A perfect $(n,k)$ shuffle function Suppose you have a deck of $n$ cards; e.g., $n{=}12$:
$$
(1,2,3,4,5,6,7,8,9,10,11,12) \;.
$$
Cut the deck into $k$ equal-sized pieces, where $k|n$;
e.g., for $k{=}4$, the $12$ cards are partitioned into
$4$ piles, each of $m=n/k=3$ cards:
$$
\left(
\begin{array}{ccc}
 1 & 2 & 3 \\
 4 & 5 & 6 \\
 7 & 8 & 9 \\
 10 & 11 & 12 \\
\end{array}
\right) \;.
$$
Now perfectly shuffle them by selecting the top card from stack $1$,
the top card from stack $2$, and so on, 
walking down the columns of the matrix above,
resulting in this shuffled deck of cards:
$$
(1,4,7,10,2,5,8,11,3,6,9,12) \;.
$$
Continue in this manner until the deck of cards returns to its initial sorting:
$$
\left(
\begin{array}{cccccccccccc}
 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
 1 & 4 & 7 & 10 & 2 & 5 & 8 & 11 & 3 & 6 & 9 & 12 \\
 1 & 10 & 8 & 6 & 4 & 2 & 11 & 9 & 7 & 5 & 3 & 12 \\
 1 & 6 & 11 & 5 & 10 & 4 & 9 & 3 & 8 & 2 & 7 & 12 \\
 1 & 5 & 9 & 2 & 6 & 10 & 3 & 7 & 11 & 4 & 8 & 12 \\
 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\end{array}
\right) \;.
$$
Here, for $n{=}12$ cards partitioned into $k{=}4$ parts, it requires $s{=}5$ 
perfect shuffles to
cycle.
Let us say that $f(n,k)=s$, i.e., $f(12,4)=5$.
Similarly I can calculate that 
$$
f(8,2)=3,\; f(18,3)=16,\; f(33,3)=8, \; f(52,2)=8,
$$
etc. The last represents a perfect "outer-shuffle" of a standard
$52$-card deck, which is known to take $8$ shuffles to cycle.
It seems likely this function is known to combinatorialists:

Q. What is $f(n,k)$?


This is just my comment above, which seems to answer the question.  Label the cards from $0$ to $n-1$.  Then, with $m=n/k$, the shuffle in the question corresponds to multiplying card $i$ by $m$ (taken mod $n-1$).  Thus repeating the shuffle $r$ times  amounts to multiplying by $m^r \pmod{n-1}$, which returns us to the original configuration after the order of $m \pmod{n-1}$ times.  Since $mk\equiv 1\pmod{n-1}$, this is also the order of $k \pmod{n-1}$.  For more information on perfect shuffles see Diaconis, Graham, Kantor.
