The Best Way to Shuffle a Deck of Cards

How to ensure a statistically random distribution

Whenever I play cards there’s a complaint commonly uttered by friends and family alike:

Who shuffled these?

Whether the disgust in their voice is genuine or they are merely bluffing to conceal a good hand I wonder about the art of shuffling. Is there a method to guarantee a statistically random distribution of cards throughout the deck?

Since a deck is made up of 52 unique cards there are 52 factorial possible arrangements the deck could be in (52 factorial is the same as 80,658 vigintillion, a number with 68 digits), guaranteeing virtually no chance that you will ever play the same game twice. Shuffling disorganizes the cards into these unique arrangements, but how can a high degree of randomness in the deck be ensured?

Randomness: the quality or state of lacking a pattern or principle of organization; unpredictability

It is important to note that the appearance of order may still arise from a randomized deck of cards. If the order is truly random, nothing prevents pairs, straights, or even a perfectly ordered deck resulting from a good shuffle. No outcome is more or less probable than another.

Since every possible random arrangement of 52 cards is equally probable, the perceived degree of disorder is not the goal. Instead, the method of randomizing the deck should aim to maximize the potential for disorder.

So, how good is shuffling at creating disorder?

Shuffling

One of the most common methods of shuffling is called riffling. The basic steps are:

  1. Divide the deck in half
  2. Successively overlay the cards, alternating between each half of the deck
  3. Repeat

Shuffling like this has a very predictable outcome, the opposite of our definition of randomness. If executed perfectly (dividing the deck exactly in half and never skipping a card when alternating from left and right), this method will actually return the cards to how they started after either 8 or 52 shuffles, depending on which half starts interleaving. Any degree of randomness is solely due to inaccuracies while manipulating the deck.

In practice, a sufficient level of randomness is achieved after 7 shuffles.

Besides being predictable, there is also a limited amount of disorder that can occur during a single shuffle. Since the deck is halved before being shuffled back together, a card can only move halfway through the deck at most.

For instance, if 10 cards are shuffled using the riffle technique, when the bottom of the top half, the 5th card, is laid on top of the bottom of the bottom half, the 10th card, it becomes the new 9th card. The 10th card doesn’t change position at all, but the new 9th card changes 4 positions.

A similar pattern continues through the rest of the deck, with the old 9th and 4th becoming the new 8th and 7th (changing 1 and 3 positions respectively), the old 8th and 3rd becoming the new 6th and 5th (2 positions each), and so on.

At the end of a single perfect shuffle of 10 cards the most a card will shift is 4 positions. With a standard 52-card deck the most a card will shift is 25 positions. Either way, the most a card can move is about halfway through the deck per shuffle.

To reach the maximum potential for disorder a card must be allowed to shift to any of the 52 positions after a single shuffle. Of the other methods to shuffle cards, such as the Overhand, Faro, and Hindi, each follows a pattern and relies on inaccuracies in execution to create randomness, save one notable exception.

Smooshing

Also known as a scramble, smooshing is a method of shuffling where the cards are placed face down and spread around over each other before being gathered back into a stack.

What this method lacks in finesse it makes up for in effectiveness. Any card can end up in any position in the deck after a single shuffle.

Unlike the riffle method, there is no semblance of order in how the cards are shuffled. The way the cards slide over or under each other is unpredictable and it is impossible to repeat the exact smooshing movement again and again. The potential for disorder is maxed out.

In practice, 1 shuffle lasting about a minute is more than enough to ensure a statistically random arrangement of cards.

So, what’s the best way to shuffle a deck of cards? Ultimately, the choice is yours: shuffle several times in an orderly fashion or only once with chaotic abandon. Ignore people when they lament their hand and just have fun playing yours.

Designer and developer who occasionally writes.

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