<PRE><B>Question 1199921</B>
.


The solution by @greenestamps is obviously incorrect,

since it allows the paths INSIDE the cube,

while, according to the condition, only the paths on the cube surface are allowed.



For a correct solution, see this link

https://www.maa.org/sites/default/files/pdf/pubs/Rubiks6.pdf


or this link

https://math.stackexchange.com/questions/740960/paths-on-a-rubiks-cube



The answer is 384 paths.



</PRE>