An out-shuffle, also known as a perfect shuffle (Golomb 1961), is a riffle shuffle in which the top half of the deck is placed in the right hand, and cards
are then alternatively interleaved from the right and left hands. In other words,
an out-shuffle on a deck of 
cards separates the bottom 
cards from the top 
cards and precisely interleaves them, with the bottom card
remaining on the bottom (Golomb 1961).
Using an out-shuffle, a deck originally arranged as 1 2 3 4 5 6 7 8 would become 1 5 2 6 3 7 4 8. The ordering of a deck of 52 cards after an out-shuffle is given by 1, 27, 2, 28, 3, 29, ... (OEIS A059953).
Out-shuffling an even number 
cards 
times when 
is prime results in the original order (Conway and Guy 1996).
The numbers of out-shuffles needed to return a deck of 
, 4, ... to its original order are 1, 2, 4, 3, 6, 10, 12,
4, 8, 18, 6, 11, ... (OEIS A002326), which
is simply the multiplicative order of 2 (mod
). For example, a deck of 52 cards
therefore is returned to its original state after eight out-shuffles, since 
(Golomb 1961). The smallest
numbers of cards 
that require 1, 2, 3, ... out-shuffles to return to the deck's original state are
1, 2, 4, 3, 16, 5, 64, 9, 37, 6, ... (OEIS A114894).
An out-shuffle on an infinite deck was considered by Gale (1992). Take an infinite deck of cards labeled 1, 2, 3, 4, 5, 6, .... At step 
, pick up the top 
cards and interlace them with the next 
cards. This is called a perfect 
-shuffle. For example, after step two, we have 3, 2, 4, 1,
5, 6, 7, .... For step three, pick up 3, 2, 4 and shuffle them in, giving 1, 3, 5,
2, 6, 4, 7, 8, 9, .... Iterate this process. It is conjectured that eventually every
number appears on top of the deck.
The cards on top of deck at the 
th step are 1, 2, 3, 1, 6, 5, 9, 1, 4, 2, 16, 10, 12, ... (OEIS
A035485). The step at which card 
first appears on top the deck is given by 0, 1, 2, 8, 5, 4,
78, 37, ... (OEIS A035490). The position of
the first card after the 
th
shuffle is 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, 7, 14, 28, ... (OEIS A035492).
The order in which new cards appear on top for the first time is 1, 2, 3, 6, 5, 9,
4, 16, 10, ... (OEIS A035493). The order in
which record new high cards appear on top for the first time is 1, 2, 3, 6, 9, 16,
... (OEIS A035494).
