w r ,. , math PUBLISHED BY THE MATHEMATICAL ASSOCIATION OF AMERICA In this issue 3 Experiencecan't Be Beat 8 1996-A Triple Anniversary Math Hmizons is for undergraduates and otherswho areinterested in mathe- 14 The Mathematician Who Became matics. Its purpose is to expand both the career and intellectual horizons of Secretary of Defense students. 22 Mathematics in Unexpected Places DONALD J. ALBERS Editor 26 Beyond the Last Theorem CAROL BAXTER Managing Editor 32 Problems JANE D'ALELIO Art Director 35 A Mathematical Word Puzzle Math Horizons (ISSN 1072-4117) is pub- lished four times a year; September, November, February, and April by the Mathematical Association of America, 1529 Eighteenth Street, NW, Washing- ton, DC 20036. September 1996 Vol- ume IV,Issue 1.Periodicalspostage paid atwashington, DC and additional mail- ing offices. Annual subscription rates are $20.00 for MAAmembersand $35.00 About the cover: the Honorable WilliamJ Perry, Secretav of Dqense, stands for nonmembers and libraries. Bulk outside the przmaryflight control bndge of the USS Dwight D. Eisenhower. subscriptions sent to a single address are encouraged. The minimum order is 20 copies ($120.00); additional sub- scriptions may be ordered in units of 10 for $60.00 per unit. For advertisingrates or editorial inquiries call (202) 387- 5200. Printed in the United States of Cover photograph of Secreta?y Pqby RD. Ward courtesy of the Department America. Copyright O 1996 The Math- of Defense. ematical Association ofAmerica. POST- MASTER: Send address changes to Math Horizons, MAA Service Center, PO Box 9 11 12, Washington, DC 20090-11 12. THE MATHEMATICAL ASSOCIATION OF AMERICA 1529 Eighteenth Street, N.W. Washington, D.C. 20036 Math Horizons September 1996 3 1 I I mathHORIZONS ADVISORY BOARD First Four Winners I GERALDL. ALEXANDERSON I Santa Clara University TOMAPOSTOL of Trevor Evans Awards I California Institute of Tecl~nolog)' GEORGEBERZSENM I Rose-Hulmun Instilute of Technology ROBERTBOZEMAN Four outstanding authors of articles in Math Horizons were honored at the I Morehouse College summer meeting of the MAA in Seattle this August. They enjoy the distinction of MARVINBRUBAKER being the first group of Trevor Evans Awards winners. We salute them for their I Messiah College exemplary writing and eagerly await their next articles. BmmFAIRES I Westminster College The Awards are named for Trevor Evans, a distinguished mathematician, DEBORAHFRANTZ teacher and writer at Emory University. I Kutztown University JOEGALLTAN I University of Minnesota-Duluth I APARNAHIGGINS Joel Chan Underwood Dudley University of Dayton Student, Professor of I ROBERTHOOD Editor Em.tus, Boys' Life University of Mathematics, 1 HAROLDJACOBS Toronto. DePauw University U.S. Grant High School, Van Nuys, CA "As Easy as Pi" 'W'hy History?" ' I SANDRAKEITH St. Cluud State University Winter 1993. November 1994. I LEONARDKLOSINSKI Sanla Clara University I JOEMALKWTCH Ywk College I CLEOPATRIAMARTINEZ I ScotL~duleCommunity CoUege JosephGallian Alan Tucker ROBERTMEGGINSON Professor of SUNY I UniversiQ of Michigan RICHARDNEAL Mathematics Distinguished I rl University of Oklahoma University of Teaching Professor HENRYPOLLAK Minnesota, Dulutf SUNY-Stony Brook 1 Teachers College, Columbia University 'Weird Dice" "The Parallel FREDAPORTER-LOCKLWR February 1995. Climbers Puzzle" n Pembroke State University PETERRENZ November 1995. , 1 il Academic Press I V.FREDERICK ~CKY il Bowling Crea State University n RONALDC. ROSIER q Conference Board of tlw Mathematical Sciences If you happened to miss any of these articles, read them now! They are MARK SAUL absolutely first rate. Bronxville School, NY :I ANITASOLOM' Ginnell College h DREW STERRETT,JR. The Mathematical Associalion of Ama'ca n IANSTEM'ART Warwick University Don Albers n JUDITHT~UR State University of Neru Yo&, Stony Brook IRVINVANCE Michigan State University 1 PETERWALTHER I1 Salish-Kootmai College How to Reach Us ANNWATKINS e-mail: [email protected] Call: (202) 387-5200 Fax: (202) 265-2384 n Califmrzia State University, Nurthridge Write: Math Horizons, The Mathematical Association of America, ROBINWILSON The Open University 1529 Eighteenth Street, N.W., Washington, D.C. 20036. !n1 4 Math Horizons September 1996 /Ir-1 -1 AMY J. HEFFELFINGER, BRENDA R. HORST, AND MELANIE B. STARR Experience Can't Be Beat chool is about learning. However, erized data entry systems, preparing our supervisors carefully explain the it is also about preparing for a datasets for statistical analysis, and per- nature of the data and how the pro- S future in the professional com- forming statistical analyses as directed posed analysis will accurately answer munity. Bridging the gap between aca- by statisticians. We are also involved in the question under investigation. demic endeavors and a career in the the preparation of reports for investiga- Though some procedures are familiar "real wor1d"is a challenge facing almost tors (usually physicians) or hospital to us through our classroom studies, all college seniors, especially during administrators. That involvement may many are beyond the scope of our col- the final months precedinggraduation. include anything from creating tables lege courses; these include survival For us, the key to the transition has analysis, repeated measures, and test- been our internship experiences. ing for equivalence. Encountering pre- We are senior mathematics majors viously studied techniques reinforces with minors in statistics at Messiah Col- our classroom experience; exposure to lege, located in Grantham, Pennsylva- Bridging the gap new ones prepares us for further gradu- nia. This year the three of us have had ate study. the opportunity to gain invaluable, prac- between academic Second, we are given a glimpse into tical experience in our field of study the medical field. We are exposed to through working as statistical analyst endeavors and a cutting edge biomedical research in- interns in the Center for Biostatistics vestigating diseases such as asthma, in- and Epidemiology (CBE) at Penn State's career in the "real terstitial cystitis, and sleep apnea. Other College of Medicine, the Milton S. studies explore bone density in young Hershey Medical Center in Hershey, world" is a challenge women, computed tomography tech- Pennsylvania. niques, methods of cholesterol control, CBE is a fairly recent and rapid de- facing almost all and the bioequivalence of drugs. In velopment. Itwas created in 1987 in an addition, we become familiar with the attempt to meet the College of college seniors. components of running clinical trials, Medicine's increasing need for biostatis- such as protocol structure, randomiza- tical and epidemiological resources. Its tion, the use of placebos, drug assign- growth has been tremendous. CBE now ments, and sample size calculations. houses over forty full-time employees at Furthermore, we are given the op- various educational levels-quite ajump and graphs to contributing our own portunity to enhance our computer from the four employees of just eight creative ideas about the format of a programmingskillsand experience, and years ago. In 1992, the structure of CBE report. to absorb knowledge of various com- was expanded to encompass three sepa- Such experience has not only been puter systems. Each of us work at an rate units: biostatistics, epidemiology, valuable for the development of skills. individual work station equippedwith a and health services research. Just as important is the exposure to the UNIX Sun SPARCstation usingasolaris How do statistical analyst interns fit numerous facets of biomedical research, Open Windows Version 3.4 operating into this picture? Most of our work an experience which cannot be repli- system. Through our daily tasks and involves designing and using comput- cated in the classroom. exploration of the system, we have in- First of all, we encounter the variety creased our knowledge and skill in of statistical analyses and procedures moving about the system and finding AMY J. HEFFELFINGER, BRENDA R. HORST being implemented in current medical ways to perform tasks more quickly and and MELANIE B. STARR are senior math- studies. Before we begin the analysis elegantly. Many of these skills are logic ematics majors at Messiah College. programming for a particular study, skills that are transferable to any sys- Math Horizons September 1996 5 Students Melanie Staw, Amy Heffeelfinnga, and Brenda Horst preparing datasets for statistical analysis at the Centerfor Biostatistics and Epidemiology at Penn State's College of Medicine, the Milton S. Hershey Medical Center in Hershey Pennsylvania. tem, while others are unique to a given in trying to accomplish the task ofmak- group continued supplementation software package. Each experiencewith ing the report aesthetically pleasing as while the other half began taking place- a new system or software package makes well as informative. bos. Likewise, half of the original pla- learning the next one easier. One of the studies that Amy works cebo group continued taking placebos, By far most of the work that we do on is part of the Penn State's Young while the other half began supplemen- involves programming in SAS, a lead- Women's Health Study. The study is a tation. The group that switched from ing statistical software package, but col- randomized, double-blind, placebo- placebos to calcium supplementation lectively we have also used other pack- controlled trial that investigates the ef- had the greatest annualized gain in all ages-S-Plus, StatXact, and MINITAB.