In-Shuffle
A riffle shuffle, in which the top half of the deck is placed in the left hand, and cards are then alternatively interleaved from the left and right hands. Using an in-shuffle, a deck originally arranged as 1 2 3 4 5 6 7 8 would become 5 1 6 2 7 3 8 4. The ordering of a deck of 52 cards after an in-shuffle is given by 27, 1, 28, 2, 29, 3, ... (OEIS A059952).
In general, in-shuffling a deck of 
cards once moves
card 
to the position originally occupied by the 
th card (mod 
) (Conway and
Guy 1996). Therefore, in-shuffling an even number 
of cards 
times when 
is prime results in the original card order. This means that
an ordinary deck of 52 cards is returned to its original order after 52 in-shuffles.
The numbers of in-shuffles needed to return a deck of 
, 4, ... to its
original order are 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, ... (OEIS A002326),
which is simply the multiplicative order
of 2 (mod 
).
Contact the MathWorld Team
cards
