Card Shuffling as a Card Shuffling as a Dynamical SystemDynamical System

Dr. Russell HermanDepartment of Mathematics and Statistics

University of North Carolina at Wilmington

How does a magician know that the eighth card in a deck of 50 cards returns to it original position after only three perfect shuffles?  How many perfect shuffles will return a full deck of cards to their original order? What is a "perfect" shuffle?  

IntroductionIntroduction

History of the Faro ShuffleHistory of the Faro ShuffleThe Perfect ShuffleThe Perfect ShuffleMathematical Models of Perfect Mathematical Models of Perfect

ShufflesShufflesDynamical Systems – The Logistic Dynamical Systems – The Logistic

ModelModelFeatures of Dynamical SystemsFeatures of Dynamical SystemsShuffling as a Dynamical SystemShuffling as a Dynamical System

A Bit of HistoryA Bit of History

History of the Faro ShuffleHistory of the Faro Shuffle CardsCards

Western Culture - 14Western Culture - 14thth Century CenturyJokers – 1860’sJokers – 1860’sPips – 1890’s added numbersPips – 1890’s added numbersFirst Card tricks by gamblersFirst Card tricks by gamblers

Origins of Perfect Shuffles not Origins of Perfect Shuffles not knownknown

Game of FaroGame of Faro1818thth Century France Century FranceNamed after face cardNamed after face cardPopular 1803-1900’s in the West Popular 1803-1900’s in the West

The Game of FaroThe Game of FaroDecks shuffled and rules are simpleDecks shuffled and rules are simple

(fâr´O) [for Pharaoh, from an old French playing card design], gambling game played with a standard pack of 52 cards. First played in France and England, faro was especially popular in U.S. gambling houses in the 19th Century. Players bet against a banker (dealer), who draws two cards–one that wins and another that loses–from the deck (or from a dealing box) to complete a turn. Bets–on which card will win or lose– are placed on each turn, paying 1:1 odds. Columbia Encyclopedia, Sixth Edition. 2001

Players bet on 13 cardsPlayers bet on 13 cardsLose Slowly!Lose Slowly!Copper Tokens – bet card to lose Copper Tokens – bet card to lose

““Coppering”, “Copper a Bet”Coppering”, “Copper a Bet”Analysis – De Moivre, Euler, …Analysis – De Moivre, Euler, …

The GameThe Game

http://www.bcvc.net/faro/rules.htm

Wichita Faro http://www.gleeson.us/faro/

Perfect (Faro or Weave) ShufflePerfect (Faro or Weave) Shuffle

Problem: Problem: Divide 52 cards into 2 equal piles Divide 52 cards into 2 equal piles Shuffle by interlacing cards Shuffle by interlacing cards Keep top card fixed (Out Shuffle)Keep top card fixed (Out Shuffle)8 shuffles => original order8 shuffles => original order

What is a typical Riffle shuffle?

What is a typical Faro shuffle?

See!See!

Period 2 @ 18 and 35!Period 2 @ 18 and 35!

History of Faro ShuffleHistory of Faro Shuffle1726 – Warning in book for first time1726 – Warning in book for first time1847 – J H Green – Stripper (tapered) 1847 – J H Green – Stripper (tapered)

CardsCards1860 – Better description of shuffle1860 – Better description of shuffle1894 – How to perform1894 – How to perform

Koschitz’s Manual of Useful InformationKoschitz’s Manual of Useful InformationMaskelyne’s Maskelyne’s Sharps and FlatsSharps and Flats – 1 – 1stst Illustration Illustration

1915 – Innis – Order for 52 Cards1915 – Innis – Order for 52 Cards1948 – Levy – O(p) for odd deck, cycles1948 – Levy – O(p) for odd deck, cycles1957 – Elmsley – Coined In/Out - shuffles1957 – Elmsley – Coined In/Out - shuffles

Mathematical ModelsMathematical Models

A Model for Card ShufflingA Model for Card ShufflingLabel the positions 0-51Label the positions 0-51ThenThen

0->0 and 26 ->10->0 and 26 ->11->2 and 27 ->31->2 and 27 ->32->4 and 28 ->52->4 and 28 ->5… … in general?in general?

2 0 25( )

2 51 26 51

x xf x

x x

Ignoring card 51: Ignoring card 51: f(x) = 2x f(x) = 2x modmod 51 51Recall Congruences:Recall Congruences:

2x 2x modmod 51 = 51 = remainder upon division by remainder upon division by 5151

The Order of a ShuffleThe Order of a Shuffle Minimum integer Minimum integer kk such that such that 2 2 k k x = x x = x modmod 51 51

for all for all xx in {0,1,…,51} in {0,1,…,51}

True for True for x = 1 x = 1 !!

Minimum integer Minimum integer kk such that such that 2 2 k k - 1= 0 - 1= 0 modmod 51 51

Thus,Thus, 51 51 dividesdivides 2 2 k k - 1 - 1 k= 6, 2 k= 6, 2 k k - 1 = 63 = 3(21)- 1 = 63 = 3(21) k= 7, 2 k= 7, 2 k k - 1 = 127 - 1 = 127 k= 8, 2 k= 8, 2 k k - 1 = 255 = 5(51)- 1 = 255 = 5(51)

Generalization to Generalization to nn cardscards

The Out ShuffleThe Out Shuffle

The In ShuffleThe In Shuffle

In ShufflesIn Shuffles

Out ShufflesOut Shuffles

Representations for n Representations for n CardsCards

mod 1, even( ) 2 1

mod , odd

n nI p p

n n

0 1p n

mod 1, even and 0 1( ) 2

mod , odd and 0 1

n n p nO p p

n n p n

Order of ShufflesOrder of Shuffles8 Out Shuffles for 52 Cards8 Out Shuffles for 52 Cards In General?In General?

o o (O,2(O,2nn-1) = -1) = o o (O,2(O,2nn) ) o o (I,2(I,2nn-1) = -1) = o o (O,2(O,2nn))

=> => o o (O,2(O,2nn-1) = -1) = o o (I,2(I,2n-1n-1) ) o o (I,2(I,2n-2n-2) = ) = o o (O,2(O,2nn))

Therefore, only need Therefore, only need o o (O,2(O,2nn))

o o (O,2(O,2nn) =) = Order for 2Order for 2nn CardsCards

One Shuffle: O(One Shuffle: O(pp) = 2) = 2pp mod (2 mod (2nn-1), -1), 00<<p<N-1p<N-1

2 2 shuffles: shuffles: O O22((pp) = 2 O() = 2 O(p)p) mod (2 mod (2nn-1) = 2-1) = 222 pp mod (2 mod (2nn-1)-1)

kk shuffles: O shuffles: Okk((pp) = 2) = 2kkpp mod (2 mod (2nn-1)-1)

Order: o Order: o (O,2(O,2nn) = smallest ) = smallest kk for 0 for 0 << p p < 2< 2n n such thatsuch that OOkk((pp) = ) = pp mod (2 mod (2nn-1)-1)

Or, 2Or, 2kk = 1 mod (2= 1 mod (2nn-1) => -1) => (2(2n n – 1) | (2– 1) | (2kk – – 1) 1)

The Orders of Perfect The Orders of Perfect ShufflesShuffles

n o(O,n) o(I,n) n o(O,n) o(I,n)

2 1 2 13 12 12

3 2 2 14 12 4

4 2 4 15 4 4

5 4 4 16 4 8

6 4 3 17 8 8

7 3 3 18 8 18

8 3 6 50 21 8

9 6 6 51 8 8

10 6 10 52 8 52

11 10 10 53 52 52

12 10 12 54 52 20

DemonstrationDemonstration

Another Model for 2n Another Model for 2n CardsCards

Example: Card 10 of 52: x = 9/51Example: Card 10 of 52: x = 9/51

0 1 2 2 10 , , , , 1.

2 1 2 1 2 1 2 1

n

n n n n

1 2 2, , , .

2 1 2 1 2 1

n

n n n In ShuffleIn Shuffle

Label positions with rationalsLabel positions with rationals

Out ShuffleOut Shuffle

Example: Card 10 of 52: x = 9/51Example: Card 10 of 52: x = 9/51

Shuffle TypesShuffle Types

Domain Endpoints Shuffle Deck Size

0 1 N -1, ,…, 0,1 out N = 2n

N -1 N -1 N -1

1 2 N, ,…, 1 in N = 2n -1

N N N

0 1 N -1, ,…, 0 out N = 2n -1

N N N

1 2 N, ,…, none in N = 2n

N +1 N +1 N +1

All denominators are odd numbers.

Doubling FunctionDoubling Function1

2 , 02( ) .

12 1, 1

2

x xS x

x x

Discrete Dynamical SystemsDiscrete Dynamical Systems

First Order System: First Order System: xxn+1n+1 = = f f ((xxnn))

Orbits: {Orbits: {xx00, , xx11, , … … }}Fixed PointsFixed PointsPeriodic OrbitsPeriodic OrbitsStability and BifurcationStability and BifurcationChaos !!!!Chaos !!!!

The Logistic MapThe Logistic MapDiscrete Population ModelDiscrete Population Model

PPn+1n+1 = a P = a Pnn

PPn+1n+1 = a = a22 P Pn-1n-1

PPn+1n+1 = a = ann P P00

a>1 => exponential growth!a>1 => exponential growth!CompetitionCompetition

PPn+1n+1 = a P = a Pnn - b P - b Pnn22

xxnn = (a/b)P = (a/b)Pnn, r=a/b => , r=a/b => xxn+1n+1 = r x = r xnn(1 - x(1 - xnn), x), xnn[0,1] and r[0,1] and r[0,4][0,4]

Example r=2.1Example r=2.1

Sample orbit for r=2.1 and x0 = 0.5

Example r=3.5Example r=3.5

Example r=3.56Example r=3.56

Example r=3.568Example r=3.568

Example r=4.0Example r=4.0

IterationsIterations

More IterationsMore Iterations

Fixed PointsFixed Pointsf(x*) = x*f(x*) = x*

x* = r x*(1-x*) x* = r x*(1-x*) => 0 = x*(1-r (1-x*) ) => 0 = x*(1-r (1-x*) ) => x* = 0 or x* = 1 – 1/r=> x* = 0 or x* = 1 – 1/r

Logistic Map - CobwebsLogistic Map - Cobwebs

Periodic Orbits for Periodic Orbits for f(x)=rx(1-x)f(x)=rx(1-x)

Period 2Period 2xx11 = r x = r x00(1- x(1- x00) and x) and x22 = r x = r x11(1- x(1- x11) = x) = x00

Or, f Or, f 2 2 (x(x00) = x) = x00

Period k Period k - smallest k - smallest k

such that f such that f k k (x*) = x*(x*) = x*Periodic CobwebsPeriodic Cobwebs

StabilityStability

Fixed PointsFixed Points|f’(x*)| < 1|f’(x*)| < 1

Periodic OrbitsPeriodic Orbits|f’(x|f’(x00)| |f’(x)| |f’(x11)| … |f’(x)| … |f’(xnn)| < 1)| < 1

BifurcationsBifurcations

BifurcationsBifurcationsr1 = 3.0r1 = 3.0

r2 = 3.449490 ...r2 = 3.449490 ...

r3 = 3.544090 ...r3 = 3.544090 ...

r4 = 3.564407 ...r4 = 3.564407 ...

r5 = 3.568759 ...r5 = 3.568759 ...

r6 = 3.569692 ...r6 = 3.569692 ...

r7 = 3.569891 ...r7 = 3.569891 ...

r8 = 3.569934 ...r8 = 3.569934 ...

Itineraries: Symbolic DynamicsItineraries: Symbolic Dynamics

Example: xExample: x00 = 1/3 = 1/3 xx00 = 1/3 => “L” = 1/3 => “L”

xx11 = 8/9 => “LR” = 8/9 => “LR”

xx22 = 32/81 => “LRL” = 32/81 => “LRL”

xx33 = … => “LRL …” = … => “LRL …”

Example: xExample: x00 = ¼ = ¼ { ¼, ¾, ¾, …}=>” LRRRR…”{ ¼, ¾, ¾, …}=>” LRRRR…”

For For G G ((xx) = 4) = 4x x ( 1-( 1-x x ) ) Assign Left “L” and Right Assign Left “L” and Right “R” “R”

Periodic Orbits Periodic Orbits ““LRLRLR …”, “RLRRLRRLRRL …”LRLRLR …”, “RLRRLRRLRRL …”

Shuffling as a Dynamical SystemShuffling as a Dynamical System1

2 , 02( ) .

12 1, 1

2

x xS x

x x

S(x) vs SS(x) vs S44(x)(x)

DemonstrationDemonstration

Iterations for 8 CardsIterations for 8 Cards

SS33(x) vs S(x) vs S22(x)(x)

SS33(x) vs S(x) vs S22(x)(x)

How can we study periodic orbits for How can we study periodic orbits for S(x)?S(x)?

Binary RepresentationsBinary Representations Binary RepresentationBinary Representation

0.10110.101122=1(=1(22-1-1)+0()+0(22-2-2)+1()+1(22-3-3)+1()+1(22-4-4) = ) = 1/2 + 1/8 + 1/16 = 10/16 = 5/81/2 + 1/8 + 1/16 = 10/16 = 5/8

xxn+1n+1 = S(x = S(xnn), given x), given x00

Represent xRepresent xnn’s in binary: x’s in binary: x00 = 0.101101 = 0.101101Then, xThen, x11 = 2 x = 2 x0 0 – 1 = 1.01101 – 1 = 0.01101– 1 = 1.01101 – 1 = 0.01101Note: S shifts binary representations! Note: S shifts binary representations!

Repeating DecimalsRepeating DecimalsS(0.101101101101…) = 0.011011011011…S(0.101101101101…) = 0.011011011011…S(0.011011011011…) = 0.110110110110…S(0.011011011011…) = 0.110110110110…

Periodic OrbitsPeriodic OrbitsPeriod 2Period 2

S(0.10101010…) = 0.01010101…S(0.10101010…) = 0.01010101…S(0.01010101…) = 0.10101010…S(0.01010101…) = 0.10101010…0.0.101022, 0., 0.010122, 0., 0.11112 2 = ?= ?

Period 3Period 30.0.10010022, 0., 0.01001022, 0.00, 0.00112 2 = ?= ?0.0.11011022, 0., 0.01101122, 0.10, 0.10112 2 = ?= ?

Maple ComputationsMaple Computations

Card Shuffling ExamplesCard Shuffling Examples8 Cards – All orbits are period 38 Cards – All orbits are period 3

52 Cards – Period 2 52 Cards – Period 2

50 Cards – Period 3 Orbit (Cycle)50 Cards – Period 3 Orbit (Cycle)

Recall:Recall:Period 2 - {1/3, 2/3}Period 2 - {1/3, 2/3}Period 3 – {1/7, 2/7, 4/7} and {3/7, 6/7, Period 3 – {1/7, 2/7, 4/7} and {3/7, 6/7,

5/7}5/7}Out Shuffles – i/(N-1) for (i+1) st cardOut Shuffles – i/(N-1) for (i+1) st card

{1/7, 2/7, 4/7} and {3/7, 6/7, 5/7} and {0/7, {1/7, 2/7, 4/7} and {3/7, 6/7, 5/7} and {0/7, 7/7}7/7}

1/3 = ?/51 and 2/3 = ?/51 1/3 = ?/51 and 2/3 = ?/51

1/7 = ?/491/7 = ?/49

Finding Specific t-CyclesFinding Specific t-Cycles

Period k: 0.Period k: 0.000 … 0001000 … 0001 22-t-t + (2 + (2-t-t)2 + (2)2 + (2-t-t) 3 + … = 2) 3 + … = 2-t-t /(1- 2 /(1- 2-t-t ) ) Or, 0.Or, 0.000 … 0001000 … 0001 = 1/(2 = 1/(2tt -1) -1)

ExamplesExamples Period 2: 1/3Period 2: 1/3 Period 3: 1/7Period 3: 1/7

In general: Select Shuffle TypeIn general: Select Shuffle Type Rationals of form i/r => (2Rationals of form i/r => (2tt –1) | r –1) | r Example r = 3(7) = 21Example r = 3(7) = 21

Out Shuffle for 22 or 21 cardsOut Shuffle for 22 or 21 cards In Shuffle for 20 or 21 cards In Shuffle for 20 or 21 cards DemonstrationDemonstration

Other TopicsOther Topics CardsCards

Alternate In/Out ShufflesAlternate In/Out Shuffles k- handed Perfect Shufflesk- handed Perfect Shuffles Random Shuffles – Diaconis, et alRandom Shuffles – Diaconis, et al ““Imperfect” Perfect ShufflesImperfect” Perfect Shuffles

Nonlinear Dynamical SystemsNonlinear Dynamical Systems Discrete (Difference Equations)Discrete (Difference Equations)

Systems in the Plane and Higher DimensionsSystems in the Plane and Higher Dimensions Continuous Dynamical Systems (ODES)Continuous Dynamical Systems (ODES)

IntegrabilityIntegrabilityNonlinear OscillationsNonlinear OscillationsMAT 463/563MAT 463/563

FractalsFractals ChaosChaos

SummarySummary

History of the Faro ShuffleHistory of the Faro ShuffleThe Perfect Shuffle – How to do it!The Perfect Shuffle – How to do it!Mathematical Models of Perfect ShufflesMathematical Models of Perfect ShufflesDynamical Systems – The Logistic Dynamical Systems – The Logistic

ModelModelFeatures of Dynamical SystemsFeatures of Dynamical SystemsSymbolic DynamicsSymbolic DynamicsShuffling as a Dynamical SystemShuffling as a Dynamical System

ReferencesReferences

K.T. Alligood, T.D. Sauer, J.A. Yorke, K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos, Chaos, An Introduction to Dynamical SystemsAn Introduction to Dynamical Systems, , Springer, 1996.Springer, 1996.

S.B. Morris, Magic Tricks, S.B. Morris, Magic Tricks, Card Shuffling and Card Shuffling and Dynamic Computer MemoriesDynamic Computer Memories, MAA, 1998, MAA, 1998

D.J. Scully, D.J. Scully, Perfect Shuffles Through Perfect Shuffles Through Dynamical Systems, Dynamical Systems, Mathematics Magazine, Mathematics Magazine, 77, 200477, 2004

WebsitesWebsites

http://i-p-c-s.org/history.html http://i-p-c-s.org/history.html http://jducoeur.org/game-hist/seaan-cardhist.html http://jducoeur.org/game-hist/seaan-cardhist.html http://www.usplayingcard.com/gamerules/http://www.usplayingcard.com/gamerules/

briefhistory.html briefhistory.html http://bcvc.net/faro/ http://bcvc.net/faro/ http://www.gleeson.us/faro/ http://www.gleeson.us/faro/

Thank you !