A perfect shuffle of a deck of cards can be represented by the following permutation $f\in S_{52}$: $f(x) =\left\{ \begin{array}{ll} 2x-1 & \text{if }x\in\{1,\ldots,26\}\\ 2(x-26) &\text{if }x\in \{27,\ldots,52\} \end{array}\right.$
I'm trying to show that if one performs 8 perfect shuffles of a deck of cards, then this returns the cards to their original position.
What I did was get some cycle decomposition:
(2 3 5 9 17 33 14 27) and (4 7 13 25 49 46 40 28) and
when I tried out starting with 6, I am getting a mess since f(31) = 10, but f(6) = 10
How many other cycle decompositions are there?