# Faro shuffle

The faro shuffle (American), weave shuffle (British), riffle shuffle, or dovetail shuffle is a method of shuffling playing cards. Mathematicians use "faro shuffle" for a shuffle in which the deck is split into equal halves of 26 cards that are then interwoven perfectly.[1]

Magicians use these terms for a particular technique (which Diaconis, Graham, and Kantor call "the technique")[2] for achieving this result. A right-handed practitioner holds the cards from above in the right and from below in the left hand. The deck is separated into two preferably equal parts by simply lifting up half the cards with the right thumb slightly and pushing the left hand's packet forward away from the right hand. The two packets are often crossed and tapped against each other to align them. They are then pushed together on the short sides and bent either up or down. The cards will then alternately fall onto each other, ideally alternating one by one from each half, much like a zipper. A flourish can be added by springing the packets together by applying pressure and bending them from above.[3]

A game of faro ends with the cards in two equal piles that the dealer must combine to deal them for the next game. According to the magician John Maskelyne, the above method was used, and he calls it the "faro dealer's shuffle".[4] Maskelyne was the first to give clear instructions, but the shuffle was used and associated with faro earlier, as discovered mostly by the mathematician and magician Persi Diaconis.[5]

A faro shuffle that leaves the original top card at the top and the original bottom card at the bottom is known as an out-shuffle; one that moves the original top card to second and the original bottom card to second from the bottom is known as an in-shuffle. These names were coined by the magician and computer programmer Alex Elmsley.[6] 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.

The faro shuffle is a controlled shuffle that does not fully randomize a deck. 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.[7] If an additional 12 cards are added to the deck, so that the deck contains 64 cards, only 6 faro shuffles are required to maintain the order of a deck, compared to 8 shuffles for a 52-card deck.

## Group theory aspects

In mathematics, a perfect shuffle can be considered an element of the symmetric group.

More generally, in ${\displaystyle S_{2n}}$, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them:

${\displaystyle {\begin{pmatrix}1&2&3&4&\cdots \\1&n+1&2&n+2&\cdots \end{pmatrix}}}$

In other words, it is the map

${\displaystyle k\mapsto {\begin{cases}\left\lceil {\frac {k}{2}}\right\rceil &k\ {\text{odd}}\\n+{\frac {k}{2}}&k\ {\text{even}}\end{cases}}}$

Analogously, the ${\displaystyle (k,n)}$-perfect shuffle permutation[8] is the element of ${\displaystyle S_{kn}}$ that splits the set into k piles and interleaves them.

The ${\displaystyle (2,n)}$-perfect shuffle, denoted ${\displaystyle \rho _{n}}$, is the composition of the ${\displaystyle (2,n-1)}$-perfect shuffle with an ${\displaystyle n}$-cycle, so the sign of ${\displaystyle \rho _{n}}$ is:

${\displaystyle {\mbox{sgn}}(\rho _{n})=(-1)^{n+1}{\mbox{sgn}}(\rho _{n-1}).}$

The sign is thus 4-periodic:

${\displaystyle {\mbox{sgn}}(\rho _{n})=(-1)^{\lfloor n/2\rfloor }={\begin{cases}+1&n\equiv 0,1{\pmod {4}}\\-1&n\equiv 2,3{\pmod {4}}\end{cases}}}$

The first few perfect shuffles are: ${\displaystyle \rho _{0}}$ and ${\displaystyle \rho _{1}}$ are trivial, and ${\displaystyle \rho _{2}}$ is the transposition ${\displaystyle (23)\in S_{4}}$.