Faro shuffle

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The faro shuffle is a method of shuffling playing cards.

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 tricks 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.

[edit] Group theory aspects

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

More generally, in S2n, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them:

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

Formally, it sends

k \mapsto \begin{cases} 2k-1 & k\leq n\\ 2(k-n) & k> n \end{cases}

Analogously, the (k,n)-perfect shuffle permutation[1] is the element of Skn that splits the set into k piles and interleaves them.

The (2,n)-perfect shuffle, denote it ρn, is the composition of the (2,n − 1)-perfect shuffle with an n-cycle, so the sign of ρn is:

sgn(ρn) = ( − 1)n + 1sgn(ρn − 1).

The sign is thus 4-periodic:

\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: ρ0 and ρ1 are trivial, and ρ2 is the transposition (23) \in S_4.

[edit] Footnotes

  1. ^ The Cycles of the Multiway Perfect Shuffle Permutation

[edit] See also

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