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# Shuffling playing cards

The riffle

A deck of playing cards is randomized by a procedure called shuffling to provide an element of chance in card games. There are several techniques for this; the most common is called a "riffle", in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table intertwined.

This can also be done by placing the halves flat on the table with their rear corner touching, then lifting the back edges with the thumbs while pushing the halves together. While this method is a bit more difficult, it is often used in casinos because it minimizes the risk of exposing cards during the shuffle.

Another procedure is called "stripping", where small groups of cards are removed from the top or bottom of a deck and replaced on the opposite side (or just assembled on the table in reverse order). This is a much less effective randomizing procedure, and is not recommended unless used in conjuction with riffling.

"Pushing" is the procedure of pushing the ends of two halves of a deck against each other in such a way that they naturally intertwine. This requires skill and practice, as does "fanning", which involves spreading the halves into fan shapes and intertwining them.

Shuffling is often followed by a cut.

The mathematician and magician Persi Diaconis is an expert on the theory and practice of card shuffling, and an author of a famous paper on the number of shuffles needed to randomize a deck, concluding that it did not start to become random until 5 good riffle shuffles, and was truly random after 7. (You would need more shuffles if your shuffling technique is poor of course.) Recently, the work of Trefethen et al. has questioned some of Diaconis' results, concluding that 6 shuffles is enough. The difference hinges on how each measured the randomness of the deck. Diaconis used a very sensitive test of randomness, and therefore needed to shuffle more. Even more sensitive measures exist and the question of what measure is best for specific card games is still open.

Here is an extremely sensitive test to experiment with. Take a standard deck without the jokers. Divide it into suits with 2 suits in ascending order from ace to king, and the other two suits in reverse. (A brand new deck already comes ordered this way.) Shuffle to your satisfaction. Then go through the deck trying to pull out each suit in the order ace, two, three.. When you reach the top of the deck start over. How many passes did it take to pull out each suit?

What you are seeing is how many rising sequences are left in each suit. It probably takes more shuffles than you think to both get rid of rising sequences in the suits which were assembled that way, and add them to the ones that weren't!

In practice the number of shuffles that you need depends both on how good you are at shuffling, and how good the people playing are at noticing and using non-randomness. 2-4 shuffles is good enough for casual play. But in club play good bridge players take advantage of non-randomness after 4 shuffles, and top blackjack players literally track aces through the deck.

## References

• D. Aldous and P. Diaconis, "Shuffling cards and stopping times", American Mathematical Monthly 93 (1986), 333-348.
• Trefethen, L. N. and Trefethen, L. M. "How many shuffles to randomize a deck of cards?" Proceedings of the Royal Society London A 456, 2561 - 2568 (2000)