We investigate the mathematics behind unshuffles, a card shuffling technique closely
related to classical perfect shuffles. To perform an unshuffle, deal all cards alternately
into two piles and then stack one pile on top of the other. There are two ways this
can be done (left pile on top or right pile on top), giving rise to the terms left shuffle
() and right
shuffle (),
respectively. We solve a generalization of Elmsley’s problem (a
classic mathematical card trick) using unshuffles for decks with
cards. We also find the structure of the permutation groups
for decks
of
cards
for all
.
The group coincides with the perfect shuffle group unless
, in which
case the group
is equal to
,
the group of centrally symmetric permutations of
elements, while the perfect shuffle group is an index-2 subgroup of
.
Keywords
perfect shuffle, permutation group, symmetric group
