Math Reveals How Many Shuffles Randomizes a Deck

Math Reveals How Many Shuffles Randomizes a Deck

Math — and some clever simulations — have revealed how many shuffles are required to randomize a deck of 52 cards, but there’s a bit more to it than that. There are different shuffling methods, and dealing methods can matter, too. [Jason Fulman] and [Persi Diaconis] are behind the research that will be detailed in an upcoming book, The Mathematics of Shuffling Cards, but the main points are easy to cover.


A riffle shuffle (pictured above) requires seven shuffles to randomize a 52-card deck. Laying cards face-down on a table and mixing them by pushing them around (a technique researchers dubbed “smooshing”) requires 30 to 60 seconds to randomize the cards. An overhand shuffle — taking sections from a deck and moving them to new positions — is a staggeringly poor method of randomizing, requiring some 10,000-11,000 iterations.


The method of dealing cards can matter as well. Back-and-forth dealing (alternating directions while dealing, such as pattern A, B, C, C, B, A) yields improved randomness compared to the more common cyclic dealing (dealing to positions in a circular repeating pattern A, B, C, A, B, C). It’s interesting to see different dealing methods shown to have an effect on randomness.


This brings up a good point: there is not really any such a thing as “more” random. A deck of cards is either randomized, or it isn’t. If even two cards have remained in the same relative positions (next to one another, for example) after shuffling, then a deck has not yet been randomized. Similarly, if seven proper riffle shuffles are sufficient to randomize a 52-card deck, there is not rea ..

Support the originator by clicking the read the rest link below.