**UMAP**

_{Publisher}^{Journal}

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**Vol. 29, No. 3**

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**Table of Contents** **Publisher’s Editorial**

**Change**

Solomon A. Garfunkel ... 185
**About This Issue... 186**

**Special Section on the MCM**

**Results of the 2008 Mathematical Contest**
**in Modeling**

Frank Giordano ... 187
**Abstracts of the Outstanding Papers**

**and the Fusaro Papers ... 223**
**The Impending Effects of North Polar Ice Cap Melt**

Benjamin Coate, Nelson Gross,

and Megan Longo... 237
**A Convenient Truth: Forecasting Sea Level Rise**

Jason Chen, Brian Choi, and ... 249
**Fighting the Waves: The Effect of North Polar**

**Ice Cap Melt on Florida**

Amy M. Evans and Tracy L. Stepien... 267
**Erosion in Florida: A Shore Thing**

Matt Thies, Bob Liu, and Zachary W. Ulissi ... 285
**Judge’s Commentary: The Polar Melt Problem Papers**

John L. Scharf ... 301
**A Difficulty Metric and Puzzle Generator for Sudoku**

Christopher Chang, Zhou Fan, and Yi Sun ... 305
**Taking the Mystery Out of Sudoku Difficulty:**

**An Oracular Model**

Sarah Fletcher, Frederick Johnson, and

David R. Morrison ... 327
**Difficulty-Driven Sudoku Puzzle Generation**

Martin Hunt, Christopher Pong, and

George Tucker ... 343
**Ease and Toil: Analyzing Sudoku**

Seth B. Chadwick , Rachel M. Krieg, and

Christopher E. Granade ... 363
**Cracking the Sudoku: A Deterministic Approach**

David Martin, Erica Cross, and Matt Alexander ... 381

**Publisher’s Editorial**

**Change**

### Solomon A. Garfunkel

Executive Director COMAP, Inc.

175 Middlesex Turnpike, Suite 3B Bedford , MA 01730–1459

s.garfunkel@mail.comap.com

This is the season of change—for good and bad. As I write this edito- rial, the election is a little less than a month away. The financial markets are imploding and the country appears more than ready to head in a new direction, even if it is unsure where that direction will take us. By the time you read this, many things will be clear. We will have a new adminis- tration, perhaps a very new administration. And we will likely be living individually and collectively on less—perhaps a lot less.

No matter. Some things still need to be done. I won’t speak in this forum about health care or infrastructure or other changes in foreign and domes- tic policy—but I will speak of mathematics education. As small as our issues may seem at times of national and international stress, education—

especially technical education—can always provide a way out and up. We cry out for mathematical and quantitative literacy, not because we are lob- byists or a special interest group trying to raise teacher salaries. We cry out for literacy because knowledge is the only way to prevent the abuses whose consequences we now endure.

How many times in these last few months have we heard about peo- ple who didn’t understand the terms of their mortgages; of managers and bankers who didn’t understand their degree of risk; of policy makers who didn’t understand how the dominos could fall? Yes, derivatives are confus- ing. And yes, derivatives of derivatives are more confusing. But isn’t this just a perfect example of why we talk about teaching mathematical mod- eling as a life skill? Mathematics education is not a zero-sum game. We don’t want our students to learn more mathematics than other countries’

students. That is just a foolish argument used to raise money, that is, the fear that another country will out perform us or another state will take our

high tech jobs.

The problem is much, much bigger. There simply are not enough mathematically-trained people in the world to run the world. The proof of that statement is all around us. And it is as much in our interest that the world’s people become more quantitatively literate as it is that the citizens of our city, our state, and our country do. In theory, now there is less money to fund changes in mathematics education. But we must. We must see the issues and problems, as global issues and problems and work together to solve them.

The good news is that the energy and commitment to do the job are here. At the recent conference on the Future of Mathematics Education, co-sponsored by Math is More, I met with mathematics and mathemat- ics education researchers, with college and high school faculty, with state and local administrators, with policy-makers, and with employers. We no longer talked about why; we talked about how. The need and desire for real change was palpable. And the energy was both exciting and challeng- ing. People kept asking, “What can I do?”—as a classroom teacher, as a supervisor of mathematics, as a staff developer, as a curriculum developer, as a policy maker.

So while the times and problems are difficult, the will for positive change is here. Now is the time for all of us to gather together to make that change a reality.

**About This Issue**

### Paul J. Campbell

Editor

This issue runs longer than a regular 92-page issue, to more than 200
pages. However, not all of the articles appear in the paper version. Some
*appear only on the Tools for Teaching 2008 CD-ROM (and at*http://www.

comap.com for COMAP members), which will reach members and sub-
*scribers later and will also contain the entire 2008 year of Journal issues.*

All articles listed in the table of contents are regarded as published in
*the Journal. The abstract of each appears in the paper version. Pagination*
of the issue runs continuously, including in sequence articles that do not
*appear in the paper version. So if, say, p. 250 in the paper version is followed*
*by p. 303, your copy is not necessarily defective! The articles on the intervening*
pages are on the CD-ROM.

We hope that you find this arrangement agreeable. It means that we do not have to procrusteanize the content to fit a fixed number of paper pages.

We might otherwise be forced to select only two or three Outstanding MCM papers to publish. Instead, we continue to bring you the full content.

**Modeling Forum**

**Results of the 2008**

**Mathematical Contest in Modeling**

### Frank Giordano, MCM Director

Naval Postgraduate School 1 University Circle

Monterey, CA 93943-5000 frgiorda@nps.navy.mil

**Introduction**

A total of 1,159 teams of undergraduates, from 338 institutions and 566 departments in 14 countries, spent the first weekend in February working on applied mathematics problems in the 24th Mathematical Contest in Modeling.

The 2008 Mathematical Contest in Modeling (MCM) began at 8:00 P.M. EST on Thursday, February 14 and ended at 8:00 P.M. EST on Monday, February 18.

During that time, teams of up to three undergraduates were to research and submit an optimal solution for one of two open-ended modeling problems.

Students registered, obtained contest materials, downloaded the problems at
the appropriate time, and entered completion data through COMAP’s MCM
Website. After a weekend of hard work, solution papers were sent to COMAP
on Monday. The top papers appear in this issue of*The UMAP Journal.*

Results and winning papers from the first 23 contests were published in
special issues of*Mathematical Modeling (1985–1987) and The UMAP Journal*
(1985–2007). The 1994 volume of*Tools for Teaching, commemorating the tenth*
anniversary of the contest, contains the 20 problems used in the first 10 years
of the contest and a winning paper for each year. That volume and the special
MCM issues of the*Journal for the last few years are available from COMAP. The*
1994 volume is also available on COMAP’s special*Modeling Resource CD-ROM.*

Also available is*The MCM at 21 CD-ROM, which contains the 20 problems*
from the second 10 years of the contest, a winning paper from each year, and
advice from advisors of Outstanding teams. These CD-ROMs can be ordered
from COMAP athttp://www.comap.com/product/cdrom/index.html.

This year’s Problem A asked teams to consider the effects on land from the melting of the North Polar ice cap due to the predicted increase in global temperatures. Specifically, teams were asked to model the effects on the coast of Florida due to the melting every 10 years for the next 50 years, with particular attention to large metropolitan areas. Additionally, they were asked to propose appropriate responses to deal with the melting.

Problem B asked teams to develop an algorithm to construct Sudoku puz- zles of varying difficulty. The problem required teams to develop metrics to define a difficulty level. Further, the team’s algorithm and metrics were to be extensible to a varying number of difficulty levels, and they should illustrate their algorithm with at least four difficulty levels. The team’s solution had to analyze the complexity of their algorithm.

The 9 Outstanding solution papers are published in this issue of*The UMAP*
*Journal, along with relevant commentaries.*

In addition to the MCM, COMAP also sponsors the Interdisciplinary Con- test in Modeling (ICM) and the High School Mathematical Contest in Modeling (HiMCM). The ICM runs concurrently with MCM and offers a modeling prob- lem involving concepts in operations research, information science, and inter- disciplinary issues in security and safety. The 2009 problem will have an envi- ronmental science theme. Results of this year’s ICM are on the COMAP Website at http://www.comap.com/undergraduate/contests; results and Out- standing papers appeared in Vol. 29 (2008), No. 2. The HiMCM offers high school students a modeling opportunity similar to the MCM. Further details about the HiMCM are athttp://www.comap.com/highschool/contests.

**Problem A: Take a Bath**

Consider the effects on land from the melting of the North Polar ice cap due to the predicted increase in global temperatures. Specifically, model the effects on the coast of Florida every 10 years for the next 50 years due to the melting, with particular attention given to large metropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the data used is an important part of the answer.

**Problem B: Creating Sudoku Puzzles**

Develop an algorithm to construct Sudoku puzzles of varying difficulty.

Develop metrics to define a difficulty level. The algorithm and metrics should be extensible to a varying number of difficulty levels. You should illustrate the algorithm with at least 4 difficulty levels. Your algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your objective should be to minimize the complexity of the algorithm and meet the above requirements.

**The Results**

The solution papers were coded at COMAP headquarters so that names and affiliations of the authors would be unknown to the judges. Each paper was then read preliminarily by two “triage” judges at either Appalachian State University (Polar Melt Problem) or at the National Security Agency (Sudoku Problem). At the triage stage, the summary and overall organization are the basis for judging a paper. If the judges’ scores diverged for a paper, the judges conferred; if they still did not agree, a third judge evaluated the paper.

Additional Regional Judging sites were created at the U.S. Military Academy and at the Naval Postgraduate School to support the growing number of contest submissions.

Final judging took place at the Naval Postgraduate School, Monterey, CA.

The judges classified the papers as follows:

Honorable Successful

Outstanding Meritorious Mention Participation Total

Polar Melt Problem 4 64 182 315 565

Sudoku Problem 5 95 296 198 594

9 159 378 513 1159

The 9 papers that the judges designated as Outstanding appear in this spe-
cial issue of *The UMAP Journal, together with commentaries. We list those*
teams and the Meritorious teams (and advisors) below; the list of all partici-
pating schools, advisors, and results is in the**Appendix.**

**Outstanding Teams**

**Institution and Advisor** **Team Members**

**Polar Melt Papers**

“The Impending Effects of North Polar Ice Cap Melt”

College of Idaho Caldwell, ID

Michael P. Hitchman

Benjamin Coate Nelson Gross Megan Longo

“A Convenient Truth: Forecasting Sea Level Rise”

“Fighting the Waves: The Effect of North Polar Ice Cap Melt on Florida”

University at Buffalo Buffalo, NY

John Ringland

Amy M. Evans Tracy L. Stepien

“Erosion in Florida: A Shore Thing”

University of Delaware Newark, DE

Louis Frank Rossi

Matt Thies Bob Liu

Zachary W. Ulissi
**Sudoku Papers**

“A Difficulty Metric and

Puzzle Generator for Sudoku”

Harvard University Cambridge, MA Clifford H. Taubes

Christopher Chang Zhou Fan

Yi Sun

“Taking the Mystery out of Sudoku Difficulty: An Oracular Model”

Harvey Mudd College Claremont, CA

Jon Jacobsen

Sarah Fletcher Frederick Johnson David R. Morrison

“Difficulty-Driven Sudoku Puzzle Generation”

Harvey Mudd College Claremont, CA

Zach Dodds

Martin Hunt Christopher Pong George Tucker

“Ease and Toil: Analyzing Sudoku”

University of Alaska Fairbanks Fairbanks, AK

Orion S. Lawlor

Seth B. Chadwick Rachel M. Krieg

Christopher E. Granade

“Cracking the Sudoku:

A Deterministic Approach”

Youngstown State University Youngstown, OH

George T. Yates

David Martin Erica Cross Matt Alexander

**Meritorious Teams**
**Polar Melt Problem (65 teams)**

Ann Arbor Huron High School, Mathematics, Ann Arbor, MI (Peter A. Collins) Beihang University, Beijing, China (HongYing Liu)

Beijing Normal University, Beijing, Beijing, China (Li Cui) Beijing Normal University, Beijing, (Laifu Liu)

Beijing University of Posts & Telecommunications, Electronic Engineering, Beijing, China (Zuguo He)

Beijing University of Posts & Telecommunications, Applied Mathematics, Beijing, China (Hongxiang Sun)

Central South University, Mechanical Design and Manufacturing Automation, Changsha, Hunan, China (Xinge Liu)

Central University of Finance and Economics, Applied Mathematics, Beijing, China (Donghong Li)

China University of Mining and Technology, Beijing, China (Lei Zhang) (two teams) China University of Petroleum (Beijing), Beijing, China (Ling Zhao)

China University of Petroleum (East China), Qingdao, Shandong, China (Ziting Wang)

Chongqing University, Applied Chemistry, Chongqing, China (Zhiliang Li) College of Charleston, Charleston, SC (Amy Langville)

Concordia College–New York, Bronxville, NY (Karen Bucher)

Dalian University of Technology, Software, Dalian, Liaoning, China (Zhe Li) Donghua University, Shanghai China (Liangjian Hu)

Duke University, Durham, NC ( Mark Huber)

East China University of Science and Technology, Physics, Shanghai, China (Lu ,hong)

Gannon University, Mathematics, Erie, PA (Jennifer A. Gorman)

Hangzhou Dianzi Unniversity, Information and Mathematics Science, Hangzhou, Zhejiang, China (Wei Li)

Harbin Institute of Technology Shiyan School, Mathematics, Harbin, Heilongjiang, China (Yunfei Zhang)

Hiram College, Hiram, OH (Brad S. Gubser)

McGill University, Mathematics and Statistics, Montreal, Quebec, Canada (Nilima Nigam)

Nankai University, Management Science and Engineering, Tianjin, Tianjin, China (Wenhua Hou)

National University of Defense Technology, Mathematics and Systems Science, Changsha, Hunan, China (Xiaojun Duan)

National University of Defense Technology, Mathematics and Systems Science, Changsha, Hunan, China (Yi Wu)

National University of Ireland, Galway, Galway, Ireland (Niall Madden)

National University of Ireland, Galway, Mathematical Physics, Galway, Ireland (Petri T. Piiroinen)

Northwestern Polytechnical University, Natural and Applied Science, Xi´an, Shaanxi, China (Zhao Junfeng)

Oregon State University, Corvallis, OR (Nathan L. Gibson) Pacific University, Physics, Forest Grove, OR (Juliet Brosing) Peking University, Beijing, China (Sharon Lynne Murrel) Providence College, Providence, RI, (Jeffrey T. Hoag)

Rensselaer Polytechnic Institute, Troy, NY (Peter R. Kramer)

University of Electronic Science and Technology of China, Applied Mathematics, Chengdu, Sichuan, China (Li Mingqi)

Shanghai Foreign Language School, Computer Science, Shanghai, China (Yue Sun) Shanghai University of Finance & Economics, Applied Mathematics, Shanghai,

China (Zhenyu Zhang)

Sichuan University, Electrical Engineering and Information, Chengdu, Sichuan, China (Yingyi Tan)

Slippery Rock University, Slippery Rock, PA (Richard J. Marchand)

South China Agricultural University, GuangZhou, Guangdong (ShaoMei Fang) South China University of Technology, Guangzhou, Guangdong, China

(Qin YongAn)

Sun Yat-Sen (Zhongshan) Univerisity, Guangzhou, Guangdong, China (GuoCan Feng)

Tsinghua University, Beijing, China (Jun Ye) Tsinghua University, Beijing, China (Zhiming Hu) Union College, Schenectady, NY (Jue Wang)

U.S. Military Academy, West Point, NY (Edward Swim) University College Cork, Cork, Ireland (Benjamin W. McKay) University College Cork, Cork, Ireland (Liya A. Zhornitskaya)

University of Guangxi, Mathematics & Information Science, Nanning, Guangxi, China (Ruxue Wu)

University of Guangxi, Mathematics & Information Science, Nanning, Guangxi, China (Zhongxing Wang)

University of Science and Technology Beijing, Beijing, China (Hu Zhixing) University of Technology Jamaica, Chemical Engineering, Kingston, Jamaica,

West Indies (Nilza G. Justiz-Smith)

Worcester Polytechnic Institute, Worcester, MA (Suzanne L. Weekes) Wuhan University, Wuhan, Hubei, China (Yuanming Hu)

Xi’an Jiaotong University, Xian, Shaanxi, China (Jing Gao)

Xi’an Jiaotong University, Center for Mathematics Teaching and Experiment, Xian, Shaanxi, China ( Xiaoe Ruan)

Xuzhou Institute of Technology, Xuzhou, Jiangsu, (Li Subei)

York University, Mathematics and Statistics, Toronto, ON, Canada, (Hongmei Zhu) Yunnan University, Computer Science, Kunming, China (Shunfang Wang)

Zhejiang University, Hangzhou, Zhejiang, China (Zhiyi Tan)

Zhuhai College of Jinan University, Computer Science, Zhuhai, Guangdong, China (Zhang YunBiu)

**Sudoku Problem (96 teams)**

Beihang University, Beijing, China (Sun Hai Yan)

Beijing Institute of Technology, Beijing, China (Guifeng Yan) Beijing Institute of Technology, Beijing, China (Houbao Xu)

Beijing Normal University, Beijing, China (Laifu Liu)

Beijing University of Posts & Telecommunications, Electronics Infomation Engineering, Beijing, China (Jianhua Yuan)

Bethel University, Arden Hills, MN (Nathan M. Gossett)

Cal Poly San Luis Obispo, San Luis Obispo, CA (Lawrence Sze) Carroll College, Chemistry, Helena, MT (John C. Salzsieder) Cheshire Academy, Cheshire, CT (Susan M Eident)

Clarkson University, Computer Science, Potsdam, NY (Katie Fowler) College of Wooster, Wooster, OH (John R. Ramsay)

Dalian Maritime University, Dalian, Liaoning, China (Naxin Chen)

Dalian University of Technology, Software School, Dalian, Liaoning, China (Zhe Li) (two teams)

Daqing Petroleum Institute, Daqing, Heilongjiang, China (Kong Lingbin) Daqing Petroleum Institute, Daqing, Heilongjiang, China (Yang Yunfeng) Davidson College, Davidson NC (Richard D. Neidinger) (two teams) East China Normal University, Shanghai, China (Yongming Liu)

East China University of Science and Technology, Shanghai, China (Su Chunjie) Hangzhou Dianzi University, Information and Mathematics Science, Hangzhou,

Zhejiang, China (Zheyong Qiu)

Harbin Institute of Technology, School of Astronautics, Management Science, Harbin, Heilongjiang, China (Bing Wen)

Harbin Institute of Technology, School of Science, Mathematics, Harbin, Heilongjiang, China (Yong Wang)

Harvey Mudd College, Computer Science, Claremont, CA (Zach Dodds) Humboldt State University, Environmental Resources Engineering, Arcata, CA

(Brad Finney)

James Madison University, Harrisonburg, VA (David B. Walton) Jilin University, Changchun, Jilin, China (Huang Qingdao) Jilin Universit, Changchun, Jilin, China (Xianrui Lu)

Korea Advanced Institute of Science & Technology, Daejeon, Korea (Yong-Jung Kim)

Luther College, Computer Science, Decorah, IA (Steven A. Hubbard) Nanjing Normal University, Computer Science, Nanjing, Jiangsu, China

(Wang Qiong)

Nanjing University, Nanjing, Jiangsu, China (Ze-Chun Hu)

Nanjing University of Posts & Telecommunications, Nanjing, Jiangsu, China (Jin Xu)

Nanjing University of Posts & Telecommunications, Nanjing, Jiangsu, China (Jun Ye)

National University of Defense Technology, Mathematics and Systems Science, Changsha, Hunan, China (Dan Wang)

National University of Defense Technology Mathematics and Systems Science, Changsha, Hunan, China (Meihua Xie)

National University of Defense Technology, Mathematics and Systems Science,

Northwestern Polytechnical University, Xi’an, Shaanxi, China (Yong Xu) Northwestern Polytechnical University, Xi’an, Shaanxi, China (Zhou Min)

Oxford University, Oxford, United Kingdom (Jeffrey H. Giansiracusa) (two teams) P¨aiv¨ol¨a College of Mathematics, Tarttila, Finland (Janne Puustelli)

Peking University, Beijing, China (Xin Yi) Peking University, Beijing, China (Xufeng Liu) Peking University, Beijing, China (Yulong Liu)

Peking University, Financial Mathematics, Beijing, China (Shanjun Lin)

PLA University of Science and Technology, Meteorology, Nanjing, Jiangsu, China (Shen Jinren)

Princeton University, Operations Research and Financial Engineering, Princeton, NJ (Warren B. Powell)

Princeton University, Princeton, NJ (Robert Calderbank)

Renmin University of China, Finance, Beijing, China (Gao Jinwu) Rensselaer Polytechnic Institute, Troy, NY (Donald Drew)

Shandong University, Software, Jinan, Shandong, China (Xiangxu Meng) Shandong University, Mathematics & System Sciences, Jinan, Shandong, China

(Bao Dong Liu)

Shandong University, Mathematics & System Sciences, Jinan, Shandong, China (Xiao Xia Rong)

Shandong University at Weihai, Weihai, Shandong, China (Yang Bing and Song Hui Min)

Shandong University at Weihai, Weihai, Shandong, China (Cao Zhulou and Xiao Hua)

Shanghai Foreign Language School, Shanghai, China (Liang Tao) Shanghai Foreign Language School, Shanghai, China (Feng Xu)

Shanghai Sino European School of Technology, Shanghai, China (Wei Huang) Shanghai University of Finance and Economics, Shanghai, China (Wenqiang Hao) Shijiazhuang Railway Institute, Engineering Mechanics, Shijiazhuang, Hebei, China

(Baocai Zhang)

Sichuan University, Chengdu, China (Qiong Chen)

Slippery Rock University, Physics, Slippery Rock, PA ( Athula R Herat)

South China Normal University, Science of Information and Computation, Guangzhou, Guangdong, China (Tan Yang)

noindent South China University of Technology, Guangzhou, Guangdong, China (Liang ManFa)

South China University of Technology, Guangzhou, Guangdong, China (Liang ManFa)

South China University of Technology, Guangzhou, Guangdong, China (Qin YongAn)

Southwest University, Chongqing, China (Lei Deng) Southwest University, Chongqing, China (Xianning Liu)

Southwestern University of Finance and Economics, Economics and Mathematics, Chengdu, Sichuan, China (Dai Dai)

Sun Yat-Sen (Zhongshan) University, Guangzhou, Guangdong, China (XiaoLong Jiang)

Tsinghua University, Beijing, China (Jun Ye)

University of Califonia–Davis, Davis, CA (Eva M. Strawbridge) University of Colorado–Boulder, Boulder, CO (Anne M. Dougherty)

University of Colorado–Boulder, Boulder, CO (Luis Melara) University of Delaware, Newark, DE (Louis Frank Rossi) University of Iowa, Iowa City, IA (Ian Besse)

University of New South Wales, Sydney, NSW, Australia (James W. Franklin) University of Puget Sound, Tacoma, WA (Michael Z. Spivey)

University of Science and Technology Beijing, Computer Science and Technology, Beijing, China (Zhaoshun Wang)

University of Washington, Applied and Computational Mathematical Sciences, Seattle, WA (Anne Greenbaum)

University of Western Ontario, London, ON, Canada (Allan B. MacIsaac) University of Wisconsin–La Crosse, La Crosse, WI (Barbara Bennie)

University of Wisconsin–River Falls, River Falls, WI (Kathy A. Tomlinson) Wuhan University, Wuhan, Hubei, China (Liuyi Zhong)

Wuhan University, Wuhan, Hubei, China (Yuanming Hu)

Xi’an Communication Institute, Xi’an, Shaanxi, China (Xinshe Qi) Xidian University, Xi’an, Shaanxi, China (Guoping Yang)

Xidian University, Xi’an, Shaanxi, China (Jimin Ye)

Xidian University, Industrial and Applied Mathematics, Xi’an, Shaanxi, China (Qiang Zhu)

Zhejiang University, Hangzhou, Zhejiang, China (Yong Wu)

Zhejiang University City College, Information and Computing Science, Hangzhou, Zhejiang, China (Gui Wang)

Zhejiang University of Finance and Economics, Hangzhou, Zhejiang, China (Ji Luo)

**Awards and Contributions**

Each participating MCM advisor and team member received a certificate signed by the Contest Director and the appropriate Head Judge.

INFORMS, the Institute for Operations Research and the Management Sciences, recognized the teams from the College of Idaho (Polar Melt Prob- lem) and University of Alaska Fairbanks (Sudoku Problem) as INFORMS Outstanding teams and provided the following recognition:

*•* a letter of congratulations from the current president of INFORMS to
each team member and to the faculty advisor;

*•* a check in the amount of $300 to each team member;

*•* a bronze plaque for display at the team’s institution, commemorating
their achievement;

*•* individual certificates for team members and faculty advisor as a per-
sonal commemoration of this achievement;

The Society for Industrial and Applied Mathematics (SIAM) designated one Outstanding team from each problem as a SIAM Winner. The teams were from the University at Buffalo (Polar Melt Problem) and Harvard Uni- versity (Sudoku Problem). Each of the team members was awarded a $300 cash prize and the teams received partial expenses to present their results in a special Minisymposium at the SIAM Annual Meeting in San Diego, CA in July. Their schools were given a framed hand-lettered certificate in gold leaf.

The Mathematical Association of America (MAA) designated one Out- standing North American team from each problem as an MAA Winner. The teams were from Duke University (Polar Melt Problem) and Harvey Mudd College Team (Hunt, Pong, and Tucker; advisor Dodds) (Sudoku Problem).

With partial travel support from the MAA, the Duke University team pre- sented their solution at a special session of the MAA Mathfest in Madison, WI in August. Each team member was presented a certificate by Richard S. Neal of the MAA Committee on Undergraduate Student Activities and Chapters.

**Ben Fusaro Award**

One Meritorious or Outstanding paper was selected for each problem for the Ben Fusaro Award, named for the Founding Director of the MCM and awarded for the fifth time this year. It recognizes an especially creative approach; details concerning the award, its judging, and Ben Fusaro are in Vol. 25 (3) (2004): 195–196. The Ben Fusaro Award winners were the University of Buffalo (Polar Melt Problem) and the University of Puget Sound (Sudoku Problem).

**Judging**

*Director*

Frank R. Giordano, Naval Postgraduate School, Monterey, CA
*Associate Director*

William P. Fox, Dept. of Defense Analysis, Naval Postgraduate School, Monterey, CA

**Polar Melt Problem**
*Head Judge*

Marvin S. Keener, Executive Vice-President, Oklahoma State University, Stillwater, OK

*Associate Judges*

William C. Bauldry, Chair, Dept. of Mathematical Sciences,

Appalachian State University, Boone, NC (Head Triage Judge)

Patrick J. Driscoll, Dept. of Systems Engineering, U.S. Military Academy, West Point, NY

Ben Fusaro, Dept. of Mathematics, Florida State University, Tallahassee, FL (SIAM Judge)

Jerry Griggs, Mathematics Dept., University of South Carolina, Columbia, SC (Problem Author)

Mario Juncosa, RAND Corporation, Santa Monica, CA (retired)

Michael Moody, Olin College of Engineering, Needham, MA (MAA Judge) David H. Olwell, Naval Postgraduate School, Monterey, CA

(INFORMS Judge)

John L. Scharf, Mathematics Dept., Carroll College, Helena, MT (Ben Fusaro Award Judge)

**Sudoku Problem**
*Head Judge*

Maynard Thompson, Mathematics Dept., University of Indiana, Bloomington, IN

*Associate Judges*

Peter Anspach, National Security Agency, Ft. Meade, MD (Head Triage Judge)

Kelly Black, Mathematics Dept., Union College, Schenectady, NY

Karen D. Bolinger, Mathematics Dept., Clarion University of Pennsylvania, Clarion, PA

Jim Case (SIAM Judge)

Veena Mendiratta, Lucent Technologies, Naperville, IL (Problem Author) Peter Olsen, Johns Hopkins Applied Physics Laboratory, Baltimore, MD Kathleen M. Shannon, Dept. of Mathematics and Computer Science,

Salisbury University, Salisbury, MD (MAA Judge)

Dan Solow, Mathematics Dept., Case Western Reserve University, Cleveland, OH (INFORMS Judge)

Michael Tortorella, Dept. of Industrial and Systems Engineering, Rutgers University, Piscataway, NJ

Marie Vanisko, Dept. of Mathematics, Carroll College, Helena MT (Ben Fusaro Award Judge)

Richard Douglas West, Francis Marion University, Florence, SC Dan Zwillinger, Raytheon Company, Sudbury, MA

**Regional Judging Session at U.S. Military Academy**
*Head Judge*

Patrick J. Driscoll, Dept. of Systems Engineering, United States Military Academy (USMA), West Point, NY

*Associate Judges*

Tim Elkins, Dept. of Systems Engineering, USMA
Michael Jaye, Dept. of Mathematical Sciences, USMA
Tom Meyer, Dept. of Mathematical Sciences, USMA
Steve Henderson, Dept. of Systems Engineering, USMA
**Regional Judging Session at Naval Postgraduate School**
*Head Judge*

William P. Fox, Dept. of Defense Analysis, Naval Postgraduate School (NPS), Monterey, CA

*Associate Judges*
William Fox, NPS
Frank Giordano, NPS

**Triage Session for Polar Melt Problem**
*Head Triage Judge*

William C. Bauldry, Chair, Dept. of Mathematical Sciences, Appalachian State University, Boone, NC

*Associate Judges*

Jeff Hirst, Rick Klima, and and Ren´e Salinas

—all from Dept. of Mathematical Sciences, Appalachian State University, Boone, NC

**Triage Session for Sudoku Problem**
*Head Triage Judge*

Peter Anspach, National Security Agency (NSA), Ft. Meade, MD
*Associate Judges*

Other judges from inside and outside NSA, who wish not to be named.

**Sources of the Problems**

The Polar Melt Problem was contributed by Jerry Griggs (Mathemat- ics Dept., University of South Carolina, Columbia, SC), and the Sudoku Problem by Veena Mendiratta (Lucent Technologies, Naperville, IL).

**Acknowledgments**

Major funding for the MCM is provided by the National Security Agency (NSA) and by COMAP. We thank Dr. Gene Berg of NSA for his coordinat- ing efforts. Additional support is provided by the Institute for Operations Research and the Management Sciences (INFORMS), the Society for Indus- trial and Applied Mathematics (SIAM), and the Mathematical Association of America (MAA). We are indebted to these organizations for providing judges and prizes.

We also thank for their involvement and support the MCM judges and MCM Board members for their valuable and unflagging efforts, as well as

*•* Two Sigma Investments. (This group of experienced, analytical, and
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**Cautions**

*To the reader of research journals:*

Usually a published paper has been presented to an audience, shown
to colleagues, rewritten, checked by referees, revised, and edited by a jour-
nal editor. Each paper here is the result of undergraduates working on
a problem over a weekend. Editing (and usually substantial cutting) has
taken place; minor errors have been corrected, wording altered for clarity
*or economy, and style adjusted to that of The UMAP Journal. The student*
authors have proofed the results. Please peruse their efforts in that context.

*To the potential MCM Advisor:*

It might be overpowering to encounter such output from a weekend of work by a small team of undergraduates, but these solution papers are highly atypical. A team that prepares and participates will have an enrich- ing learning experience, independent of what any other team does.

COMAP’s Mathematical Contest in Modeling and Interdisciplinary Con- test in Modeling are the only international modeling contests in which students work in teams. Centering its educational philosophy on mathe-

**Appendix: Successful Participants**

KEY:

P = Successful Participation H = Honorable Mention M = Meritorious

O = Outstanding (published in this special issue)

INSTITUTION DEPT. CITY ADVISOR

ALASKA

U. Alaska Fairbanks CS Fairbanks Orion S. Lawlor B H

U. Alaska Fairbanks CS Fairbanks Orion S. Lawlor B O

ARIZONA

Northern Arizona U. Math & Stats Flagstaff Terence R. Blows A H CALIFORNIA

Cal Poly San Luis Obispo Math San Luis Obispo Lawrence Sze B M

Cal Poly San Luis Obispo Math San Luis Obispo Lawrence Sze B H

California State Poly. U. Physics Pomona Kurt Vandervoort B P

California State Poly. U. Math & Stats Pomona Joe Latulippe B P Calif. State U. at Monterey Bay Math & Stats Seaside Hongde Hu A H Calif. State U. at Monterey Bay Math & Stats Seaside Hongde Hu A H Calif. State U. Northridge Math Northridge Gholam-Ali Zakeri B P

Cal-Poly Pomona Math Pomona Hubertus F. von Bremen A H

Cal-Poly Pomona Physics Pomona Nina Abramzon B P

Harvey Mudd C. Math Claremont Jon Jacobsen A H

Harvey Mudd C. Math Claremont Jon Jacobsen B O

Harvey Mudd C. CS Claremont Zach Dodds B M

Harvey Mudd C. CS Claremont Zach Dodds B O

Humboldt State U. Env’l Res. Eng. Arcata Brad Finney A H

Humboldt State U. Env’l Res. Eng. Arcata Brad Finney B M

Irvine Valley C. Math Irvine Jack Appleman A P

Pomona C. Math Claremont Ami E. Radunskaya A H

Saddleback C. Math Mission Viejo Karla Westphal A P

U. of Califonia Davis Math Davis Eva M. Strawbridge A P

U. of Califonia Davis Math Davis Eva M. Strawbridge B M

U. of California Merced Natural Sci. Merced Arnold D. Kim B H

U. of San Diego Math San Diego Cameron C. Parker A P

U. of San Diego Math San Diego Cameron C. Parker B H

COLORADO

U. of Colorado - Boulder Appl. Math. Boulder Anne M. Dougherty A H U. of Colorado - Boulder Appl. Math. Boulder Bengt Fornberg A H U. of Colorado - Boulder Appl. Math. Boulder Anne Dougherty B M U. of Colorado - Boulder Appl. Math. Boulder Bengt Fornberg B H

U. of Colorado - Boulder Appl. Math. Boulder Luis Melara B M

U. of Colorado Denver Math Denver Gary A. Olson A P

CONNECTICUT

Cheshire Acad. Math Cheshire Susan M. Eident B M

Connecticut C. Math New London Sanjeeva Balasuriya A P

Sacred Heart U. Math Fairfield Peter Loth B P

Southern Connecticut State U. Math New Haven Ross B. Gingrich A H Southern Connecticut State U. Math New Haven Ross B. Gingrich B H DELAWARE

U. of Delaware Math Sci. Newark Louis Frank Rossi A O

U. of Delaware Math Sci. Newark John A. Pelesko B P

U. of Delaware Math Sci, Newark Louis Rossi B M

FLORIDA

Bethune-Cookman U. Math Daytona Beach Deborah Jones A P

Jacksonville U. Math Jacksonville Robert A. Hollister A H

INSTITUTION DEPT. CITY ADVISOR GEORGIA

Georgia Southern U. Math Sci. Statesboro Goran Lesaja A P

Georgia Southern U. Math Sci. Statesboro Goran Lesaja B H

U. of West Georgia Math Carrollton Scott Gordon A H

IDAHO

C. of Idaho Math/Phys. Sci. Caldwell Michael P. Hitchman A O

ILLINOIS

Greenville C. Math Greenville George R. Peters A P

INDIANA

Goshen C. Math Goshen Patricia A. Oakley B H

Rose-Hulmann Inst. of Tech. Chemistry Terre Haute Michael Mueller B H Rose-Hulman Inst. of Tech. Chemistry Terre Haute Michael Mueller B H Rose-Hulman Inst. of Tech. Math Terre Haute William S. Galinaitis A H Rose-Hulman Inst. of Tech. Math Terre Haute William S. Galinaitis B P

Saint Mary’s C. Math Notre Dame Natalie K. Domelle A H

Saint Mary’s C. Math Notre Dame Natalie K. Domelle B H

IOWA

Coe C. Math Sci. Cedar Rapids Calvin R. Van Niewaal B H

Grand View C. Math & CS Des Moines Sergio Loch A H

Grand View C. Math & CS Des Moines Sergio Loch A H

Grinnell C. Math & Stats Grinnell Karen L. Shuman A H

Luther C. CS Decorah Steven A. Hubbard B H

Luther C. CS Decorah Steven A. Hubbard B M

Luther C. Math Decorah Reginald D. Laursen B H

Luther C. Math Decorah Reginald D. Laursen B H

Simpson C. Comp. Sci. Indianola Paul Craven A H

Simpson C. Comp. Sci. Indianola Paul Craven A P

Simpson C. Math Indianola William Schellhorn A H

Simpson C. Math Indianola Debra Czarneski A P

Simpson C. Math Indianola Rick Spellerberg A P

Simpson C. Physics Indianola David Olsgaard A P

Simpson C. Math Indianola Murphy Waggoner B P

Simpson C. Math Indianola Murphy Waggoner B H

U. of Iowa Math Iowa City Benjamin J. Galluzzo A H

U. of Iowa Math Iowa City Kevin Murphy A H

U. of Iowa Math Iowa City Ian Besse B M

U. of Iowa Math Iowa City Scott Small B H

U. of Iowa Math Iowa City Benjamin Galluzzo B H

KANSAS

Kansas State U. Math Manhattan David R. Auckly B H

Kansas State U. Math Manhattan David R. Auckly B H

KENTUCKY

Asbury C. Math & CS Wilmore David L. Coulliette A H

Asbury C. Math & CS Wilmore David L. Coulliette B H

Morehead State U. Math & CS Morehead Michael Dobranski B P

Northern Kentucky U. Math Highl& Heights Lisa Joan Holden A H

Northern Kentucky U. Math Highl& Heights Lisa Holden B P

INSTITUTION DEPT. CITY ADVISOR MARYLAND

Hood C. Math Frederick Betty Mayfield A P

Loyola C. Math Sci. Baltimore Jiyuan Tao A H

Loyola C. Math Sci. Baltimore Jiyuan Tao B H

Mount St. Mary’s U. Math Emmitsburg Fred Portier B P

Salisbury U. Math & CS Salisbury Troy V. Banks B P

Villa Julie C. Math Stevenson Eileen C. McGraw A H

Washington C. Math & CS Chestertown Eugene P. Hamilton A P

MASSACHUSETTS

Bard C./Simon’s Rock Math Great Barrington Allen B. Altman A P

Bard C./Simon’s Rock Math Great Barrington Allen Altman B P

Bard C./Simon’s Rock Physics Great Barrington Michael Bergman A P

Harvard U. Math Cambridge Clifford H. Taubes B O

Harvard U. Math Cambridge Clifford H. Taubes B H

U. of Mass. Lowell Math Sci. Lowell James Graham-Eagle B P

Worcester Poly. Inst. Math Sci. Worcester Suzanne L. Weekes A M

Worcester Poly. Inst. Math Sci. Worcester Suzanne L. Weekes A H

MICHIGAN

Ann Arbor Huron HS Math Ann Arbor Peter A. Collins A M

Lawrence Tech. U. Math & CS Southfield Ruth G. Favro A H

Lawrence Tech. U. Math & CS Southfield Guang-Chong Zhu A P

Lawrence Tech. U. Math & CS Southfield Guang-Chong Zhu A P

Lawrence Tech. U. Math & CS Southfield Ruth Favro B H

Siena Heights U. Math Adrian Jeff C. Kallenbach A P

Siena Heights U. Math Adrian Tim H. Husband A P

Siena Heights U. Math Adrian Tim H. Husband B P

MINNESOTA

Bethel U. Math & CS Arden Hills Nathan M. Gossett B M

Carleton C. Math Northfield Laura M. Chihara A H

Northwestern C. Sci. & Math. St. Paul Jonathan A. Zderad A P MISSOURI

Drury U. Math & CS Springfield Keith James Coates A H

Drury U. Math & CS Springfield Keith James Coates A P

Drury U. Physics Springfield Bruce W. Callen A P

Drury U. Physics Springfield Bruce W. Callen A H

Saint Louis U. Math & CS St. Louis David A. Jackson B H

Saint Louis U. Eng., Aviation & Tech. St. Louis Manoj S. Patankar A H

Truman State U. Math & CS Kirksville Steve Jay Smith B H

U. of Central Missouri Math & CS Warrensburg Nicholas R. Baeth A P U. of Central Missouri Math & CS Warrensburg Nicholas R. Baeth B P MONTANA

Carroll C. Chemistry Helena John C. Salzsieder B M

Carroll C. Chemistry Helena John C. Salzsieder A P

Carroll C. Math., Eng. , & CS Helena Holly S. Zullo B H

Carroll C. Math., Eng., & CS Helena Mark Parker A H

NEBRASKA

Nebraska Wesleyan U. Math & CS Lincoln Melissa Claire Erdmann A P

Wayne State C. Math Wayne Tim Hardy A P

NEW JERSEY

Princeton U. Math Princeton Robert Calderbank B M

Princeton U. OR & Fin. Eng. Princeton Robert J. Vanderbei B H Princeton U. OR & Fin. Eng. Princeton Robert J. Vanderbei B H

Princeton U. OR & Fin. Eng. Princeton Warren B. Powell B P

Princeton U. OR & Fin. Eng. Princeton Warren B. Powell B M

INSTITUTION DEPT. CITY ADVISOR

Richard Stockton C. Math Pomona Brandy L. Rapatski A H

Rowan U. Math Glassboro Paul J. Laumakis B P

Rowan U. Math Glassboro Christopher Jay Lacke B H

NEW MEXICO

NM Inst. Mining & Tech. Math Socorro John D. Starrett B P

New Mexico State U. Math Sci. Las Cruces Caroline P. Sweezy A P NEW YORK

Clarkson U. Comp. Sci. Potsdam Katie Fowler B H

Clarkson U. Comp. Sci. Potsdam Katie Fowler B M

Clarkson U. Math Potsdam Joseph D. Skufca A H

Clarkson U. Math Potsdam Joseph D. Skufca B P

Colgate U. Math Hamilton Dan Schult B H

Concordia C. Bio. Chem. Math. Bronxville Karen Bucher A M

Concordia C. Math Bronxville John F. Loase A H

Concordia C. Math Bronxville John F. Loase B H

Cornell U. Math Ithaca Alexander Vladimirsky B H

Cornell U. OR & Ind’l Eng. Ithaca Eric Friedman B H

Ithaca C. Math Ithaca John C. Maceli B H

Ithaca C. Physics Ithaca Bruce G. Thompson B H

Nazareth C. Math Rochester Daniel Birmajer A P

Rensselaer Poly. Inst. Math Sci. Troy Peter R. Kramer A M

Rensselaer Poly. Inst. Math Sci. Troy Peter R. Kramer B H

Rensselaer Poly. Inst. Math Sci. Troy Donald Drew B M

Rensselaer Poly.Inst. Math Sci. Troy Donald Drew B H

Union C. Math Schenectady Jue Wang A M

U.S. Military Acad. Math Sci. West Point Edward Swim A M

U.S. Military Acad. Math Sci. West Point Robert Burks B H

U. at Buffalo Math Buffalo John Ringland A O

U. at Buffalo Math Buffalo John Ringland B H

Westchester Comm. Coll. Math Valhalla Marvin Littman B P

NORTH CAROLINA

Davidson C. Math Davidson Donna K. Molinek B H

Davidson C. Math Davidson Donna K. Molinek B H

Davidson C. Math Davidson Richard D. Neidinger B M

Davidson C. Math Davidson Richard D. Neidinger B M

Duke U. Math Durham Scott McKinley A O

Duke U. Math Durham Mark Huber A M

Duke U. Math Durham David Kraines B H

Duke U. Math Durham Dan Lee B P

Duke U. Math Durham Lenny Ng B H

Duke U. Math Durham Bill Pardon B H

Meredith C. Math & CS Raleigh Cammey Cole Manning A H

NC Schl of Sci. & Math. Math Durham Daniel J. Teague B M

NC Schl of Sci. & Math. Math Durham Daniel J. Teague B H

U. of North Carolina Math Chapel Hill Sarah A. Williams A H

U. of North Carolina Math Chapel Hill Brian Pike A H

Wake Forest U. Math Winston Salem Miaohua Jiang A H

Western Carolina U. Math & CS Cullowhee Jeff Lawson A H

Western Carolina U. Math & CS Cullowhee Erin K. McNelis B H

INSTITUTION DEPT. CITY ADVISOR

Miami U. Math & Stats Oxford Doug E. Ward A P

Miami U. Math & Stats Oxford Doug E. Ward B H

U. of Dayton Math Dayton Youssef N. Raffoul B H

Xavier U. Math & CS Cincinnati Bernd E. Rossa A H

Xavier U. Math & CS Cincinnati Bernd E. Rossa B H

Youngstown State U. Math & Stats Youngstown George T. Yates A H Youngstown State U. Math & Stats Youngstown Angela Spalsbury A H Youngstown State U. Math & Stats Youngstown Angela Spalsbury A H Youngstown State U. Math & Stats Youngstown Gary J. Kerns A H Youngstown State U. Math & Stats Youngstown Paddy W. Taylor A P Youngstown State U. Math & Stats Youngstown Paddy W. Taylor A H Youngstown State U. Math & Stats Youngstown George Yates B O Youngstown State U. Math & Stats Youngstown Gary Kerns B H OKLAHOMA

Oklahoma State U. Math Stillwater Lisa A. Mantini B H

SE Okla. State U. Math Durant Karl H. Frinkle A P

OREGON

Lewis & Clark Coll. Math Sci. Portland Liz Stanhope A P

Linfield C. Comp. Sci. McMinnville Daniel K. Ford B H

Linfield C. Math McMinnville Jennifer Nordstrom A H

Linfield C. Math McMinnville Jennifer Nordstrom B H

Oregon State U. Math Corvallis Nathan L. Gibson A M

Oregon State U. Math Corvallis Nathan L. Gibson A P

Oregon State U. Math Corvallis Vrushali A. Bokil B H

Pacific U. Math Forest Grove Michael Boardman B H

Pacific U. Math Forest Grove John August A H

Pacific U. Physics Forest Grove Juliet Brosing A M

Pacific U. Physics Forest Grove Steve Hall A P

PENNSYLVANIA

Bloomsburg U. Math, CS, & Stats Bloomsburg Kevin Ferland A H

Bucknell U. Math Lewisburg Peter McNamara B H

Gannon U. Math Erie Jennifer A. Gorman A M

Gettysburg C. Math Gettysburg Benjamin B. Kennedy B H

Gettysburg C. Math Gettysburg Benjamin B. Kennedy B P

Juniata C. Math Huntingdon John F. Bukowski A H

Shippensburg U. Math Shippensburg Paul T. Taylor A H

Slippery Rock U. Math Slippery Rock Richard J. Marchand A M

Slippery Rock U. Math Slippery Rock Richard J. Marchand B H

Slippery Rock U. Physics Slippery Rock Athula R. Herat B M

U. of Pittsburgh Math Pittsburgh Jonathan Rubin B H

Westminster C. Math & CS New Wilmington Barbara T. Faires A H Westminster C. Math & CS New Wilmington Barbara T. Faires A H Westminster C. Math & CS New Wilmington Warren D. Hickman B H Westminster C. Math & CS New Wilmington Carolyn K. Cuff B P RHODE ISLAND

Providence C. Math Providence Jeffrey T. Hoag A M

SOUTH CAROLINA

C. of Charleston Math Charleston Amy Langville A M

C. of Charleston Math Charleston Amy Langville B H

Columbia C. Math & Comp. Columbia Nieves A. McNulty B H

Francis Marion U. Math Florence David W. Szurley B H

Midlands Technical Coll. Math Columbia John R. Long A H

Midlands Technical Coll. Math Columbia John R. Long B P

Wofford C. Comp. Sci. Spartanburg Angela B. Shiflet B H