Card Shuffling as a Dynamical System

1
Card Shuffling as a Dynamical System
Dr. Russell HermanDepartment of Mathematics and
StatisticsUniversity 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?  
2
Introduction

• History of the Faro Shuffle
• The Perfect Shuffle
• Mathematical Models of Perfect Shuffles
• Dynamical Systems The Logistic Model
• Features of Dynamical Systems
• Shuffling as a Dynamical System

3
A Bit of History
4
History of the Faro Shuffle

• Cards
• Western Culture - 14th Century
• Jokers 1860s
• Pips 1890s added numbers
• First Card tricks by gamblers
• Origins of Perfect Shuffles not known
• Game of Faro
• 18th Century France
• Named after face card
• Popular 1803-1900s in the West

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The Game of Faro

• Decks shuffled and rules are simple
• (fârO) 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
  cardsone that wins and another that losesfrom
  the deck (or from a dealing box) to complete a
  turn. Betson which card will win or lose are
  placed on each turn, paying 11 odds. Columbia
  Encyclopedia, Sixth Edition. 2001
• Players bet on 13 cards
• Lose Slowly!
• Copper Tokens bet card to lose
• Coppering, Copper a Bet
• Analysis De Moivre, Euler,

6
The Game
Wichita Faro http//www.gleeson.us/faro/
http//www.bcvc.net/faro/rules.htm
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Perfect (Faro or Weave) Shuffle

• Problem
• Divide 52 cards into 2 equal piles
• Shuffle by interlacing cards
• Keep top card fixed (Out Shuffle)
• 8 shuffles gt original order

What is a typical Riffle shuffle?
What is a typical Faro shuffle?
8
See!
Period 2 _at_ 18 and 35!
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History of Faro Shuffle

• 1726 Warning in book for first time
• 1847 J H Green Stripper (tapered) Cards
• 1860 Better description of shuffle
• 1894 How to perform
• Koschitzs Manual of Useful Information
• Maskelynes Sharps and Flats 1st Illustration
• 1915 Innis Order for 52 Cards
• 1948 Levy O(p) for odd deck, cycles
• 1957 Elmsley Coined In/Out - shuffles

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Mathematical Models
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A Model for Card Shuffling

• Label the positions 0-51
• Then
• 0-gt0 and 26 -gt1
• 1-gt2 and 27 -gt3
• 2-gt4 and 28 -gt5
• in general?

• Ignoring card 51 f(x) 2x mod 51
• Recall Congruences
• 2x mod 51 remainder upon division by 51

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The Order of a Shuffle

• Minimum integer k such that 2 k x x mod 51 for
  all x in 0,1,,51
• True for x 1 !
• Minimum integer k such that 2 k - 1 0 mod 51
• Thus, 51 divides 2 k - 1
• k 6, 2 k - 1 63 3(21)
• k 7, 2 k - 1 127
• k 8, 2 k - 1 255 5(51)

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Generalization to n cards
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The Out Shuffle
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The In Shuffle
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Representations for n Cards

• In Shuffles
• Out Shuffles

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Order of Shuffles

• 8 Out Shuffles for 52 Cards
• In General?
• o (O,2n-1) o (O,2n)
• o (I,2n-1) o (O,2n)
• gt o (O,2n-1) o (I,2n-1)
• o (I,2n-2) o (O,2n)
• Therefore, only need o (O,2n)

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o (O,2n) Order for 2n Cards

• One Shuffle O(p) 2p mod (2n-1), 0ltpltN-1
• 2 shuffles O2(p) 2 O(p) mod (2n-1) 22
  p mod (2n-1)
• k shuffles Ok(p) 2kp mod (2n-1)
• Order o (O,2n) smallest k for 0 lt p lt 2n such
  that Ok(p) p mod (2n-1)
• Or, 2k 1 mod (2n-1) gt (2n 1) (2k 1)

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The Orders of Perfect Shuffles
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
Demonstration
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Another Model for 2n Cards

• Label positions with rationals
• Out Shuffle

• Example Card 10 of 52 x 9/51

• In Shuffle

• Example Card 10 of 52 x 9/51

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Shuffle Types
All denominators are odd numbers.
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Doubling Function
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Discrete Dynamical Systems

• First Order System xn1 f (xn)
• Orbits x0, x1,
• Fixed Points
• Periodic Orbits
• Stability and Bifurcation
• Chaos !!!!

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The Logistic Map

• Discrete Population Model
• Pn1 a Pn
• Pn1 a2 Pn-1
• Pn1 an P0
• agt1 gt exponential growth!
• Competition
• Pn1 a Pn - b Pn2
• xn (a/b)Pn, ra/b gt
• xn1 r xn(1 - xn), xne0,1 and re0,4

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Example r2.1
Sample orbit for r2.1 and x0 0.5
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Example r3.5
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Example r3.56
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Example r3.568
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Example r4.0
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Iterations

• More Iterations

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Fixed Points

• f(x) x
• x r x(1-x)
• gt 0 x(1-r (1-x) )
• gt x 0 or x 1 1/r
• Logistic Map - Cobwebs

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

• Period 2
• x1 r x0(1- x0) and x2 r x1(1- x1) x0
• Or, f 2 (x0) x0
• Period k
• - smallest k such that f k (x) x
• Periodic Cobwebs

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Stability

• Fixed Points
• f(x) lt 1
• Periodic Orbits
• f(x0) f(x1) f(xn) lt 1
• Bifurcations

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Bifurcations

• r1 3.0
• r2 3.449490 ...
• r3 3.544090 ...
• r4 3.564407 ...
• r5 3.568759 ...
• r6 3.569692 ...
• r7 3.569891 ...
• r8 3.569934 ...

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Itineraries Symbolic Dynamics

• For G (x) 4x ( 1-x ) Assign Left L and Right
  R

• Example x0 1/3
• x0 1/3 gt L
• x1 8/9 gt LR
• x2 32/81 gt LRL
• x3 gt LRL
• Example x0 ¼
• ¼, ¾, ¾, gt LRRRR

• Periodic Orbits
• LRLRLR , RLRRLRRLRRL

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Shuffling as a Dynamical System
S(x) vs S4(x)
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Demonstration
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Iterations for 8 Cards
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S3(x) vs S2(x)
S3(x) vs S2(x)
How can we study periodic orbits for S(x)?
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Binary Representations

• Binary Representation
• 0.101121(2-1)0(2-2)1(2-3)1(2-4)
• 1/2 1/8 1/16 10/16 5/8
• xn1 S(xn), given x0
• Represent xns in binary x0 0.101101
• Then, x1 2 x0 1 1.01101 1 0.01101
• Note S shifts binary representations!
• Repeating Decimals
• S(0.101101101101) 0.011011011011
• S(0.011011011011) 0.110110110110

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Periodic Orbits

• Period 2
• S(0.10101010) 0.01010101
• S(0.01010101) 0.10101010
• 0.102, 0.012, 0.112 ?
• Period 3
• 0.1002, 0.0102, 0.0012 ?
• 0.1102, 0.0112, 0.1012 ?
• Maple Computations

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Card Shuffling Examples

• 8 Cards All orbits are period 3
• 52 Cards Period 2
• 50 Cards Period 3 Orbit (Cycle)
• Recall
• Period 2 - 1/3, 2/3
• Period 3 1/7, 2/7, 4/7 and 3/7, 6/7, 5/7
• Out Shuffles i/(N-1) for (i1) st card

1/7, 2/7, 4/7 and 3/7, 6/7, 5/7 and 0/7,
7/7 1/3 ?/51 and 2/3 ?/51 1/7 ?/49
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Finding Specific t-Cycles

• Period k 0.000 0001
• 2-t (2-t)2 (2-t) 3 2-t /(1- 2-t )
• Or, 0.000 0001 1/(2t -1)
• Examples
• Period 2 1/3
• Period 3 1/7
• In general Select Shuffle Type
• Rationals of form i/r gt (2t 1) r
• Example r 3(7) 21
• Out Shuffle for 22 or 21 cards
• In Shuffle for 20 or 21 cards

Demonstration
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Other Topics

• Cards
• Alternate In/Out Shuffles
• k- handed Perfect Shuffles
• Random Shuffles Diaconis, et al
• Imperfect Perfect Shuffles
• Nonlinear Dynamical Systems
• Discrete (Difference Equations)
• Systems in the Plane and Higher Dimensions
• Continuous Dynamical Systems (ODES)
• Integrability
• Nonlinear Oscillations
• MAT 463/563
• Fractals
• Chaos

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Summary

• History of the Faro Shuffle
• The Perfect Shuffle How to do it!
• Mathematical Models of Perfect Shuffles
• Dynamical Systems The Logistic Model
• Features of Dynamical Systems
• Symbolic Dynamics
• Shuffling as a Dynamical System

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References

• K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos, An
  Introduction to Dynamical Systems, Springer,
  1996.
• S.B. Morris, Magic Tricks, Card Shuffling and
  Dynamic Computer Memories, MAA, 1998
• D.J. Scully, Perfect Shuffles Through Dynamical
  Systems, Mathematics Magazine, 77, 2004

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Websites

• http//i-p-c-s.org/history.html
• http//jducoeur.org/game-hist/seaan-cardhist.html
  
• http//www.usplayingcard.com/gamerules/briefhistor
  y.html
• http//bcvc.net/faro/
• http//www.gleeson.us/faro/

Thank you !