In shuffle

In shuffle

An in shuffle is a type of "perfect" shuffle done in two steps:
#Split the cards exactly in half (a bottom half and a top half) and then
#Interweave each half of the deck such that every-other card came from the same half of the deck.

If this shuffle moves the top card to be 2nd from the top then it is an in shuffle, otherwise it is known as an out shuffle.


For simplicity, we will use a deck of six cards.

The following shows the order of the deck after each in shuffle. Notice that a deck of this size returns to its original order after 3 in shuffles.::


The number of in shuffles required to return a deck of cards of even size "N", to original order is given by the multiplicative order of 2 modulo ("N" + 1).

For example, for a deck size of "N" = 2, 4, 6, 8, 10, 12 ..., the number of out shuffles needed are: 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, ... (This is the integer sequence [ A002326] ).

For a standard deck of 52 playing cards, the number of in shuffles required to return the deck to its original order is 52.


*cite journal
last = Diaconis
first = P.
authorlink =
coauthors = R.L. Graham, and W.M. Kantor
title = The mathematics of perfect shuffles
journal = Advances in Applied Mathematics
volume = 4
issue = 2
pages = 175–196
publisher =
date = 1983
url =
doi = 10.1016/0196-8858(83)90009-X
id =
accessdate =

*cite journal
last = Kolata
first = Gina
authorlink =
coauthors =
title = Perfect Shuffles and Their Relation to Math
journal = Science
volume = 216
issue = 4545
pages = 505–506
publisher =
year = 1982
month = April
url =
doi =
id =
accessdate =

*cite book
last = Morris, S.B.
first = S. Brent
authorlink =
coauthors =
title = Magic Tricks, Card Shuffling and Dynamic Computer Memories
publisher = The Mathematical Association of America
date = 1998
location =
pages =
url =
doi =
isbn = 0883855275
ISBN status = May be invalid - please double check

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