1. Card Shuffling as a Dynamical System
Dr. Russell Herman Department 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?  
We learn about continuous systems .. We derive models by starting discrete .. We simulate using discrete … here we start discrete and generalize to continuous .. There are many continuous models .. But we will concentrate on discrete

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
In this talk we will review some of the history and mathematics of the perfect shuffle. We will explore models of the perfect shuffle of a deck of arbitrary size, leading to a discrete dynamical system. In particular, we will look at the dynamics of the doubling map and the logistic map as a way of introducing standard notions from nonlinear dynamics, such as fixed points, periodic orbits, symbolic dynamics and chaos. This talk will be at a level accessible by undergraduates and is meant as an introduction to discrete dynamical systems via card shuffling

3. A Bit of History

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

5. The Game of Faro Decks 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 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/

7. 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 => original order
What is a typical Riffle shuffle?
What is a typical Faro shuffle?

8. See!
Period 18 and 35!

9. 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
Koschitz’s Manual of Useful Information
Maskelyne’s 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

10. Mathematical Models

11. A Model for Card Shuffling
Label the positions 0-51
Then
0->0 and 26 ->1
1->2 and 27 ->3
2->4 and 28 ->5
… in general?
Ignoring card 51: f(x) = 2x mod 51
Recall Congruences:
2x mod 51 = remainder upon division by 51

12. 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)

13. Generalization to n cards

14. The Out Shuffle

15. The In Shuffle

16. Representations for n Cards
In Shuffles
Out Shuffles

17. 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)
=> o (O,2n-1) = o (I,2n-1)
o (I,2n-2) = o (O,2n)
Therefore, only need o (O,2n)

18. o (O,2n) = Order for 2n Cards
One Shuffle: O(p) = 2p mod (2n-1), 0<p<N-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 < p < 2n such that Ok(p) = p mod (2n-1)
Or, 2k = 1 mod (2n-1) => (2n – 1) | (2k – 1)

19. The Orders of Perfect Shuffles
n o(O,n) o(I,n) n o(O,n) o(I,n)
Demonstration

20. 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

21. Shuffle Types
All denominators are odd numbers.

22. Doubling Function

23. Discrete Dynamical Systems
First Order System: xn+1 = f (xn)
Orbits: {x0, x1, … }
Fixed Points
Periodic Orbits
Stability and Bifurcation
Chaos !!!!

24. The Logistic Map Discrete Population Model Competition Pn+1 = a Pn
Pn+1 = an P0
a>1 => exponential growth!
Competition
Pn+1 = a Pn - b Pn2
xn = (a/b)Pn, r=a/b =>
xn+1 = r xn(1 - xn), xne[0,1] and re[0,4]

25. Example r=2.1
Sample orbit for r=2.1 and x0 = 0.5

26. Example r=3.5

27. Example r=3.56

28. Example r=3.568

29. Example r=4.0

30. Iterations
More Iterations

31. Fixed Points f(x*) = x* Logistic Map - Cobwebs x* = r x*(1-x*)
=> x* = 0 or x* = 1 – 1/r
Logistic Map - Cobwebs

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

33. Stability Fixed Points Periodic Orbits Bifurcations |f’(x*)| < 1
|f’(x0)| |f’(x1)| … |f’(xn)| < 1
Bifurcations

34. Bifurcations
r1 = 3.0
r2 =
r3 =
r4 =
r5 =
r6 =
r7 =
r8 =

35. Itineraries: Symbolic Dynamics
For G (x) = 4x ( 1-x ) Assign Left “L” and Right “R”
Example: x0 = 1/3
x0 = 1/ => “L”
x1 = 8/ => “LR”
x2 = 32/ => “LRL”
x3 = … => “LRL …”
Example: x0 = ¼
{ ¼, ¾, ¾, …}=>” LRRRR…”
Subscripts, itinerary.mws????
Periodic Orbits
“LRLRLR …”, “RLRRLRRLRRL …”

36. Shuffling as a Dynamical System
S(x) vs S4(x)

37. Demonstration

38. Iterations for 8 Cards

39. S3(x) vs S2(x) S3(x) vs S2(x)
How can we study periodic orbits for S(x)?

40. Binary Representations
=1(2-1)+0(2-2)+1(2-3)+1(2-4) =
1/2 + 1/8 + 1/16 = 10/16 = 5/8
xn+1 = S(xn), given x0
Represent xn’s in binary: x0 =
Then, x1 = 2 x0 – 1 = – 1 =
Note: S shifts binary representations!
Repeating Decimals
S( …) = …
S( …) = …

41. Periodic Orbits Period 2 Period 3 Maple Computations
0.102, 0.012, = ?
Period 3
0.1002, , = ?
0.1102, , = ?
Maple Computations

42. 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 (i+1) 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

43. Finding Specific t-Cycles
Period k: … 0001
2-t + (2-t)2 + (2-t) 3 + … = 2-t /(1- 2-t )
Or, … 0001 = 1/(2t -1)
Examples
Period 2: 1/3
Period 3: 1/7
In general: Select Shuffle Type
Rationals of form i/r => (2t –1) | r
Example r = 3(7) = 21
Out Shuffle for 22 or 21 cards
In Shuffle for 20 or 21 cards
Demonstration

44. Other Topics Cards Nonlinear Dynamical Systems
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
Lyupanov exponents .. Period 3 => chaos, stability issues and other types of riffle shuffles

45. 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

46. 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

47. Websites Thank you ! http://i-p-c-s.org/history.html
Thank you !