{"id":14924,"date":"2023-04-21T17:21:00","date_gmt":"2023-04-22T00:21:00","guid":{"rendered":"https:\/\/dornsife.usc.edu\/news\/?p=14924"},"modified":"2024-12-17T21:48:53","modified_gmt":"2024-12-18T05:48:53","slug":"link-between-math-and-card-shuffling","status":"publish","type":"post","link":"https:\/\/dornsife.usc.edu\/news\/stories\/link-between-math-and-card-shuffling\/","title":{"rendered":"Expert reveals the fascinating link between math and card shuffling"},"content":{"rendered":"\n\n\n    \n                                          \n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--article-hero \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--article-hero\"\n    \n      >\n\n    \n<div class=\"inner-wrapper\">\n          \n<div class=\"f--field f--image\">\n\n    \n    \n    \n    \n    \n    \n              \n      <img\n                            data-src=\"https:\/\/dornsife.usc.edu\/news\/wp-content\/uploads\/sites\/7\/2023\/04\/card-shuffling-math_central-768x432.jpg\"\n          data-srcset=\"https:\/\/dornsife.usc.edu\/news\/wp-content\/uploads\/sites\/7\/2023\/04\/card-shuffling-math_central-1280x720.jpg 1280w,https:\/\/dornsife.usc.edu\/news\/wp-content\/uploads\/sites\/7\/2023\/04\/card-shuffling-math_central-768x432.jpg 768w\"          data-sizes=\"(min-width:1200px) 75vw, (min-width:768px) 83vw, 100vw\"          class=\"lazyload\"\n        \n                  alt=\"Close up of hands shuffling a deck of cards.\"\n        \n        \n                                      \/>\n\n    \n    \n  \n  \n\n<\/div>\n  \n      <div class=\"image-caption\">\n          \n<div class=\"f--field f--description\">\n\n    \n  The mathematics involved in card shuffling may point to answers for problems in biology, computer science and even freeway traffic. (Images source: iStock.)\n\n\n<\/div>\n    <\/div>\n  \n  <div class=\"text-wrapper\">\n          <nav aria-label=\"Breadcrumb\" class=\"breadcrumbs\">\n        <ul>\n                      <li><a href=\"\/news\/stories\/\">News<\/a><\/li>\n                      <li><a href=\"\/news\/stories\/\/?category=faculty\">Faculty<\/a><\/li>\n                  <\/ul>\n      <\/nav>\n    \n              \n<div class=\"f--field f--page-title\">\n\n    \n  <h1>Expert reveals the fascinating link between math and card shuffling<\/h1>\n\n\n<\/div>\n    \n          <div class=\"subtitle\">\n            \n<div class=\"f--field f--description\">\n\n    \n  USC Dornsife math professor explores ideal techniques and real-world applications of finding the perfect shuffle.\n\n\n<\/div>\n      <\/div>\n    \n           <strong class=\"author-field\"><span >By<\/span><a href=\"mailto:communication@dornsife.usc.edu\">Ileana Wachtel<\/a><\/strong>\n    \n          <span class=\"post-date-field\">April 21, 2023<\/span>\n      <\/div>\n<\/div>\n\n\n  <\/div><\/div>\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--social-share \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--social-share\"\n    \n      >\n\n    \n  <div class=\"content-wrapper\">\n    <span class=\"a2a_kit a2a_kit_size_32 addtoany_list\" style=\"line-height: 32px;\">\n      <span class=\"title\">\n        Share\n      <\/span>\n                        <a class=\"a2a_button_copy_link\" target=\"_blank\" href=\"\/#copy_link\" rel=\"nofollow noopener\" title=\"Link\">\n            <span class=\"a2a_svg a2a_s__default a2a_s_copy_link\">\n              <svg height=\"19\" viewBox=\"0 0 19 19\" width=\"19\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"m7.43475275 9.52380952-2.17490843 2.26076008c-1.08745421 1.058837-1.68841575 2.518315-1.68841575 4.0350275 0 1.5167124.60096154 2.9475732 1.68841575 4.0350274 1.058837 1.0874543 2.51831502 1.6884158 4.03502747 1.6884158 1.44087681 0 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<p>Mathematics sometimes impacts our lives in seemingly unsuspecting ways, including card shuffling.<\/p>\n<p>Math can answer the age-old question of how many times a deck of cards needs to be shuffled to ensure the cards are thoroughly mixed. It can even reveal the best method for dealing cards.<\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/cf\/faculty-and-staff\/faculty.cfm?pid=1012559\">Jason Fulman<\/a>, professor of\u00a0<a href=\"https:\/\/dornsife.usc.edu\/mathematics\/\">mathematics<\/a>\u00a0at the USC Dornsife College of Letters, Arts and Sciences, studies card shuffling using math. He shares what is known on the topic in an upcoming book,\u00a0<em>The Mathematics of Shuffling Cards\u00a0<\/em>(<a href=\"https:\/\/www.barnesandnoble.com\/s\/%22American+Mathematical+Society%22?Ntk=Publisher&amp;Ns=P_Sales_Rank&amp;Ntx=mode+matchall\" target=\"_blank\" rel=\"noopener\">American Mathematical Society<\/a>), which he co-wrote with acclaimed mathematician Persi Diaconis. The book is due out in June.<\/p>\n<h2>Card shuffling is a numbers game<\/h2>\n<p>Among the many insights Fulman provides is that the number of shuffles required to thoroughly mix a deck of 52 cards depends on the shuffle type used.<\/p>\n<p>The riffle shuffle \u2014 splitting the deck roughly in half then using the thumbs to quickly interleave the cards \u2014 is the most efficient. It requires just seven shuffles to mix a deck well.<\/p>\n<p>Scattering the cards out flat on the table and randomly spreading them over each other, called \u201csmooshing,\u201d requires 30 to 60 seconds for thorough mixing.<\/p>\n<p>The overhand method \u2014 taking sections of a stacked deck and moving them over to make a new stack \u2014 must be repeated a whopping 10,000 times to mix the cards well.<\/p>\n<p>What the cards are being used for makes a difference, too. In blackjack, for example, card suits don\u2019t matter, and certain cards are equivalent, so just four or five riffle shuffles are plenty for mixing.<\/p>\n<p>Then there\u2019s magic. Perfect shuffles can restore a deck to its original order, and specific sequences of shuffles can move a card to a desired position, enabling a magician to control the cards in a way that seems magical.<\/p>\n<div class=\"video-embed\"><iframe loading=\"lazy\" title=\"The Best and Worst Ways to Shuffle A Deck of Cards\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ySRX1_B6qlA?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/div>\n<h2>For mathematicians, fairness is a big deal<\/h2>\n<p>Fulman also explores card dealing, a key to ensuring fairness in card games.<\/p>\n<p>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 one, two, three, four, one, two, three, four. Back-and-forth uses alternating directions such as one, two, three, four followed by four, three, two, one.<\/p>\n<p>Back and forth dealing is faster\u00a0and improves the cards\u2019 randomness, thus requiring fewer shuffles for a well-mixed deck.<\/p>\n<h2>Card shuffling is not just fun and games<\/h2>\n<p>Card shuffling has practical applications beyond card games, magic tricks and gambling.<\/p>\n<p>Analyzing the mixing time of shuffling helps computer scientists determine the optimal distribution of files and folders in databases.\u00a0And biologists have considered the mixing time of shuffles to study the order of genes, which can help them estimate the evolutionary distance between two organisms, Fulman says.<\/p>\n<p>Studying \u201cpatience sorting,\u201d dealing cards into piles, sheds light on passenger airline boarding, and researchers study card shuffling in hopes of understanding and improving traffic flow.<\/p>\n<p>But mathematicians still puzzle over many questions about card shuffling, Fulman says.<\/p>\n<p>For instance,\u00a0they want to know the number of shuffles required to thoroughly mix a deck using the almost perfect shuffle technique employed by Las Vegas casino dealers, who perform \u201cneater\u201d riffle shuffles achieving near-perfect alternation from one hand to the other.<\/p>\n<p>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 riffle shuffles. An answer to this question is also of interest\u00a0to 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.<\/p>\n<p>But given enough time, the odds are that mathematicians like Fulman will sort out these and many other card-shuffling conundrums.<\/p>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n\n\n\n  \n        \n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--article-related-stories \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--article-related-stories\"\n    \n      >\n\n    \n  <div class=\"inner-wrapper\">\n              \n<div class=\"f--field f--section-title\">\n\n    \n  <h2>\n          Related News\n      <\/h2>\n\n\n<\/div>\n    \n                  <article>\n              \n<div class=\"f--field f--cta-title\">\n\n    \n  <h3>\n          <a 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