A riffle shuffle is a shuffle in which a deck is divided into two packets whose cards are interleaved while preserving their original order within each packet. A perfect riffle shuffle, also called a Faro shuffle, divides a deck of cards into equal halves and interleaves their cards in strict alternation. Reading the shuffled deck from top to bottom, an in-shuffle begins with a card from the bottom half, while an out-shuffle begins with a card from the top half. Using an in-shuffle, a deck originally arranged from top to bottom as 1 2 3 4 5 6 7 8 would become 5 1 6 2 7 3 8 4. Using an out-shuffle, the deck order would become 1 5 2 6 3 7 4 8. Riffle shuffles are used in card tricks (Marlo 1958a,b, Adler 1973), and also in the theory of parallel processing (Stone 1971, Chen et al. 1981).

For a positive even number of cards, an out-shuffle is implemented in the Wolfram Language using Riffle, for example:

  Riffle @@ Partition[Range[n], n/2, n/2, 1, {}]

For a perfect riffle shuffle, card moves to the position originally occupied by the th card modulo . For an in-shuffle, the first card is numbered 1 and the multiplication is done modulo . For an out-shuffle, the first card is numbered 0 and the multiplication is done modulo . Note that (in the out-shuffle case) this maps the first and last card to 0, but this makes sense, because they are both fixed points.

Therefore, in-shuffling an even number cards times when is prime results in the original card order. Similarly, out-shuffling an even number cards times when is prime results in the original order (Diaconis et al. 1983, Conway and Guy 1996). An ordinary deck of 52 cards is returned to its original order after 52 in-shuffles, but after only eight out-shuffles!

In the Gilbert-Shannon-Reeds model, the cut size has a binomial distribution with parameters and , and each interleaving that preserves the order within the two packets is chosen with equal probability. For independent shuffles in this model, Aldous (1983, p. 266) showed that the mixing time at any fixed total variation distance threshold strictly between 0 and 1 is asymptotic to as . Aldous and Diaconis (1986, p. 345) suggested seven riffle shuffles to get close to random for a deck of 52 cards. For this deck size, Bayer and Diaconis (1992, p. 296) give total variation distance from the uniform distribution of approximately 0.334 after seven shuffles, 0.167 after eight, and 0.085 after nine.

Morris (1994) discusses aspects of the perfect riffle shuffle (in which the deck is cut exactly in half and cards are perfectly interlaced). Ramnath and Scully (1996) give an algorithm for the shortest sequence of in- and out-shuffles to move a card from arbitrary position to position . This algorithm works for any deck with an even number of cards and is .