Faro shuffle
Encyclopedia
The faro shuffle is a method of shuffling playing card
s.
In a perfect shuffle or perfect faro shuffle, the deck is split into equal halves of 26 cards which are then pushed together in a certain way so as to make them perfectly interweave.
The faro shuffle is performed by cutting the deck into two, preferably equal, packs in both hands as follows (right-handed): The cards are held from above in the right and from below in the left hand. Separation of the deck is done simply lifting up half the cards with the right hand thumb slightly and pushing the left hand's packet forward away from the right hand. The two packets are often crossed and slammed into each other as to align them. They are then pushed together by the short sides and bent (either up or down). The cards will then alternately fall into each other, much like a zipper
. A flourish can be added by springing the packets together by applying pressure and bending them from above. The faro shuffle is a controlled shuffle which does not fully randomize a deck.
A faro shuffle which leaves the original top card at the top and the original bottom card at the bottom is known as an out shuffle
; one which moves the original top card to second and the original bottom card to second bottom is known as an in shuffle
.
A perfect faro shuffle, where the cards are perfectly alternated, is considered one of the most difficult sleights of card manipulation
, because it requires the shuffler to cut the deck into two equal stacks and apply just the right pressure when pushing the half decks into each other. If one manages to perform eight perfect faro out-shuffles in a row, then the deck of 52 cards will be restored to its original order. If one can do perfect in-shuffles, then 26 shuffles will reverse the order of the deck and 26 more will restore it to its original order.
, a perfect shuffle can be considered to be an element of the symmetric group
.
More generally, in
, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them:
Formally, it sends
Analogously, the
-perfect shuffle permutation is the element of 
that splits the set into k piles and interleaves them.
The
-perfect shuffle, denote it 
, is the composition of the 
-perfect shuffle with an 
-cycle, so the sign of 
is:
The sign is thus 4-periodic:
The first few perfect shuffles are:
and 
are trivial, and 
is the transposition 
.
Playing card
A playing card is a piece of specially prepared heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic, marked with distinguishing motifs and used as one of a set for playing card games...
s.
In a perfect shuffle or perfect faro shuffle, the deck is split into equal halves of 26 cards which are then pushed together in a certain way so as to make them perfectly interweave.
The faro shuffle is performed by cutting the deck into two, preferably equal, packs in both hands as follows (right-handed): The cards are held from above in the right and from below in the left hand. Separation of the deck is done simply lifting up half the cards with the right hand thumb slightly and pushing the left hand's packet forward away from the right hand. The two packets are often crossed and slammed into each other as to align them. They are then pushed together by the short sides and bent (either up or down). The cards will then alternately fall into each other, much like a zipper
Zipper
A zipper is a commonly used device for temporarily joining two edges of fabric...
. A flourish can be added by springing the packets together by applying pressure and bending them from above. The faro shuffle is a controlled shuffle which does not fully randomize a deck.
A faro shuffle which leaves the original top card at the top and the original bottom card at the bottom is known as an out shuffle
Out shuffle
An out shuffle is a type of perfect shuffle done in two steps:#Split the cards exactly in half and then#Interweave each half of the deck such that every-other card came from the same half of the deck....
; one which moves the original top card to second and the original bottom card to second bottom is known as an in shuffle
In shuffle
An in shuffle is a type of perfect shuffle done in two steps:#Split the cards exactly in half and then#Interweave each half of the deck such that every-other card came from the same half of the deck....
.
A perfect faro shuffle, where the cards are perfectly alternated, is considered one of the most difficult sleights of card manipulation
Card manipulation
Card magic is the branch of conjuring that deals with creating magical effects using a deck of playing cards. Card magic is commonplace in magical performances, especially in close up magic or parlor magic and street magic. Some of the most recognized names in this field include John Scarne, Juan...
, because it requires the shuffler to cut the deck into two equal stacks and apply just the right pressure when pushing the half decks into each other. If one manages to perform eight perfect faro out-shuffles in a row, then the deck of 52 cards will be restored to its original order. If one can do perfect in-shuffles, then 26 shuffles will reverse the order of the deck and 26 more will restore it to its original order.
Group theory aspects
In mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a perfect shuffle can be considered to be an element of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
.
More generally, in
, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them:Formally, it sends
Analogously, the
-perfect shuffle permutation is the element of that splits the set into k piles and interleaves them.The
-perfect shuffle, denote it , is the composition of the -perfect shuffle with an -cycle, so the sign of is:The sign is thus 4-periodic:
The first few perfect shuffles are:
and are trivial, and is the transposition .