Perfect Shuffles: A Shortest Sequence to Move a Card Katie Glacey ...

Perfect Shuffles: A Shortest Sequence to Move a CardKatie GlaceyMaster’s Expository PaperIn partial fulfillment of the MAT DegreeDepartment of MathematicsUniversity of Nebraska-LincolnJuly, 2011

A deck of cards can be found in many households in the world. The earliestreference to playing cards can be dated to the year 1377. The ubiquitous nature ofcards leads some to take the card shuffling for granted.The fascination with card shuffling is not a recent occurrence but has beenstudied for many years. The study of card shuffling has enveloped the minds ofmagicians, mathematicians, and even a rancher. Magician and mathematician PersiDiaconis wanted to determine the perfect number of shuffles for a deck of cards. Hedetermined that seven shuffles was the perfect amount. Mathematical writer MartinGardner pondered an efficient algorithm to bring any card to the top of a deckthrough a combination of shuffles. Even a Nebraska rancher was enthralled with theidea of card shuffling. Rancher Fred Black practiced shuffling on horseback andgained recognition by Ripley’s Believe It or Not for his ability to deal four Bridgehands, collect them, shuffle the deck, cut and shift the deck, and deal identical handsto the same players in less than one minute.When one shuffles cards the goal is to randomize the cards, but in fact thecards permute in a systematic fashion. The randomizing of a deck of cards can bedone improperly or precisely, and that knowledge can be used to either theshuffler’s disadvantage or advantage. A card dealer in a casino would receive thedisadvantage of an improper card shuffle, because a very astute gambler can use theknowledge to his or her advantage. A magician can take advantage of a card shuffledone is a precise manner. The magician is actually employing mathematics toamaze the audience.The Cards and the ShufflesThis examination of card shuffling employs a deck of 2n cards. The positionsof the cards are labeled 0 through 2n-1. Position 0 is the top position in the deck.The bulk of the analysis of shuffling revolves around two very precise variations of ashuffle referred to as the riffle shuffle: the perfect in-shuffle and the perfect outshuffle.In the perfect in-shuffle, a deck is cut in half into stacks of n cards andinterlaced with each other so the original top card moves into the second position of

the shuffled pile. In the perfect out-shuffle, the two stacks of n cards are interlacedso that the original top card of the deck remains in the top position after shuffling.The Position FunctionsTo begin this investigation, I observed the nature of the cards when shuffling,to develop position functions to help determine the position of a card originally in xposition to describe its position after one perfect in-shuffle and one perfect outshuffle.The Perfect In-ShuffleGiven a stack of cards with 2n = 12, I observed and noted the shift in positions asseen below.Position 0 1 2 3 4 5 6 7 8 9 10 11Card A B C D E F G H I J K LAfter one perfect in-shufflePosition 0 1 2 3 4 5 6 7 8 9 10 11Card G A H B I C J D K E L FNumberofPositionsMoved+6 -1 +5 -2 +4 -3 +3 -4 +2 -5 +1 -6I noticed that after one in-shuffle that the movement of the cards created a mirrorimage of movement. After n cards the pattern of movement begins to reflect butwith opposite signs. To create a function to describe the movement in position, Ihad to consider two separate cases based on where the card was originally in thedeck.

Case One: x < nWhen x < n, a card’s position will move down 2x + 1 positions after oneperfect in-shuffle.Observe:Position 0 1 2 3 4 5 6 7 8 9 10 11Card A B C D E F G H I J K LAfter one perfect in-shufflePosition 0 1 2 3 4 5 6 7 8 9 10 11Card G A H B I C J D K E L FAfter one perfect in-shuffle, a card originally in the second position will be in thefifth position.The variables used in the function to describe this movement are:x = card’s positionf(x) = card’s new position after one perfect shuffleThe function to describe this movement is: f(x) = 2x + 1 when x < n.Case Two: n < x < 2n-1When n < x < 2n-1, a card’s position moves up the difference between 2x and2n in the shuffled pile after one perfect in-shuffle.

Observe:Position 0 1 2 3 4 5 6 7 8 9 10 11Card A B C D E F G H I J K LAfter one perfect in-shufflePosition 0 1 2 3 4 5 6 7 8 9 10 11Card G A H B I C J D K E L FAfter one perfect in-shuffle, a card originally in the seventh position will be in thesecond position, which is a movement of five places from its original position.The variables used in the function to describe this movement are:x = card’s position2n = the total number of cards in a deckf(x) = card’s new position after one perfect shuffleThe function to describe this movement is: f(x) = 2x – 2n when n < x < 2n-1.The Perfect Out-ShuffleGiven a stack of cards with 2n = 12, I observed and noted the shift in positions asseen below.Position 0 1 2 3 4 5 6 7 8 9 10 11Card A B C D E F G H I J K L

After one perfect out-shufflePosition 0 1 2 3 4 5 6 7 8 9 10 11Card A G B H C I D J E K F LNumberofPositionsMoved0 +5 -1 +4 -2 +3 -3 +2 -4 +1 -5 0I noticed that after one out-shuffle that the movement of the cards created a mirrorimage of movement. After n cards the pattern of movement begins to reflect butwith opposite signs. Also, the positions of the first and last card do not change afterone perfect out-shuffle. To create a function to describe the movement in position, Ihad to consider two separate cases based on where the card was originally in thedeck.Case One: x < nout-shuffle.When x < n a card’s position will move down 2x positions after one perfectObserve:Position 0 1 2 3 4 5 6 7 8 9 10 11Card A B C D E F G H I J K LAfter one perfect out-shufflePosition 0 1 2 3 4 5 6 7 8 9 10 11Card A G B H C I D J E K F L

After one perfect out-shuffle, a card originally in the second position will be in thefourth position.The variables used in the function to describe this movement are:x = card’s positionf(x) = card’s new position after one perfect shuffleThe function to describe this movement is: f(x) = 2x when x < n.Case Two: n < x < 2n-1When n < x < 2n-1, a card’s position moves up the difference between 2x and2n+1 in the shuffled pile after one perfect out-shuffle.Observe:Position 0 1 2 3 4 5 6 7 8 9 10 11Card A B C D E F G H I J K LAfter one perfect out-shufflePosition 0 1 2 3 4 5 6 7 8 9 10 11Card A G B H C I D J E K F LAfter one perfect out-shuffle, a card originally in the seventh position will be in thethird position, which is a movement of four places from its original position.The variables used in the function to describe this movement are:x = card’s position2n = the total number of cards in a deckf(x) = card’s new position after one perfect shuffle

The function to describe this movement is: f(x) = 2x – 2n + 1 when n < x < 2n-1.Shortest Possible SequenceIn the paper “Moving Card i to Position j with Perfect Shuffles” by SarnathRamnath and Daniel Scully, the authors used the pictorial representation of trees topresent perfect shuffles. The trees create a visual path to follow to get a card fromone position to another. The authors used the following functions in base 2 tocreate the trees.They then let Dm(x) = 2x mod 2n and Em(x) = (2x + 1) mod 2n. Starting at position 0they applied the functions branching out to create the following.

In this diagram in order to go from position 0 to position 4, the cards must beshuffled by applying one Em followed by two Dm’s.To streamline the process, the authors decided it was unnecessary to construct thewhole tree but only to construct the outermost path from a position i to a position jsince dealing with a standard 52 card deck would result in a giant tree. To constructthe outermost path the following is used:(Dm(i), D 2 m (i), . . . D t m(i) and Em (i), E 2 m (i),. . . E t m(i)) until j falls in the interval[D t m(i), E t m(i)]. (In Z2n , [a, b] = {a, a + 1. b} if a < b, and [a, b] = {a, a + 1. 2n - 1, 0, 1,... bif b < a.)

ExampleIn the following example, I will illustrate moving a card from i to j using thetheorem and steps set forth by Ramnath and Scully. i = 10 and j = 33.In the article the authors illustrated the theorem: Label the positions in adeck of 2n cards 0 through 2n – 1 consecutively with 0 representing the top position.To determine the shortest possible sequence of perfect in and out shuffles that willmove a card in position i to position j, proceed as follows:1. Calculate the sequences Dm(i), D 2 m(i), . . ., D t m(i) and Em(i), E 2 m(i), . . ., E t m(i) until j ∈[D t m(i), E t m(i)]The number 33 falls between the interval [28, 35] in Z52. I had to apply the function3 times to find the interval that contains j so t = 3.2. Let s = (j – D t m (i))mod 2n and write s as a t-digit base two numeral s1s2. . . st.s = (j – D t m (i))mod 2n = (33-28) mod 52 = 5 mod 52 = 5This now needs to be written as a 3-digit base two numeral so it becomes 101.3. Reading the digits s1 s2. . . . st from left to right, apply consecutively Dm to i if sk is0 and Em if sk is 1.

So 101 becomesEmDmEm4. Make the translation Dm = O if Dm is being applied to an integer in [0, n-1] and Dm = Iif applied to an integer in [n, 2n - 1]. Similarly, Em = I in [0, n - 1] and Em = O in [n, 2n -1]. The resulting sequence of in and out shuffles (I's and O's) moves the card in positioni to position j in a minimum number of perfect shuffles.Card’s Position 10 21 42 33Function Applied to Achieve Position Em Dm EmTranslation of Position and Applied Function In Out OutSo to move a card in the 10 th position to the 33 rd position in the least amountof perfect shuffles, apply one perfect in-shuffle followed by two perfect out-shuffles.A Classroom ActivityThe activity of card shuffling can be brought into the classroom through thestudy of modular arithmetic. I teach fifth and sixth grade, so this activity is gearedtoward the intermediate level (grades 3 through 6) of elementary school.The activity begins with an introduction of modular arithmetic through thestudy of time on a clock.1) Ask students what time it will be 3 hours after noon? 10 hours afternoon? 14 hours after noon? 28 hours after noon?2) Ask student to explain how they were able to arrive at the variousanswers.3) Focus on the remainder and introduce the notation for modulararithmetic.Once the ideas of modular arithmetic and correct notation have been established,shift to the everyday object of cards.

4) Explain the concept of perfect in-shuffle.5) Distribute to the students 10 playing cards.6) Students work and apply perfect in-shuffles to their 10-card deck of cards.Students record the movement of the cards into chart.7) Discuss results.8) Wrap up with describing movement of cards in modular arithmetic form.

ReferencesS. Ramnath and D. Scully, Moving Card I to Position j with Perfect Shuffles,Mathematics Magazine, Volume 69, Number 5 (1996), pp. 361-365.