I'm considering a loop to be a sequence of moves that brings a Rubik's cube configuration back to its original state. One example is if you turn any face a quarter turn four times, or if you turn one face a quarter turn and then reverse it. My question is, is there a loop that consists of an odd number of moves (considering a half turn to be two moves, obviously), and if there is, what is its minimum size?
Is there a loop consisting of an odd number of moves for the Rubik’s cube
group-theoryrubiks-cube
Best Answer
recapping the comments:
One can view the rubik's group as a subgroup of $S_{54}$ and within the context of $S_{54}$ you can view a quarter turn of a face move as the composition of five length-4 cycles.
(More specifically, using blindfold notation of the stickers and capitals to indicate corners versus lowercase to indicate edges, you could view the move R as the permutation $(C~Q~W~K)(B~T~V~J)(M~N~O~P)(b~t~v~j)(m~n~o~p)$)
It is well known that in the context of permutation groups you may refer to the parity of the permutation (the even-ness or the odd-ness) and one of the results of such study is that if you were to express an even permutation (such as the identity) as the composition of several other permutations, the number of odd permutations appearing in such a composition must have been even and could not have been odd. This is just like how in numbers, if you add an odd number of odd numbers together (and any number of even numbers) then the result will be odd and not even.
Further, it is known that a permutation is odd if and only if the number of even-length cycles in its disjoint cyclic decomposition is odd. As such, the move R (and indeed all other quarter-turn face moves) are odd permutations.
It follows then that the answer to your question is that no, there do not exist any sequences of moves consisting only of quarter face turns that are of odd-length which are equivalent to the identity.