AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 43

Mathematics and Sports

Joseph A. Gallian, Editor

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Mathematics and Sports

10.1090/dol/043

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c 2010 by

The Mathematical Association of America (Incorporated)

Library of Congress Catalog Card Number 2010931026

Print ISBN 978-0-88385-349-8

Electronic ISBN 978-1-61444-200-4

Printed in the United States of America

Current Printing (last digit):

10 9 8 7 6 5 4 3 2 1

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The Dolciani Mathematical Expositions

NUMBER FORTY-THREE

Mathematics and Sports

Edited by

Joseph A. Gallian

University of Minnesota Duluth

Published and Distributed by

The Mathematical Association of America

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DOLCIANI MATHEMATICAL EXPOSITIONS

Committee on Books

Gerald Bryce, Chair

Dolciani Mathematical Expositions Editorial Board

Underwood Dudley, Editor

Jeremy S. Case

Rosalie A. Dance

Tevian Dray

Patricia B. Humphrey

Virginia E. Knight

Michael J. McAsey

Mark A. Peterson

Jonathan Rogness

Thomas Q. Sibley

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The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical As-

sociation of America was established through a generous gift to the Association from

Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University

of New York. In making the gift, Professor Dolciani, herself an exceptionally talented

and successful expositor of mathematics, had the purpose of furthering the ideal of

excellence in mathematical exposition.

The Association, for its part, was delighted to accept the gracious gesture initiat-

ing the revolving fund for this series from one who has served the Association with

distinction, both as a member of the Committee on Publications and as a member of

the Board of Governors. It was with genuine pleasure that the Board chose to name

the series in her honor.

The books in the series are selected for their lucid expository style and stimulating

mathematical content. Typically, they contain an ample supply of exercises, many

with accompanying solutions. They are intended to be sufficiently elementary for the

undergraduate and even the mathematically inclined high-school student to under-

stand and enjoy, but also to be interesting and sometimes challenging to the more

advanced mathematician.

1. Mathematical Gems, Ross Honsberger

2. Mathematical Gems II, Ross Honsberger

3. Mathematical Morsels, Ross Honsberger

4. Mathematical Plums, Ross Honsberger (ed.)

5. Great Moments in Mathematics (Before 1650), Howard Eves

6. Maxima and Minima without Calculus, Ivan Niven

7. Great Moments in Mathematics (After 1650), Howard Eves

8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette

9. Mathematical Gems III, Ross Honsberger

10. More Mathematical Morsels, Ross Honsberger

11. Old and New Unsolved Problems in Plane Geometry and Number Theory,

Victor Klee and Stan Wagon

12. Problems for Mathematicians, Young and Old, Paul R. Halmos

13. Excursions in Calculus: An Interplay of the Continuous and the Discrete,Robert

M. Young

14. The Wohascum County Problem Book, George T. Gilbert, Mark Krusemeyer,

and Loren C. Larson

15. Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics,

Verse, and Stories by Ralph P. Boas, Jr., edited by Gerald L. Alexanderson and

Dale H. Mugler

16. Linear Algebra Problem Book, Paul R. Halmos

17. From Erdos to Kiev: Problems of Olympiad Caliber, Ross Honsberger

18. Which Way Did the Bicycle Go? . . . and Other Intriguing Mathematical Mys-

teries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon

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19. In Polya’s Footsteps: Miscellaneous Problems and Essays, Ross Honsberger

20. Diophantus and Diophantine Equations, I. G. Bashmakova (Updated by Joseph

Silverman and translated by Abe Shenitzer)

21. Logic as Algebra, Paul Halmos and Steven Givant

22. Euler: The Master of Us All, William Dunham

23. The Beginnings and Evolution of Algebra, I. G. Bashmakovaand G. S. Smirnova

(Translated by Abe Shenitzer)

24. Mathematical Chestnuts from Around the World, Ross Honsberger

25. Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures,

Jack E. Graver

26. Mathematical Diamonds, Ross Honsberger

27. Proofs that Really Count: The Art of Combinatorial Proof, Arthur T. Benjamin

and Jennifer J. Quinn

28. Mathematical Delights, Ross Honsberger

29. Conics, Keith Kendig

30. Hesiod’s Anvil: falling and spinning through heaven and earth, Andrew J.

Simoson

31. A Garden of Integrals, Frank E. Burk

32. A Guide to Complex Variables (MAA Guides #1), Steven G. Krantz

33. Sink or Float? Thought Problems in Math and Physics, Keith Kendig

34. Biscuits of Number Theory, Arthur T. Benjamin and Ezra Brown

35. Uncommon Mathematical Excursions: Polynomia and Related Realms, Dan

Kalman

36. When Less is More: Visualizing Basic Inequalities, Claudi Alsina and Roger B.

Nelsen

37. A Guide to Advanced Real Analysis (MAA Guides #2), Gerald B. Folland

38. A Guide to Real Variables (MAA Guides #3), Steven G. Krantz

39. Voltaire’s Riddle: Micromegas and the measure of all things, Andrew J.

Simoson

40. A Guide to Topology, (MAA Guides #4), Steven G. Krantz

41. A Guide to Elementary Number Theory, (MAA Guides #5), Underwood Dudley

42. Charming Proofs: A Journey into Elegant Mathematics, Claudi Alsina and

Roger B. Nelsen

43. Mathematics and Sports, edited by Joseph A. Gallian

MAA Service Center

P.O. Box 91112

Washington, DC 20090-1112

1-800-331-1MAA FAX: 1-301-206-9789

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Preface

Each year the Joint Policy Board for Mathematics (JPBM) sponsors Math-

ematics Awareness Month to increase public understanding of and appre-

ciation for mathematics. This is achieved through a web page, posters, re-

source materials, and theme essays. With its abundance of data, great vari-

ety, numerous strategies, and widespread popularity, sports is an ideal venue

to demonstrate the illuminating power of mathematics to a larger audience.

This book is an eclectic compendium of the essays solicited for the 2010

Mathematics Awareness Month on the theme Mathematics and Sports.

In keeping with the goal of promoting mathematics awareness to a broad

audience, all of the articles are accessible to college level mathematics stu-

dents and many are accessible to the general public.

The book is divided into sections by the kind of sports. The section on

football includes articles on ranking college football teams; counting the

number of defensive formations an NFL team can deploy; and evaluating

a method for reducing the advantage of the winner of a coin flip in an NFL

overtime game.

The section on track and field examines the ultimate limit on how fast a

human can run 100 meters; ranking college track and field conferences; ways

to determine the winning team in a cross-country race; the effects of wind

and altitude in a 400 meter race; and modeling the biomechanics of running

and walking.

The section on baseball has essays on ways to measure the performance

of baseball players; deciding if humidifying baseballs reduces the number of

home runs; and the probability of batting streaks.

The section on golf has articles on strategies in golf; modeling a golf

swing; breaking down Tiger Woods game; and modeling Tiger Woods ca-

reer.

For basketball there are articles about strategies in basketball at the end of

the game; modeling jump shots; and ranking college basketball teams.

For tennis there is an article about strategies in tennis and an essay about

using tennis to teach mathematics.

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viii Preface

The last section features articles about how to schedule a tournament; the

aerodynamics of a free kick in soccer; and tire design for NASCAR cars.

I am grateful to Woody Dudley for meticulously reading every article and

providing valuable advice for improving the exposition.

Joe Gallian

University of Minnesota Duluth

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Contents

Preface vii

I Baseball 1

1 Sabermetrics: The Past, the Present, and the Future

Jim Albert 3

2 Surprising Streaks and Playoff Parity: Probability Problems in

a Sports Context

Rick Cleary 15

3 Did Humidifying the Baseball Decrease the Number of Homers

at Coors Field?

Howard Penn 23

4 Streaking: Finding the Probability for a Batting Streak

Stanley Rothman and Quoc Le 31

II Basketball 53

5 Bracketology: How can math help?

Tim Chartier & Erich Kreutzer & Amy Langville

& Kathryn Pedings 55

6 Down 4 with a Minute to Go

G. Edgar Parker 71

7 Jump Shot Mathematics

Howard Penn 85

ix

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x Contents

III Football 91

8 How Deep Is Your Playbook?

Tricia Muldoon Brown and Eric B. Kahn 93

9 A Look at Overtime in the NFL

Chris Jones 109

10 Extending the Colley Method to Generate Predictive Football

Rankings

R. Drew Pasteur 117

11 When Perfect Isn’t Good Enough: Retrodictive Rankings

in College Football

R. Drew Pasteur 131

IV Golf 147

12 The Science of a Drive

Douglas N. Arnold 149

13 Is Tiger Woods a Winner?

Scott M. Berry 157

14 G. H. Hardy’s Golfing Adventure

Roland Minton 169

15 Tigermetrics

Roland Minton 179

V NASCAR 187

16 Can Mathematics Make a Difference? Exploring Tire Troubles

in NASCAR

Cheryll E. Crowe 189

VI Scheduling 201

17 Scheduling a Tournament

Dalibor Froncek 203

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Contents xi

VII Soccer 217

18 Bending a Soccer Ball with Math

Tim Chartier 219

VIII Tennis 225

19 Teaching Mathematics and Statistics Using Tennis

Reza Noubary 227

20 Percentage Play in Tennis

G. Edgar Parker 241

IX Track and Field 257

21 The Effects of Wind and Altitude in the 400m Sprint

with Various IAAF Track Geometries

Vanessa Alday and Michael Frantz 259

22 Mathematical Ranking of the Division III Track and

Field Conferences

Chris Fisette 279

23 What is the Speed Limit for Men’s 100 Meter Dash

Reza Noubary 287

24 May the Best Team Win: Determining the Winner of a Cross

Country Race

Stephen Szydlik 295

25 Biomechanics of Running and Walking

Anthony Tongen and Roshna E. Wunderlich 315

About the Editor 329

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About the Editor

Joseph A. Gallian was born in Arnold, Pennsylvania on January 5, 1942.

He obtained a B.A. from Slippery Rock University in 1966, an M.A. from

the University of Kansas in 1968 and a Ph.D. from the University of Notre

Dame in 1971. After serving as a visiting Assistant Professor at Notre Dame

for one year, he went to the University of Minnesota Duluth where he is a

University Distinguished Professor of Teaching.

Among his honors are the MAA’s Haimo Award for distinguished teach-

ing, the MAA Allendoerfer and Evans awards for exposition, MAA Polya

Lecturer, MAA Second Vice President, MAA President, co-director of the

MAA Project NExT, associate editor of the American Mathematical Monthly

and the Mathematics Magazine, advisory board member for Math Horizons,

the Carnegie Foundation for the Advancement of Teaching Minnesota Pro-

fessor of the Year, and recipient of the University of Minnesota Duluth Chan-

cellor’s Award for Distinguished Research.

Over 150 research papers written under Gallian’s supervision by under-

graduates have been published in mainstream journals. He has given more

than 250 invited lectures at conferences and colleges and universities and

written more than 100 articles. He is the author of Contemporary Abstract

Algebra, Cengage, 7th edition, coauthor of For All Practical Purposes, W.H.

Freeman, 8th edition, and coauthor of Principles and Practices of Mathe-

matics, Springer. He is the editor of two conference preceedings published

by the American Mathematical Society and the Executive Producer of the

documentary film “Hard Problems: The Road to the World’s Toughest Math

Contest.” Gallian has received more than $4,000,000 in grants.

Besides the usual math courses, Gallian has taught a Humanities course

called the “The Lives and Music of the Beatles” for more than 30 years and

a liberal arts course on math and sports. In 2000 a Duluth newspaper cited

him as one of the “100 Great Duluthians of the 20th Century.”

329

AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS

Mathematics and Sports | Edited by Joseph A. Gallian

This book is an eclectic compendium of the essays solicited for the 2010 Mathematics Awareness Month web page on the theme of Mathematics and Sports. In keeping with the goal of promoting mathematics awareness to a broad audience, all of the articles are accessible to college level mathematics students and many are accessible to the general public.

The book is divided into sections by the kind of sports. The section on football includes an article that evaluates a method for reducing the advantage of the winner of a coin fl ip in an NFL overtime game; the section on track and fi eld examines the ulti-mate limit on how fast a human can run 100 meters; the section on baseball includes an article on the likelihood of streaks; the section on golf has an article that describes the double-pendulum model of a golf swing, and an article on modeling Tiger Wood’s career.

The articles provide source material for classroom use and student projects. Many students will fi nd mathematical ideas motivated by examples taken from sports more interesting than the examples selected from traditional sources.