An out-shuffle, also known as a perfect shuffle (Golomb 1961), is a riffle shuffle in which a deck is divided into two equal halves whose cards are interleaved in their original order. Reading the shuffled deck from top to bottom, the first card comes from the top half. The original top and bottom cards remain in their respective positions (Golomb 1961).
Using an out-shuffle, a deck originally arranged from top to bottom 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 
of 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). For 
, this is the multiplicative
order of 2 modulo 
;
the two-card case is the identity shuffle. For example, a deck of 52 cards therefore
is returned to its original state after eight out-shuffles, since 
(Golomb 1961). The smallest half-deck sizes 
for which a deck of 
cards has exact out-shuffle period 1, 2, 3, ... are 1, 2,
4, 3, 16, 5, 64, 9, 37, 6, ... (OEIS A114894).
A related infinite-deck problem uses in-shuffles (Gale 1992). Begin with an infinite deck of cards labeled 1, 2, 3, 4, 5, 6, .... At step
, interleave the top 
cards with the next 
cards, placing the first card from the lower packet on top.
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 the deck at steps 
, 1, 2, ..., where step 0 is the initial state, 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 original first card
after 
,
1, 2, ... shuffles 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).
