Card Shuffling as a Dynamical System Dr Russell
- Slides: 47
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 l History of the Faro Shuffle l The Perfect Shuffle l Mathematical Models of Perfect Shuffles l Dynamical Systems – The Logistic Model l Features of Dynamical Systems l Shuffling as a Dynamical System
A Bit of History
History of the Faro Shuffle Cards l Western Culture - 14 th Century l Jokers – 1860’s l Pips – 1890’s added numbers l First Card tricks by gamblers l Origins of Perfect Shuffles not known l Game of Faro l 18 th Century France l Named after face card l Popular 1803 -1900’s in the West l
The Game of Faro l Decks shuffled and rules are simple l (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 19 th 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 l Players bet on 13 cards l Lose Slowly! l Copper Tokens – bet card to lose l “Coppering”, “Copper a Bet” l 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: l Divide 52 cards into 2 equal piles l Shuffle by interlacing cards l Keep top card fixed (Out Shuffle) l 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 l 1726 – Warning in book for first time l 1847 – J H Green – Stripper (tapered) Cards l 1860 – Better description of shuffle l 1894 – How to perform l Koschitz’s Manual of Useful Information l Maskelyne’s Sharps and Flats – 1 st Illustration l 1915 – Innis – Order for 52 Cards l 1948 – Levy – O(p) for odd deck, cycles l 1957 – Elmsley – Coined In/Out - shuffles
Mathematical Models
A Model for Card Shuffling l Label the positions 0 -51 l Then l 0 ->0 and 26 ->1 l 1 ->2 and 27 ->3 l 2 ->4 and 28 ->5 l … in general? l Ignoring card 51: f(x) = 2 x mod 51 l Recall Congruences: l 2 x mod 51 = remainder upon division by 51
The Order of a Shuffle l Minimum integer k such that 2 k x = x mod 51 for all x in {0, 1, …, 51} l True for x = 1 ! l Minimum integer k such that 2 k - 1= 0 mod 51 l Thus, 51 divides 2 k - 1 k= 6, 2 k - 1 = 63 = 3(21) l k= 7, 2 k - 1 = 127 l k= 8, 2 k - 1 = 255 = 5(51) l
Generalization to n cards
The Out Shuffle
The In Shuffle
Representations for n Cards l In Shuffles l Out Shuffles
Order of Shuffles l 8 Out Shuffles for 52 Cards l In General? lo (O, 2 n-1) = o (O, 2 n) l o (I, 2 n-1) = o (O, 2 n) l => o lo (O, 2 n-1) = o (I, 2 n-1) (I, 2 n-2) = o (O, 2 n) l Therefore, only need o (O, 2 n)
o (O, 2 n) = Order for 2 n Cards l One Shuffle: O(p) = 2 p mod (2 n-1), 0<p<N-1 l 2 shuffles: O 2(p) = 2 O(p) mod (2 n-1) = 22 p mod (2 n-1) l k shuffles: l Order: o (O, 2 n) = smallest k for 0 < p < 2 n such that Ok(p) = p mod (2 n-1) l Or, 2 k = 1 mod (2 n-1) => (2 n – 1) | (2 k – 1) Ok(p) = 2 kp mod (2 n-1)
The Orders of Perfect Shuffles n o(O, n) o(I, n) 2 13 12 12 3 2 2 14 12 4 4 2 4 15 4 4 16 4 8 6 4 3 17 8 8 7 3 3 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 2 n Cards l Label positions with rationals l Out Shuffle l Example: Card 10 of 52: x = 9/51 l In Shuffle l Example: Card 10 of 52: x = 9/51
Shuffle Types All denominators are odd numbers.
Doubling Function
Discrete Dynamical Systems l First Order System: xn+1 = f l Orbits: {x 0, x 1, … } l Fixed Points l Periodic Orbits l Stability and Bifurcation l Chaos !!!! ( xn )
The Logistic Map l Discrete Population Model l Pn+1 = a Pn l Pn+1 = a 2 Pn-1 l Pn+1 = an P 0 l a>1 => exponential growth! l Competition l Pn+1 = a Pn - b Pn 2 l xn = (a/b)Pn, r=a/b => l xn+1 = r xn(1 - xn), xne[0, 1] and re[0, 4]
Example r=2. 1 Sample orbit for r=2. 1 and x 0 = 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* l x* = r x*(1 -x*) l => 0 = x*(1 -r (1 -x*) ) l => x* = 0 or x* = 1 – 1/r l Logistic Map - Cobwebs
Periodic Orbits for f(x)=rx(1 -x) l Period 2 l x 1 = r x 0(1 - x 0) and x 2 = r x 1(1 - x 1) = x 0 l Or, f 2 (x 0) = x 0 l Period k l - smallest k such that f k (x*) = x* l Periodic Cobwebs
Stability l Fixed Points l |f’(x*)| < 1 l Periodic Orbits l |f’(x 0)| |f’(x 1)| … |f’(xn)| < 1 l Bifurcations
Bifurcations r 1 = 3. 0 r 2 = 3. 449490. . . r 3 = 3. 544090. . . r 4 = 3. 564407. . . r 5 = 3. 568759. . . r 6 = 3. 569692. . . r 7 = 3. 569891. . . r 8 = 3. 569934. . .
Itineraries: Symbolic Dynamics l. For G (x) = 4 x ( 1 -x ) Assign Left “L” and Right “R” l Example: x 0 = 1/3 l x 0 = 1/3 => “L” l x 1 = 8/9 => “LR” l x 2 = 32/81 => “LRL” l x 3 = … => “LRL …” l Example: x 0 = ¼ l { ¼, ¾, ¾, …}=>” LRRRR…” l. Periodic Orbits l“LRLRLR …”, “RLRRLRRLRRL …”
Shuffling as a Dynamical System S(x) vs S 4(x)
Demonstration
Iterations for 8 Cards
S 3(x) vs S 2(x) S 3(x) vs S 2(x) How can we study periodic orbits for S(x)?
Binary Representations l Binary Representation l 0. 10112=1(2 -1)+0(2 -2)+1(2 -3)+1(2 -4) = l 1/2 + 1/8 + 1/16 = 10/16 = 5/8 l xn+1 = S(xn), given x 0 l Represent xn’s in binary: x 0 = 0. 101101 l Then, x 1 = 2 x 0 – 1 = 1. 01101 – 1 = 0. 01101 l Note: S shifts binary representations! l Repeating Decimals l S(0. 101101…) = 0. 011011… l S(0. 011011…) = 0. 110110…
Periodic Orbits l Period 2 l. S(0. 1010…) = 0. 0101… l. S(0. 0101…) = 0. 1010… l 0. 102, 0. 012, 0. 112 = ? l Period 3 l 0. 1002, 0. 0102, 0. 0012 = ? l 0. 1102, 0. 0112, 0. 1012 = ? l Maple Computations
Card Shuffling Examples l 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} l 52 Cards – Period 2 1/3 = ? /51 and 2/3 = ? /51 l 50 Cards – Period 3 Orbit (Cycle) 1/7 = ? /49 l Recall: l Period 2 - {1/3, 2/3} l Period 3 – {1/7, 2/7, 4/7} and {3/7, 6/7, 5/7} l Out Shuffles – i/(N-1) for (i+1) st card
Finding Specific t-Cycles l Period k: 0. 000 … 0001 2 -t + (2 -t)2 + (2 -t) 3 + … = 2 -t /(1 - 2 -t ) l Or, 0. 000 … 0001 = 1/(2 t -1) l l Examples Period 2: 1/3 l Period 3: 1/7 l l In general: Select Shuffle Type Rationals of form i/r => (2 t – 1) | r l Example r = 3(7) = 21 l Out Shuffle for 22 or 21 cards l In Shuffle for 20 or 21 cards l Demonstration
Other Topics l Cards Alternate In/Out Shuffles l k- handed Perfect Shuffles l Random Shuffles – Diaconis, et al l “Imperfect” Perfect Shuffles l l Nonlinear Dynamical Systems l Discrete (Difference Equations) l l Systems in the Plane and Higher Dimensions Continuous Dynamical Systems (ODES) Integrability l Nonlinear Oscillations l MAT 463/563 l Fractals l Chaos l
Summary l History of the Faro Shuffle l The Perfect Shuffle – How to do it! l Mathematical Models of Perfect Shuffles l Dynamical Systems – The Logistic Model l Features of Dynamical Systems l Symbolic Dynamics l Shuffling as a Dynamical System
References K. T. Alligood, T. D. Sauer, J. A. Yorke, Chaos, An Introduction to Dynamical Systems, Springer, 1996. l S. B. Morris, Magic Tricks, Card Shuffling and Dynamic Computer Memories, MAA, 1998 l D. J. Scully, Perfect Shuffles Through Dynamical Systems, Mathematics Magazine, 77, 2004 l
Websites l l l 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 !
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