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?
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

A Bit of History
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 1803-1900’s in the West

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, …
The Game
Wichita Faro
http://www.gleeson.us/faro/
http://www.bcvc.net/faro/rules.htm
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?
See!
Period 2 @ 18 and 35!
History of Faro Shuffle
1726
 1847
 1860
 1894

–
–
–
–
Warning in book for first time
J H Green – Stripper (tapered) Cards
Better description of shuffle
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

Mathematical Models
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?
0  x  25
 2x
f ( x)  
2 x  51 26  x  51
Ignoring card 51: f(x) = 2x mod 51
 Recall Congruences:

 2x
mod 51 = remainder upon division by 51
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)
Generalization to n cards
The Out Shuffle
The In Shuffle
Representations for n Cards
0  p  n 1


In Shuffles
mod n  1, n even
I ( p)  2 p  1 
n odd
 mod n,
Out Shuffles
mod n  1, n even and 0  p  n  1
O( p )  2 p 
n odd and 0  p  n  1
 mod n,
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 (O,2n-1) = o (I,2n-1)
(I,2n-2) = o (O,2n)
Therefore, only need o (O,2n)
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:

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)
Ok(p) = 2kp mod (2n-1)
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
Another Model for 2n Cards


Label positions with rationals
Out Shuffle
 Example:

In Shuffle
 Example:
0
1
2
0
,
,
,
2n  1 2n  1 2n  1
2n  1
,
 1.
2n  1
Card 10 of 52: x = 9/51
1
2
,
,
2n  1 2n  1
2n
,
.
2n  1
Card 10 of 52: x = 9/51
Shuffle Types
Domain
1
N -1
 0
,
,…,


N -1
N -1 N -1
N
1 2
 , ,…, 
N
N N
N -1
0 1
 , ,…,

N 
N N
2
N 
 1
,
,…,


N +1
N +1 N +1
Endpoints Shuffle Deck Size
0,1
out
N = 2n
1
in
N = 2n - 1
0
out
N = 2n - 1
none
in
N = 2n
All denominators are odd numbers.
Doubling Function
1

0 x
 2 x,
2
S ( x)  
.
2 x  1, 1  x  1

2
Discrete Dynamical Systems
First Order System: xn+1 = f (xn)
 Orbits: {x0, x1, … }
 Fixed Points
 Periodic Orbits
 Stability and Bifurcation
 Chaos !!!!

The Logistic Map

Discrete Population Model
 Pn+1
= a Pn
 Pn+1 = a2 Pn-1
 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]
Example r=2.1
Sample orbit for r=2.1 and x0 = 0.5
Example r=3.5
Example r=3.56
Example r=3.568
Example r=4.0
Iterations
More Iterations
Fixed Points
f(x*) = x*
 x*
= r x*(1-x*)
 => 0 = x*(1-r (1-x*) )
 => x* = 0 or x* = 1 – 1/r

Logistic Map - Cobwebs
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
Stability

Fixed Points
 |f’(x*)|

Periodic Orbits
 |f’(x0)|

<1
|f’(x1)| … |f’(xn)| < 1
Bifurcations
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 ...
Itineraries: Symbolic Dynamics
For
G (x) = 4x ( 1-x )
Assign Left “L” and Right “R”

Example: x0 = 1/3





x0
x1
x2
x3
=
=
=
=
1/3
8/9
32/81
…
=>
=>
=>
=>
“L”
“LR”
“LRL”
“LRL …”
Example: x0 = ¼

{ ¼, ¾, ¾, …}=>” LRRRR…”
Periodic
Orbits
“LRLRLR
…”, “RLRRLRRLRRL …”
Shuffling as a Dynamical System
1

0 x
 2 x,
2
S ( x)  
.
2 x  1, 1  x  1

2
S(x)
vs
S4(x)
Demonstration
Iterations for 8 Cards
S3(x) vs S2(x)
S3(x)
vs
S2(x)
How can we study periodic orbits for S(x)?
Binary Representations

Binary Representation
 0.10112=1(2-1)+0(2-2)+1(2-3)+1(2-4)
 1/2
 xn+1
=
+ 1/8 + 1/16 = 10/16 = 5/8
= S(xn), given x0
 Represent
xn’s 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…
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
Card Shuffling Examples

8 Cards – All orbits are period 3
{1/7, 2/7, 4/7} and {3/7, 6/7, 5/7} and {0/7, 7/7}
52 Cards – Period 2
1/3 = ?/51 and 2/3 = ?/51
 50 Cards – Period 3 Orbit (Cycle)
1/7 = ?/49
 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
Finding Specific t-Cycles

Period k: 0.000 … 0001



Examples



2-t + (2-t)2 + (2-t) 3 + … = 2-t /(1- 2-t )
Or, 0.000 … 0001 = 1/(2t -1)
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
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)


Continuous Dynamical Systems (ODES)





Systems in the Plane and Higher Dimensions
Integrability
Nonlinear Oscillations
MAT 463/563
Fractals
Chaos
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

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

Websites
http://i-p-c-s.org/history.html
 http://jducoeur.org/game-hist/seaan-cardhist.html
 http://www.usplayingcard.com/gamerules/briefhistory.html
 http://bcvc.net/faro/
 http://www.gleeson.us/faro/

Thank you !