Mathematics can have unexpected effects on our lives,
such as in the way we shuffle cards.
Mathematics can tell us how many times a deck of cards
needs to be shuffled to ensure it's mixed well.
It can also show us the best way to deal cards.
Jason Fulman, a math professor
at the USC Dornsyth College
of Letters, Arts and Sciences,
uses math to study card shuffling In his new book,
“The Mathematics of Shuffling Cards”
（American Mathematical Society）
co-written with renowned mathematician Percy Diaconus,
he shares what is known about the topic.
The book explains the mathematics needed to answer
the question how many times should a deck
of cards be shuffled to mix it up?
The shuffles examined are the usual ones
that people use, such as the rifle,
overhand and Smooshing methods.
Card shuffling is a numbers game.
One of the insights the book provides is that the
number of shuffles required to mix a deck of 52
cards depends on the type of shuffle used.
The rifle shuffle,
which involves splitting the deck in half and quickly
interleaving the cards with your thumbs,
is the most efficient method.
It only requires seven shuffles to mix the deck well.
The Smooshing method,
where the cards are spread out on a
table and randomly mixed together,
takes 30 to 60 seconds to mix thoroughly.
The overhand method,
which involves taking sections of a stack deck
and moving them to make a new stack,
needs to be repeated 10,000 times to mix the cards well.
The purpose of the cards makes a difference, too.
In blackjack, for example,
card suits don't matter,
and certain cards are equivalent,
so just four or five rifle shuffles are enough for mixing.
Magic is another aspect of card shuffling.
Perfect shuffles can restore a
deck to its original order,
and specific sequences of shuffles can
move a card to a desired position,
allowing a magician to control the
cards in a seemingly magical way.
The book also examines card dealing,
which is crucial to ensuring fairness in card games.
Two commonly used methods of card dealing are
the cyclic method and Back and forth.
In the cyclic method,
cards are dealt in a repeating sequence,
such as 1234-1234.
Back and forth uses alternating directions,
such as 1234 followed by 4321.
Back and forth dealing
is faster and improves the randomness of the cards,
which means fewer shuffles are required for a
wellmixed deck. Card shuffling isn't just for fun.
Card shuffling has practical applications beyond
card games, magic tricks and gambling.
Analyzing the mixing time of shuffling helps computer
scientists determine the optimal distribution
of files and folders and databases.
Biologists have also considered the mixing time
of shuffles to study the order of genes.
Which can help them estimate the evolutionary distance
between two organisms, according to Fulman,
Studying patient sorting or dealing cards into piles
sheds light on passenger airline boarding,
and researchers study card shuffling in hopes
of understanding and improving traffic flow.
But mathematicians are still puzzled by many questions
about card shuffling. For example,
they want to know the number of shuffles required to
mix a deck using the almost perfect shuffle technique
used by Las Vegas casino dealers who
perform "neater" rifle shuffles,
achieving near perfect alternation
from one hand to the other.
They also remain stumped by the optimal guessing
strategy to maximize the expected number of correct
guesses when turning up cards one at a time
after a series of rifle shuffles.
An answer to this question is also of interest to
gamblers who want to be able to guess
as many correct cards as possible,
and to casino executives who want gamblers to be
able to guess as few correct cards as possible.