endgroup$ – Jaap Scherphuis Nov 4 '20 at 12:01
  • $\begingroup$ Wow, that means that "almost every" (not in a rigorous mathematical sense!) configuration is significantly affected by all 48 symmetries of the cube! $\endgroup$ – lesath82 Nov 4 '20 at 14:51
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    $\begingroup$ @lesath82 Yes. Basically, almost all arrangements are asymmetric - the asymmetric positions vastly outnumber the symmetric ones. In any symmetric arrangement, the locations of (at least) half the pieces is determined by the other half. So out of all the ways to arrange the second half, only one has that particular symmetry, while the rest do not. Of course this is complicated by symmetries interacting, but does give you a feel for why symmetric arrangements are so rare. $\endgroup$ – Jaap Scherphuis Nov 4 '20 at 15:11
  • $\begingroup$ If someone like me is interested if the cube can be used to send secret messages, the information contents is slightly above 63bits. $\endgroup$ – Gyro Gearloose Nov 19 '20 at 17:32

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