  We investigate the mathematics behind unshuffles, a type of card shuffle
closely related to classical perfect shuffles. To perform an unshuffle, deal
all the cards alternately into two piles and then stack the one pile on top of
the other. There are two ways this stacking can be done (left stack on top or
right stack on top), giving rise to the terms left shuffle ($L$) and right
shuffle ($R$), respectively. We give a solution to a generalization of
Elmsley's Problem (a classic mathematical card trick) using unshuffles for
decks with $2^k$ cards. We also find the structure of the permutation groups
$\langle L, R \rangle$ for a deck of $2n$ cards for all values of $n$. We prove
that the group coincides with the perfect shuffle group unless $n\equiv 3 \pmod
4$, in which case the group $\langle L, R \rangle$ is equal to $B_n$, the group
of centrally symmetric permutations of $2n$ elements, while the perfect shuffle
group is an index 2 subgroup of $B_n$.
