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Card Shuffling as a Dynamical System Dr. Russell Herman
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 !