# Luke Morgan: Shuffling cards with groups

Source: Discrete mathematics seminar

**Povzetek.** There are two standard ways to shuffle a deck of cards, the in and out
shuffles. For the in shuffle, divide the deck into two piles, hold one
pile in each hand and then perfectly interlace the piles, with the top
card from the left hand pile being on top of the resulting stack of
cards. For the out shuffle, the top card from the right hand pile ends
up on top of the resulting stack.
Standard card tricks are based on knowing what permutations of the
deck of cards may be achieved just by performing the in and out
shuffles.

Mathematicians answer this question by solving the problem
of what permutation group is generated by these two shuffles.
Diaconis, Graham and Kantor were the first to solve this problem in
full generality - for decks of size 2**n*. The answer is usually “as big
as possible”, but with some rather beautiful and surprising
exceptions. In this talk, I’ll explain how the number of permutations
is limited, and give some hints about how to obtain different
permutations of the deck. I’ll also present a more general question
about a “many handed dealer” who shuffles *k***n* cards divided into *k
*piles.