5 a Student Guide to the National Math Meetings 8 the Use and Care of Numbers

Coulll on Controversy 20

Use and Care of Numbers 8 In this issue

5 A Student Guide to the National Math Meetings 8 The Use and Care of Numbers

Math Horizons is for undergraduates and others who are interested in mathematics. 12 Internships Its purpose is to expand both the career and intellectual horizons of students. 14 Art Benjamin-Mathemagician DONALD J. ALBERS Editor

BEVERLY JOY RUED1 21 Census 2000: Count on Controversy Managing Editor CAROL BAXTER 24 Exploring Moduli Spaces Associate Manapng Editor JANE D'ALELIO 26 A Dozen Questions About a Donut Art Director Math Horizons (ISSN 1072-4 1 17) is pub- Problem Section lished four times a year; September, 32 November, February, and April by the Mathematical Association of America, 1529 Eighteenth Street, NU!, Washington, DC 20036. September 1998 Volume VI, Issue 1. Periodicals postage paid at Washington, DC and additional mailing of- fices. Annual subscription rates are S20.00 for kL4A members and S33.00 for non- members and libraries. Bulk subscriptions sent to a single address are encouraged. The minimum order is 20 copies (3120.00); additional subscriptions may be ordered in units of 10 for 360.00 per unit. For advertising rates or editorial in- quiries call (202) 387-5200. Printed in the United States of America. Copyright 01998 The Mathematical Association of ,h~erica.POSTMASTER: Send address changes to Math Horizons, MAA Service Center, PO Box 91 112, Washington, DC 20090-1 112. Cower b~ Tom Mann of Mann & Mann Inc.

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Math Horizons November 1998 3 math HORIZONS

ADVISORY BOARD GERALDL.ALEXANDERSON Fasten Your Seat Belts! Santa Cla,m University TOMAPOSTOL California Institute of Technology Math Horizons (MH) is now five years old. It has been nl\ GEOKCEBERZSEW great pleasure to serve as founding editor and to watch ir Rose-H,ftlmnn Institute of Technology ROBERTBOZEX~N develop. Thanks to the thousands of subscribers like you Morehozcse College who have given support to MH during its infancy. Special M~R\'INBRURAKER thanks to Marcia Sward, Executive Director of the MAA, Messiah College who collaborated with me on the original idea of a maga- BARBARAFAIRES Westmi~uterCollege zine for students. The backing of the MAA's Executive and DEBOM FRA~UT'L Finance Committees; the Board of Governors, the Commit- K~ctztownUniversity tee on Publications, and the MH Advisory Board has been JOEG.AI.I.IAN central to its success. Carol Baxter and Beverly Ruedi have University of Minnesota-Df~luth APARNAHIGGINS served magnificently as managing editors, and Jane University of Daj~ton D'Alelio has provided inspired designs as art director. ROBERTHOOD Editor Em.eritzls, Boys' Life I've had a wonderful time, but the MAA changes editors HAROLDJACOBS U.S. Grant High School, Van Nzqs, CA every five years and it's time now for new editors. In 1999, SANDRAKEITH Professors Deanna Haunsperger and Steve Kennedy of St. Cloud State Uniuersity Carleton College take over as co-editors. They are loaded LEONARD~LOS~NSK~ with ideas, full of energy, fine teachers, and are very Santa Clara Univer.xity JOEMALKWITCH involved with student activities. Their ideas for MH are so York College exciting that it's clear that readers are in for a great ride. CLEOPATRIAMARTINEZ So, fasten your seat belts! Phoenix College ROBERTMEGGINSON University of Michigan RICHARDNEAL University of Oklahomn HENRYPOLLAK Teachers College, Columbia University FREDAPORTER-LOCKLEAR Pembroke State University PETER RENZ Free-lc~.nceEditor V. FREDERICKRICKEY Bowling Green State University RONALDC. ROSIER Zonference Boa.rd of the Mathematical Sciences MARKSAUL Bronxville School, NY ANITASOLOJV Randolph-Macon Woman's College ASDRE!~ST ERR^, JR. The Mathematical Association of America IAN STWVARI. Wanuick University JUDITHTANUR State University of New York, Stony Brook IRVINVANCE Michigan State University PETERWALTHER Salislt-Kootenui College ANN~VATKINS California State University, Northridge ROBINWILSON TIE Open University

Math Horizons November 1998 DAN KALMAN A Student Guide to the National Math Meetings

he two largest professional societies for mathematics are the Mathematical Association of America (MAA) T and the American Mathematical Society (AMS). Each January, these organizations hold a joint convention, called the "national meetings" by most members of the organizations. The MAA also holds a national meeting in the summer. Attending a national meeting can be a rewarding experience for college math students. You can hear interesting talks about math, meet people from around the U.S. who share an enthusiasm for the subject, and get a glimpse into the professional world of mathematicians. But to get the most out of the math meetings you have to be prepared. That'swhere this guide comes in. It gives you inside information on what to expect, how to pick events, and even how to keep your expenses to a minimum.

Kinds of Talks Let's start with the program. One of the main activities at the ers ~vhoare experts in the area that is the focus of the session. math meetings is attending talks. These come in three dif- A variation on a special session is a panel discussion. This ferent flavors: invited addresses (hour talks), invited special kind of session often has more closely coordinated presenta- session talks (usually on the order of 20 minutes), and con- tions, and more time for discussion with the audience. To illus- tributed papers (10 to 15 minutes). The invited addresses are trate the differences, here is an example. There might be a the main attractions for a general audience. The speakers are special session on dynamical systems (which is the mathemati- asked to choose topics that will make sense to a large audi- cal branch of chaos). The organizer would invite outstanding ence. Usually, they are nationally recognized mathematicians researchers in the field to make presentations, but would not who are known for outstanding research, for outstanding necessarily try to fit the presentations together. In contrast, for speaking skills, or both. The speakers are on the program at a panel discussion on career opportunities in industry, there the invitation of the organizers of the meetings, which is why might be one spokesperson from the aerospace industry, one they are called invited addresses. (No, they don't check for from the pharmaceutical industry, and one from a special ef- invitations at the door - anyone can go and listen.) fects company in the entertainment industry. The invited special session talks are more specialized than The contributed paper sessions feature speakers who have the invited addresses. These are arranged as part of a session volunteered to give talks, as opposed to appearing by invita- that focuses on a particular topic or theme. They invite speak- tion. These speakers want to share something they have been working on with colleagues having similar interests. Most of these sessions are organized around a theme. For example, the MAA might announce a contributed paper session on applications of mathematics to environmental science. Then, anyone who has been working in this area can send in a DAN KALMAN teaches mathematics at American University. description of a brief talk for inclusion in the program. There

Math Horizons November 1998 5 are also AMS sessions of contributed papers that are not so- licited in any organized way. Rather, mathematicians simply send in a request to present a paper concerning anything they might be working on, and the AMS tries to group them according to subject and schedules sessions for them.

What to Attend So now that you knowwhat the different kinds of talks are, how do you decide what to go to? One helphl resource is a list of events expected to be of interest to students. These are often available from faculty advisors for student math clubs and MAA or Pi Mu Epsilon chapters. They can also be found on the world wide web before the meetings begin. The MAA's website, MAA Online (http://~w.maa.org),has a page for student activities. Look there for information on the national meetings. There are three levels of information about almost all of ;he presentations. The first two are contained in the main pro- presentation in the session will cover, but they do give you a gram, and can be found at both the summer and winter meet- pretty good idea of the kinds of topics that can be expected. ings. The third level is published in a separate booklet,just for the winter meeting. The information from each level can help Abstracts you choose what to attend. Here are some details about what kind of information is available. That brings us to the third level of information, abstracts for The best overview of the entire meeting is the preliminary the talks. These are generally available only at the January program. It is published before the meeting in both FOCUS, meetings, in a separate abstracts booklet, usually to be found in the newsletter of the MAA, and in the Notices of tlw AMS. You the registration area. As part of your registration fee, you are should be able to find at least one of these by askingjust about entitled to pick up a copy, but you generally have to ask for it. any faculty member in your mathematics department. Another They don't give copies away just as a matter of course. The place to find the same information is on the internet through book of abstracts includes a paragraph or two prepared by each MAA Online. The preliminary program includes information speaker about his or her presentation. There is a kind offunny about registering for the meeting and making reservations for numbering system that you use to find the abstract for a par- a place to stay. For the January meeting that means in a hotel, ticular talk, or you can look up the author's name in an index. but for summer meetings there are often inexpensive dormi- As you can see, carefully planning out what to attend can tory rooms available. be a lot ofwork. That is why a list of events for students is The preliminary program includes a brief timetable, where compiled for each meeting. But even with the list in hand, you can see at a glance what kinds of sessions are scheduled at plan to spend some time figuring out what to do. The list is different times. This will show for example that there will be an compiled by well meaning mathematics teachers, but after invited address in the main ballroom at 1 PM, and several all, they aren't students, and can only guess about what might special sessions or contributed paper sessions running from 2 interest you. At some times there may be several different to 5.You will be able to see what the title of each paper session activities listed, and you have to choose what to do. And it is is, but not the titles of the individual talks in the session. a good idea to look through the program in case there is The next level of information is the detailed program that is something of special interest to ~011. available at the meeting, or is mailed in advance to preregistrants. This gives the titles of each talk in a session, Making a Plan and the names of the presenters. Sometimes it is hnto hear a talk by someonewhose name you know. It might be the author Here is what I do to plan my time at the meetings. First, I of a text book you have used. Or maybe someone who has look at the titles of the the invited addresses, and their ab- written an article forMath Horizons or the College Mathematics stracts, and pick the ones that I think will be most interest- Journal. Many students go to hear their friends give talks in ing. Next, I look at the titles of the special sessions, and student sessions, or their teachers. You can also pick talks to figure out which ones might interest me. Then I scan the attend based on the titles. However, like judging a book by its titles of the talks in those sessions, and look up abstracts for cover, this is not always an effective approach. It is often hard any that seem especially interesting. Finally, I go through to tell what the talk will really be about based on the title. the entire list of presenters looking for familiar names, just The detailed program may also contain more extended for the fun of hearing someone I know, or someone I have descriptions about the organizational theme for contributed heard of but never met. At this point, I compile a list of the paper sessions, special sessions, panel discussions, and so sessions I want to attend, and in the case of conflicting ses- on. These descriptions will not tell you what any particular sions (there are always a few), make tentative choices. It is a

6 Math Horizons November 1998 bit ofwork, but it helps me get the most out of the meetings. there are a few things to look out for. First of all, the exhibits There are a couple more things to keep in mind. First, in area is definitely worth a look. There are often neat give-aways, a two or three hour session of short talks, you are not ex- like totebags, mugs, or scratch pads. You can see all the latest pected to attend the entire session. It is a normal part of the math books, including some that are of interest to students. meetings for people to go to a session for a particular talk, The MAA display always has some of those. You can see com- and then leave to attend a talk in a different session. Second, puter software for doing or writing about mathematics, calcula- you should not expect to understand all of the mathematics. tors, and cool graphics and visualization stuff. There are usu- A common expectation for a one hour talk is that everyone ally some miscellaneous items like tee-shirts and videos, too. should understand the first 20 minutes, specialists the next The social get-togethers are another activity that are worth 20 minutes, and the speaker's collaborators should be able to looking for. These often present an opportunity to network understand the final 20 minutes. This is an exaggeration, with new friends and old. And they often have some kind of but it is very rare to completely understand every part of a goodies: snacks, drinks, cookies, etc. Of course, I would never talk. If participants at thejoint meetings know ahead of time encourage you to go just to grab some food. But if there is that they won't understand a talk, why do they even go? I like something about the group that is of interest to you, and you to think of it as similar to hearing an unfamiliar piece of are not sure whether to go or not, its worth keeping the pos- music at a concert. You may not be able to completely follow sibility of food in mind. As you look in the program, the key the melodic line, but it is still possible to enjoy the rhythms, word to keep in mind is "reception". patterns, and dramatic effects of the performance. In the Banquets can be hn, if you have a few friends to go with. same way, when you hear a lecturer you can get a feel for the They are sometimes a little pricey, but often have very enter- rhythm and patterns in the arguments, and enjoy the taining speakers or presentations.A prize session is where awards speaker's excitement and energy. It is satisfying to make con- for outstanding research, writing, teaching, and service are nections to other parts of mathematics, and even if you don't presented. Often there is not much happening at these ses- completely understand the subject, hearing some of the main sions besides reading the names and accomplishments of the ideas provides a foundation for hture exposures. winners. But it can be interesting to see who is being recog- In many ways, mathematics is like a foreign language. Each nized, and why. little part of the subject has its own vocabulary. In most talks, the speakers make an effort to tell what the vocabulary is, but most ofus can only absorb so much new language at a time. Eats and Digs It is easy to get overwhelmed by new concepts and terms, so This brings us to the problems of finding food and lodging that you literally cannot understand what the speaker is say- without busting your budget. There are obvious cost savings ing. For students who have only been studying the language measures, like sharing a hotel room with two or three fellow for a few years, this problem is especially acute. So be on the students. But here are some tips that you might not think of. lookout for talks that involve mathematical subjects you are Bring along a small electric pot for heating water, and either familiar with. For example, in sessions devoted to teaching bring a cheap picnic cooler, or pick one up after you arrive. methods for calculus or linear algebra, you will at least know On the first day, find a small local grocery store or conve- the basic vocabulary. Often the talks in these sessions con- nience store, and get some things for breakfast and lunch: a tain new mathematical ideas that can be quite interesting. package of bagels, some instant oatmeal, instant soup, fruit, Even if you are not directly interested in the discussion of even milk and cereal. Between food items like these, and any teaching methods, you can get a lot out of the math that goodies you find at social activities, you can pretty much take comes up in the discussion. This is not true for all talks. care of everything except dinner. If you are really on a tight Some speakers will focus on aspects of course organization, budget, you might try to make a contact at a college or univer- grading, and student participation, topics that are primarily sity in the area of the meeting. If there is a math club, or a of interest to teachers. But if you check the titles and ab- student MAA chapter or Pi Mu Epsilon chapter, it might be stracts of the talks in a special session on teaching, you can possible to arrange some kind ofvisit on campus. This is prob- find talks that do involve interesting mathematics, too. ably best arranged between faculty advisors, so if you are in a math club or student chapter, get your faculty advisor involved. Other Activities Play It Safe Besides the various kinds of talks, there are lots of other things going on at the national meetings. There are social gatherings, One final comment needs to be made: be carehl where and business meetings, prize sessions, reunions. In past meetings when you go out in the area of the math meetings, especially if there have been classical piano concerts (performed by math- you are from out of town. When you leave the hotel where the ematicians!),juggling and circus acts (also performed by math- meetings are, remember to remove your registration badge, ematicians!), poetry sessions (written and read by mathemati- which identifies you as an out-of-towner. Use common sense at cians!), dramatic skits to help raise consciousness regarding night about unlit areas, or areas far from other activities. It is gender issues, and other events you might not expect to find at always a good idea to walk in groups instead of alone. If you a math meeting. You won't find all of this very exciting, but are uncertain about an excursion, checkwith the hotel staff. . Math Horizons November 1998 7 The Use and Care of Numbers

e are all numbers users. Mre at all, do not necessarily have meaning Definitional differences: virtually regularly use numbers to help or importance. Similarly, while most never is the exact same definition of us understand something else numbers are not accurate, they often still what is to be counted or measured ap- - the condition of our businesses, the have great meaning and importance. plied consistently throughout the en- dynamics of the American population, And, hopefully, everyone understands tire counting or measuring process. the health of our bodies, the money we that the most useful numbers often are This is especially true when the re- have available in our bank accounts, etc. neither accurate nor meaningful. Per- sults from multiple counters/reporters For number users it is something other haps an example will help. are being aggregated together to pro- than the numbers that is really impor- duce one, composite number for the tant; the numbers are merely very use- whole. For example, when the Cen- ful descriptors. The Missile Gap sus Bureau counts the number of To understand what it means to use Blacks in a community, how is that numbers as part of our daily lives, we In 1960 when Senator John F. Kennedy racial category defined and how many must recognize that numbers have hu- was running for the presidency, he was alternative definitions might be ap- man or social characteristics above and told that the USSR had more missiles plicable? beyond their mathematical qualities. then did the US. For Kennedy, the Measurement error: we measure only The number user is concerned about: meaning was clear - there was a Mis- to the accuracy of our measuring de- 1) the accuracy of a number, sile Gap. The importance of the num- vice, whether it be a yard stick, a sci- 2) the meaning of a number, bers was that American security was entific micrometer, or whatever. 3) the importance of a number, and threatened. The usefulness was that these Incomplete records or data: All hun- 4) the usefulness of a number. numbers provided candidate Kennedy dred cities in the state are supposed with a major campaign issue to be used to report, but only 98 do so. Number These are all different. Accuracy, is how against the incumbent administration providers sometimes impute (make closely the number that is used to de- and, particularly, Vice President Nixon. educated guesses of) missing numbers scribe reality actually does so. Meaning Later, President Kennedy learned the pre- and, then use those imputed numbers is the substantive interpretation of the vious missile numbers were not correct; as if they 11-ereexact counts. number; in other words, what the num- there was no Missile Gap; and American The timing of the numbers: The number tell us about what is being described security was not in question. The new bers are to be as of September 19th; by the number. Importance essentially numbers were not especially useful to the some reporting units actually used the embodies the answer to the question new President. required date; some use a prior date; "Who cares?". And usefulness seeks to As it turns out, many, if not virtually some use a later date; and others can- reflect how the number fits into the pur- all, of the numbers we use to describe not possibly tell you the operational poses or needs of the number user. things are inaccurate. When we read in date of the number provided. Upon reflection, you will see that a government publication that in the fall There is a second aspect to the tim- these four characteristics of numbers are of 1992 a particular state enrolled ing issue. Most meaningful, impor- not related to each other in any fixed 42,604 freshman students, that is an ap- tant, and useful numbers are chang- way. Accurate numbers, when they exist proximation, not an exact count. Each ing continuously. The accurate num- college and university in the state is sup- ber today will not be the accurate nwn- posed to provide an exact count, but ber tomorrow or even this evening or, JOSEPH ZIKMUND II is Chief Information even though most institutions expend perhaps, even five minutes from now. Officer with the Connecticut Department a great deal of time and energy to do Human blood pressure, for example, of Higher Education. He has a PhD in po- so, exact counts are very hard to come varies from moment to moment. Thus, litical science and, previously, taught at a by. Why are most numbers or counts less what do we even mean by an accurate number of colleges and universities than fully accurate? reading of a person's blood pressure?

8 Math Horizons November 1998 Human error: of the accuracy can you go back and check if 5 13 stu- in the House of Representatives is de- problems I alreadv ha1.e outlined can dents reallywere in attendance? rived from the Census; the amount of be attributed to human error, but here federal dollars flowing to each of the Even where accuracy checks do occur, I refer to human error in the sense of states is affected by the Census; the po- they do not always pick up the inaccu- an individual failing-peoplewho can- litical benefits to the Republicans ver- racy of a number. Recently, an university not count correctly, people who get sus the Democrats will depend on the reported that it had enrolled 4,527 stu- tired and quit before thejob is finished; Census. Whatever the methodology dents, exactly the same number as the people who inaccurately copy a num- used, Census 2000 will create important prior year. The U.S. Department of Edu- ber; people who forget; etc. Human er- and usehl numbers. Whether the num- cation has an accuracy check which re- ror, despite the best efforts of most bers also will be (more or less) accurate quires confirmation of any number which people, happens all of the time. After and meaningful is another matter. varies from that of the preceding year by all, we are human, not machines. more than +/- 5%. Later, it was learned Data-processingerrors: Calculators and that the university actually had sent the Historical Numbers computers are great because they take previous year's enrollment number by much of the drudgery out of doing cal- mistake. In Washington they did not Historical numbers or contemporary nurn- culations. Unfortunately, sometimes check to see if the number was the same bers about history really are not much dif- they, especially computers, are mis-pro- as last year's, only if it was different. ferent from the rest of the numbers we grammed. The more complicated our confront daily. There are serious prob- datawarehomes, the more tweaked our lems of accuracy, but the real danger for data analysis programs, the more pro- the number user is to presume either the grammers who had their hands on the essential accuracy of an historical num- creation of those programs, the more The one thing we ber (just because it is authoritative and programming steps necessary to reach oft repeated) or the inevitable gross inac- the final number, the more likely it is can be confident of curacy of such a number. Was the US that there is a data processing error. is that all of these Census of 1820 more inaccurate than the Furthermore, most often that error does upcoming Census 2000 will be just be- not appear in the initial test runs of projected numbers cause it was done a long time ago and we the program. In a sense, this is what do things better now? Y2K (the year 2000 millenium date will be wrong or Projections are numbers about the h- problem) is all about. ture. Because there will be no real num- Pure and simple fabrication: Some inaccurate to some ber which the projection can be compared numbers simply are made up to suit degree. against for some time to come, the ques- the purposes of their generators. Be- tion of accuracy changes. Every year the cause numbers can be both important US Department of Education projects the and useful, persons sometimes create number of bachelor's degrees that will be numbers out of nothing, especially .Ispecial national debate now is tak- awarded by colleges and universities in when there is virtually noway to check ing place around the expected accuracy the United States for ten years into the the accuracy or to reconstruct the ori- (or inaccura?) of the upcoming Cen- future. The one thing we can be confi- gin of such numbers. Many Ameri- sus 2000. The question is whether to try dent of is that all of these projected num- cans, for example, believe that the to count eveq.one in the country as of bers will be wrong or inaccurate to some Warren Commission's conclusion that some particular date in the year 2000 degree. It is not that the department a single "lone assassin" killed Presi- or to use statistical sampling to estimate makes bad projections. Rather, there sim- dent Kennedy was pure fabrication, some portions of the population. In ei- ply is no way to know how many students generated by the Commission for po- ther case, the final published number will be graduated ten years from nom7,or litical (national interest) purposes. will not be absolutely accurate. It is esti- evenjust next year. Only highly improb- The absence of audits or traceable pa- mated that perhaps as many as able, dumb luck would be the source of a per trails: Few of the numbers we use 15,000,000 persons were not counted projected number actually landing right are or ever could be audited to deter- in the 1990 Census. If that estimate is on target. If this is the case, why do we mine their accuracy. A business audit accurate, that would mean an error of do projections? or an IRS income tax audit might be roughly five to six percentage points. First, even though projections and considered an exception, but again Statistical sampling could reduce that most other numbers, for that matter, are how often is an audit done more than error in the year 2000 to about three inaccurate, they can still provide mean- once just to see if the same numbers percentage points. The choice would infil information. In most cases, num- result. In fact, most numbers cannot seem to be obvious. ber users do not need to know a quantity be audited. When the high school However, the Census produces politi- exactly, only a reasonable approximation. teacher reports there were 5 13 students cal numbers, notjust demographic num- In the overall scheme of things, it really at the school assembly yesterday ho~v bers. The number of seats each state has does not make much difference whether

Math Horizons November 1998 9 there are 154,097 students some fall en- What kinds of personal or organiza- rolled in the state's colleges and uni- tional motivations might be behind the versities rather than 154,964or 154,001 Successful number creation or the publication of the num- or even 153,648. What matters is that it ber? Might the estimate of that crowd is a little less than 155,000, which again user[s]. .. need both at the campaign rally be greater if you is down from each of the preceding good mathematical ask the rally organizers rather than the eight years. Or as a projected public political reporters? university system enrollment for the year skills and good What are the consequences to you (the 2020, it really does not matter if the fig- costs and benefits) of accepting andlor ure turns out to be 110,539 versus the sense. Often, good using the number? If you accept projected 112,639. What would make a 112,000 as the projected public system difference is if the real figure turned out sense is the more enrollment in the year 2020, what to be 125,997 or 96,007. In sum, ap- might be the impact on your proximate numbers which give the order dijjkult to achieve. institution's plans to seek state money of magnitude and, over time, a sense of for a new campus student center? direction usually are far more important What is the range of accuracy (margin than whether the numbers are exactly Is the number reasonable?Does it make of error) which you can tolerate for you accurate. substantive sense? If the best guess of to accept or use the number? Some- Second, making a projection, based the state's projected population in the times our needs require very accurate on carefully specified assumptions, helps year 2020 is 3,593,860, how likely is it numbers; often however, we can toler- the number user to structure his or her that the total college enrollment in the ate quite gross approximations. It is thinking about the problem and, thus, state will be 200,000? essential to understand one's own needs to increase understanding, even when the Was the number arrived at in an ap- as one evaluates a particular number. particular projected number turns out to propriate and reasonable manner? Is How does the number relate to other be inaccurate. Thus, in many cases the it procedurally reasonable? Does the numbers of interest? Often, a single act of making careful projections is more source of the number describe how the number tells you very little, but two important than whether the number pro- number came into existence? Is asking numbers in combination (a ratio, for jected is accurate or not. three political reporters to estimate the example) has considerable meaning, size of the crowd at a campaign rally a importance, and usefulness. The nurn- Advice to the Number User reasonable way to determine how many ber of hits a baseball player gets in a people were present? season, in most cases, is less interest- All of this suggests that being an effec- What kinds of hidden bias might be ing than the player's batting average (hits divided by at bats). tive number user must involve more than affecting the number? Might surveying just concern with the accuracy or inac- an advanced math class about why In sum, to be a successf~~lnumber user, curacy of a number. tVhen you confront women students tend to need moremath you need both good mathematical skills a number in your role as a number user, remediation upon enteringcollege than and good sense. Often, good sense is consider the following: men students produce a biased result? the more difficult to achieve..

The PhD Program in Mathematics at Dartmouth

The Dartmouth Teaching Fellowship. The progra1.11requires that students develop both as research mathematicians and teachers. All regular students in the program are teaching fellows. Fellows begin as tutors, usually tutoring two or three evenings a week for tn.ent\-weeks each year during the first two years of study. After admission to candidacy for the PhD degree, students take a course on teaching mathematics and then teach one ten-week course per year. Dartmouth takes teaching seriousl\; and supports its teaching fellows strongly, especially as regards the careful selection of teaching assignments. Program Features. A flexibly timed system of certification, through exams or othenvise, of knowledge of algebra, analysis, topology, and a fourth area of mathematics, replaces formal qualifying exams. There is a wide choice of fields and outstanding people to work with. Interests include algebra, analysis, topology, applied math, combinatorics, geometry, logic, probability, number theory, and set theory. For More Information. Write to Graduate Program Secretary, Department of Mathematics, Dartmouth College, 6 188 Bradley Hall, Hanover, NH 03755-355 1 or e-mail [email protected].

10 Math Horizons November 1998 JACK WILSON Internships

ne of the most frequently asked questions by student Therefore, a student seeking an internship should make members is: "How do I find an internship?" Not to be a concerted effort to cover each of these bases, i.e.: facetious, but it reminds me of the old joke about the Talk with key academic and administrative officials young man who stops an older guy on a New York street and at the school; asks "HOWdo I get to Carnegie Hall?" and the older guy says "Practice, my son, practice!" So my flippant answer to the Meet with the counselor who handles internships in internship question is: ''Work at it!" Here's how: the placementlcareer office; Getting an internship can be easy for the most talented, Check the bulletin boards, newspapers, and trade skilled, highly ranked student, who may actually be sought press for ads; for example, one of the internship out by companies through the school's placement office or ads in the New York Times this summer was by through a CS department head. But by and large, the num- Prodigy Services and gave an e-mail address of ber of students seeking internships is usually greater than [email protected]; the number of companies advertising for same. Typically, Seek out agencies (e.g., Manpower Inc., who expected companies who have an established program for interns will to fill 100,000 openings this summer) and recruit- fill their positions through one or a combination of the fol- ing firms in the area of interest, especially those who lowing: specialize in information technology (see Working Informal contact with trusted academicians at cer- with Recruiters, Math Horizons, September 1998, pp. tain schools, asking them for leads and recommen- 16-1 7); dations; Ask the leaders and members in the local student Formal requests to the schools' placementlcareer of- and professional chapters for leads and suggestions. fices; See Internship Opportunities for Students Advertising with bulletin board notices and in news- Review the listings in directories, which should be papers or trade press; available in the library, book store, or placement1 Using an agency or recruiting firm to identify can- career office. The three we know of (and their pub- didates; lishers) are: Through professional groups, societies, chapters; The National Directory of Internships Listing in an internship directory. National Society for Experiential Education 3509 Haworth Drive Raleigh, NC 27609; 9 191787-3263; America's Top Internships JACK WILSON is principal and founder of Career Sciences, a human Princeton Review, resource consulting firm specializing in career development and transi- 23 15 Broadway tion programs for the technical professions. He founded the firm in New York,NY 10024; 1989 after completing a 25-careerwith the IBM Corporation. While at 2 121874-8282);and IBM Jack was Program Director of Technical Resource Programs, a management position with responsibility for the professional growth Internships 1995 and career development of IBM's technical personnel. He currently Peterson's Guides, provides services to corporate clients and to members of ACM, the P. 0. Box 2123 Association for Computing Machinery, as a benefit of membership. Princeton, NJ 08543.

12 Math Horizons November 1998 Remember these directories are generic; they cover all brary, scouting out dynamic growth companies in such pub- kinds of internships, not just computing. lications as the CorpTech directories, you can find dozens You will notice that several of the above ideas involve or even hundreds of potential targets. Even if they have no "net~vorking,"i.e., using contacts with other people to formal program, with a little salespersonship and a good ad~anceyour interests. Statistically, networking accounts resume, you can convince them that having a high-quality, for the majority of all jobs filled across the board, and a professional intern like yourselfwould be a real boon to their similar phenomenon could apply to internships. While company. it may not account for the majority of internships, it prob- One other piece of advice in this direct contact sce- ably represents a significant number of opportunities and nario: don't write to Personnel or Human Resources; they should not be overlooked. Thus, every internship-search will just tell you they don't have any openings. Send a should include networking with personal and professional clear, crisp, convincing letter to the top manager in the contacts to unco\.er relevant information, leads, and re- functional area where you want to work. If it's an IT or IS ferrals. organization, send it to the CIO or VP of that function. If In addition to your networking and the six methods it's in software development for a technology company, above, my personal favorite (for internships, not job- send it to the executive in charge of development. You search) is making direct, unsolicited contact with a few may have to make a phone call to the company to get the selected companies. Irrespective ofwhether they have an person's name, but the personalized approach is ten times internship program, or have published openings, or have better than "To Whom It May Concern." All the latter even thought about hiring an intern, the direct approach says is that you don't care enough to find out whom that can be just as productive, if not more so, as applying for is. H an advertised position. Why? Because the latter is by definition a competitive situation. Copyright 0 199 7 Everyone else who knows about the opening can apply The Association for Computing Maclzine~y and compete for it. Whereas, you may be the only one ap- Reprinted by permission. proaching a company who has not advertised or published Visit ACMS Web site at www.acm.org. an opening. With a little time in the business reference li-

Math Horizons November 1998 13 DONALD J. ALBERS Art Benjamin-Mathemagician

athematics professor Arthur (Art) hobby, I sang, I danced, I got good at He has appeared on numerous televi- Benjamin, 37, is a friendly per- different games, and I learned to calcu- sion shows in the US, England, Canada, son, a popular teacher, happily late quickly. When I was five, I memo- and Japan, including The Today Show, married, and soon to be a father. But as rized the states and their capitals in al- Evening Magazine, Square One, Live! a child growing up in Cleveland, Ohio, phabetical order. I even learned the Dick Clark Presents, and CNN Head- all bets were off about his future. Little presidents in order." line News. He also has given hundreds Art Benjamin was so rambunctious in Today, Professor Benjamin certainly of performances for school children, his first nursery school that he was is not lacking for attention. For start- high school and college students, adults, thrown out! Ditto for his second nurs- ers, he gets lots of it from his students and even mathematicians. ery school, and the third, .... "At nap at Harvey Mudd College in Claremont, As a premier "mathemagician", he time, all of the students except me California. He is a popular teacher with regularly dazzles audiences with his cal- would lie down on their blankets," he a knack for involving students in his culating feats. He begins a typical per- recalls. "I would be running around the subject. He has written several research formance by asking members of the au- room. I was hyperactive. A lot of the papers, including many co-authored dience with calculators to assist him. In problem may have stemmed from the with his students, and he is the Editor order to test the accuracy of the calcula- fact that I was bored with many things. of the Spectrum Book series that is pub- tors, members of the audience will call I usually understood things the first lished by the Mathematical Association out two numbers for him to multiply, time they were explained, and by the of America. During his spare time, he for example, 68 and 92. "Make sure you time something was being taught the performs on numerous stages as Art get 6,256," Benjamin says, before his fourth or fifth time, I'd rather get silly Benjamin-Mathemagician. He is a assistants have even entered the num- than pay attention." During the first five regular at Hollywood's Magic Castle, the bers. The race between Benjamin and years of life, his hyperactivity resulted leading club for magicians in the world. the machines is on. He then challenges in several periods of hospitalization, them in squaring two-digit numbers, for often for weeks at a time. In today's lan- example, 47, 59, 63, 89. He beats the guage (and less understood duringkt's machnes every timewith ease. Benjamin childhood), his problem would be called then moves on to squaring three-digit ADHD, attention deficit hyperactivity and four-digit numbers, and again the disorder. He was prescribed the medi- calculatorslose. By this time, most mem- cationvalium,which he took for the next bers of the audience are thoroughly ten years. When Art was finally taken awed by his feats of lightning calcula- off Valium in the eighth grade, he tion. promptly went out for the track team. He had always liked sports, but the Valium had for years slowed him down. He Likes All Numbers As a child, Benjamin remembers wanting a lot of attention and doing lots Benjamin's interest in numbers, calcu- of things to get it. "I took up magic as a lation, and number patterns goes back to his early childhood. He says that he's never met a number that he hasn't liked. One of his first toys resembled a slot machine. When he pressed the lever of DONALD J. ALBERS is the editor of Math the machine, four arithmetic problems Honzons as well as co-author of Mathemati- Co~cldyoiik~ck thzc sroe~tlrttlr boy out ojniirwiy appeared in each of four windows - cal People. school? one in addition, one in subtraction, one

14 Math Horizons November 1998 in multiplication, and one in division. He recalls learning his multiplication The Raisin

UIILC UWUII d U~YUUILC tables with that toy. By the time he was 1 ~heery, in third grade, he was discovering nu- Far fro]& lusterlc:ss Lake 1Crie, merical properties resulting from mul- Onabeach som ewhere sc outhwest of I the Cil ty of Sing &T------..I. .-. L-J - tiplying two-digit numbers. "One day ~wdsd g~dpeLII~L II~U d notion. in school my classmates and I were in If he'd rest clost: to the ocean, the hallway ready to go to lunch. I re- He would tan without hi:j lotion, member standing up against the wall Just by resting rkear the shore, daydreaming about numbers. I was mar- And he'd meet up with adventure? never dreamt befo. veling over the fact that 2,520 is the All of this and much, much more smallest number that all numbers be- All of this had happened one day tween one and ten divide into perfectly. It wa,s c3n a sunriy Monda'Y I tried a smaller number, like 1,260, and As hc : s,oaked upI every su n ray goi ng into every pore realized that eight did not divide into Not 218 ~st of wi nd was bl reezing, A-2 - -..,.-*L "- -em.-. it. I then wondered what would be the ~IUthc W~IIIILII was, oh, au C~~III~ smallest number that one through eleven And so beautifully pleasing. would go into, and figured out that He was filled with joy galore. LVUIU rlere 011 LIIC ~IIUre. , would probably be 2,520 times 11." "Oh, how I wish," he said, "that I swy Remaining here forever more." A few years later he hung a velcro dartboard on the wall at the end of his Well, that grape who had that not ion, bed. The darts were actually velcro balls Slowly motioned to the ocean. and the object of the game was to throw He the,n discovc:red something which filled him with much gore. them at the numbered regions, 1 For he saw by h is reflection, Nature made a Ismall correction. through 10, on the dartboard. "Some- - - He was hrther from perfection, times I would lie awake at night staring Meaning worse off than before. at the dartboard. I would see the num- He had changed into a raisin while he rested on the shore, bers, and I would multiply them to- And that he'd be forever more. gether, getting various combinations as large as ten factorial (3,628,800)." Well, one day a man in yellow Gazed down at the little fellow One day he took a bus into down- Anrl .-ekedIn at him and other raisins resting on the shore. town Cleveland to visit his father at -., For you see his occupation work. To pass the time, he started think- Was to go to this location, ing about numbers that add up to 20. Meaning that it was his station He started at the centerwith 10 and 10, To pic1c raisins off the shore. and then formed the product 10 x 10 So he 1melt down towards that raisin on the beach near Singapore, toget l00;9x 11 was99;8x12was96; Picked him up with many more. 7 x 13 was 91. He then noticed that All of them were squished together 9 x 11 was one less than 100; 8 12 was x In a packet made of leather 4 less; 7 x 13 h7as9 less. The pattern was In a factory southwest of that city, Singapore. 1,4,9;he guessed that 16 and 25 would And that raisin was a snooper come next. "Oh wow," he said to him- Then he saw a spacious, super, self, "I wonder if this works for other Massive, mammoth, monstrous scooper, numbers, and then I tried it for big Scooping raisins by the score. numbers, small numbers, fractions. It Packaged them in Raisin Bran and sold them to the store, worked!" With elemen- algebra, it is Scooped him up with many more. easy to verify his discovery, but for a And that raisin now is well aware young boy with no knowledge of alge- He's in a bowl ii n Delawaire bra, it was a big accomplishment. Ready to be eatcEn by a cl ~ildnot yet four. In middle school, Benjamin's inter- And the milk was slowly dropping . . est in games blossomed, and he started And the raisin heard it plopping, to play both chess and backgammon in Then the raisin heard it stopping a serious way. When he started college, For the milk had ceased to pour. he gave up chess in favor of backgam- Au revoir sweet life, he cried, life which I truly do adore, mon because it was faster and easier to Quoth the raisin never more. play. He soon got to be very good at Benjamin wrote "ThRaisin" in his fleshman year of high backgammon. In 1997, he won the school. It is a parody of "The Raven" by Edgar Allen Poe.

Math Horizons November 1998 15 rector by night. I think he wished that down and get hit over the head with he had taken a shot at a career in the wands, and make them laugh. Thatwas theater. He passed on to all of his chil- my goal." dren his love of the stage." When he started doing shows for Each year the Scotch and Soda Soci- adults, he felt the need for more sophis- ety (a theater group) of CMU produced ticated material. He added card tricks an original musical, and over the space and mentalism-sort of fake ESP. After of a few months, Benjamin and another he got into mentalism, his father sug- freshman wrote the lyrics and music for gested that he put some of his mental the 19 songs of "Kije" (based on the calculations into his act. Young Ben- story "Lieutenant Kije Suite"). Accord- jamin was skeptical of his father's sug- ing to Benjamin, it was the biggest gestion and told him, "But that's not moneymaker that Scotch and Soda ever magic, that's real, Dad!" Nonetheless, had. "It was a big hit. I had delusions he took his father's advice, and the of Broadway." As he sat telling me of rapid calculations part of his act got the that experience, he suddenly broke into best response. "I decided that this must song, remembering perfectly a song he be the hture of my act." had written 19 years ago (the song is "I was something of a celebrity in performed by the villain as he attempts high school, and not always in a posi- to seduce the princess in the second act tive sense. Everybody knew me as the 'bli. 1~1~7~~/~~1/~1.(ill~\l/~!I,? ,Ill,. ,, of "Kije"). magician." He decided that when he started college, he wanted people to know him for himself. But fate-inter- American Backgammon Toill- I the .lBT is a series of tournament, plavecl around The Great Benjamini vened twice during that year to frustrate the countn thl-oughout the \-earl.and his goal. In his first month at CMU, a he is the all-time point leader in the During his middle school and high magic convention was held in Pitts- history of the =\BT. Benjamin cave his school days, Benjamin was known pri- burgh. He attended, impressed some early interest in gar-nei ancl calculation marily for his magic and he performed well-known magicians with his mental ties directly to his research area

16 Math Horizons November 1998 Tlze newly minted D1: Benjamin with lzis parents, La.rry and Lenore Benjamin, on the occasion of his graduation fi-om Johns Hopkzns in 1989.

ing lyrics for "Kije" and keeping up with My Turtle Pancho see the string 3876254, and try to com- classes. mit it to memory, it's much easier to do But in the Spring fate struck again. In spite of all his amazing mental cal- so if we break it up into groups: 387- Benjamin was taking a course in cogni- culating ability, Benjamin insists that his 6254. Benjamin's calculating feats in- tive psychology from Marcel Just and memory is only a little above average. volved remembering far more than 10 the topic of the day was "lightning cal- How then can he hold so many digits digits. Chase knew that he must be us- culators." Professor Just knew a few in his memory when multiplying two ing some mnemonic system. tricks himself that were designed to give distinct four-digit numbers in his head? Art can square three-digit numbers the impression of mental talent. He then in less than a second; four-digit num- asked the class, "We have lots of math bers take him about 15 seconds. Chase and science students in this room. Does asked Art to explain every step ofwhat an!.one here know any tricks?" he was doing as he squared a four-digit "I really shouldn't have," Benjamin But that's not number. He couldn't do it, because as recalls, "I !\-as tl?ing to keep my magic he explains, "The steps were so rapid under wraps but when somebody asked magic, that I couldn't follow them in detail." that question ... I couldn't resist. I did that's real, Dad! In order to slow him down, Chase asked my act from the Dove and Rabbit. The him to square a five-digit number, some- last thing I did ]\.as to square a four- thing he had never done before. And digit numbel; something I had just in- that slowed him down enough that he troduced in my nightclub act. While I could describe what he was doing. was concentrating on the calculation Most of us can hold about seven unre- For starters, Art does his calcula- and had already given out the first few lated digits in our short-term memory. tions from left to right instead of right digits of the answer, it )\,as so quiet, you When Professor Chase first tested Ben- to left. Asked to square 46,872, Art could hear a pin drop. I then gave the jamin, he found that he could only re- pinches his eyes shut, clenches his rest of the answer. Itwas right, and there call 8-10 digits, just a bit above aver- fists, and paces rapidly back and forth was an audible gasp. My impromptu age. Chase had studied other subjects, on the stage. He utters what seem like performance dazzled them." .Uter class, including one man who could recall disconnected words (his mnemonics) Professor Just introduced Benjamin to strings of 80 digits. Chase knew that -fuzz, nunnery, mover - and then Professor William Chase, 1vho was do- such feats of recall involved the use of says, "two billion one hundred ninety ing research on skilled memol?. For the the grouping of digits and mnemonics six million." After saying these digits next two years, Benjamin worked with (memory tricks). We can routinely re- out loud, he never thinks about them Chase, first as a subject and later as his call strings of seven digits, such as when again, depending on his audience to research assistant. we remember telephone numbers. Ifwe remember them, and freeing up space

Math Horizons November 1998 17 in his memory to continue with the become a full-time performer, it would calculation. After more pacing, hand become a bit repetitive after a while. I wringing, and strings of bizarre words, I wanted to be love applause, no question about that, he gives the answer: "2,196,984,384." but eventually I would say to myself, 'Yes, When Art started college, he did not someone who would I can do this and I can do that. But think that multiplying distinct three- what else?' I wanted something more digit numbers in his head was something bring math to substantial and intellectually satisfying. that he could ever do. "I co~ilclsquare I also felt that if I were going to teach them, but multiplying two different ones others, popularize mathematics professionally, I would seemed beyond me." By the time he it, and use my need to go on and earn my Ph.D." It graduated from CMU, he was multiply- appears that he made a very wise deci- ing distinct six-digit numbers in his entertaining talents sion, for today he is teaching, doing head! The problem would have to be research, performing, and receiving called out slowly to him so that he could to get more people applause for all of those activities. create the mnemonic to remember the 12 digits of the problem. The phonetic excited about it. code that he employs can be found in Secrets of Benjamin his book Matlzenzugics: How lo Look Like A particularly delightful aspect of Pro- a Geniza Without Reall: Ering: 0 is the Castle in HollFood. Before graduate fessor Benjamin is that he is eager to "s" or "z" sound. 1 is the .'t" or "d" school, Benjamin had already per- share his secrets of rapid mental calcu- sound, 2 is the "n" sound. 3 is the "m" -- . formed on several national television lations with others. Most magiciansjeal- sound, etc. Kext he converts numbers shows. It's clear that he could have been ously guard their secrets, but not him. into ~rorclsby placingvowels around or a full-time magician if he had chosen Several years ago he distributed within between the consonant sounds (spelling to do so. But he chose mathematics and the magic community his booklet, " The doesn't matter). For example, the num- teaching to be the center of his life. He Secrets of Benjamin." In 1993, he and ber 32 can become anv of the following says, "I wanted to be someonewho would Michael Shermer wrote Mathemagzcs, worcls: ?nun, men. mine. moon, money, bring math to others, popularize it, and which lays out his secrets in detail. He nzenu, etc. Here's the code he used to use my entertaining talents to get more clearly wants others to learn the art of translate the first 24 digits of pi: people excited about it. On top of that; rapid mental calculation and to come I had had some teaching experience as to love mathematics in the process. Can l4 926 j8 97 an undergraduate. Teaching was fun; it they get as good as Art Benjamin - "My turtle Pancho will.' my love, pick wasjust like performing, but better be- Mathemagician? Perhaps, but that will 9 3 2 384 6264 cause your repertoire changed with ev- require enormous practice and prodi- up my new mover, Ginger." ery lecture. I sensed that if I were to gious motivation. You can count on it!.

With just a little bit of practice, us-

in^" the I hone tic code. vou can memo- rize over a hundred digits of pi!

Choosing a Career: Math or Magic? Benjamin's reputation as a mathemagician, researcher, and teacher contin- ued to grow during his undergraduate days at CMU and graduate days at Johns Hopkins University. M'hile at Hopkins, Benjamin worked on his dis- sertation ("Turnpike Structures for Op- timal Maneuvers") under Alan Goldman, research that later earned him the prestigious Nicholson Prize from the Operations Research Society of America. Hopkins is also where he met Deena, a fellow mathematician, whom he married 4 years later at the Magic

18 Math Horizons November 1998 Graphs, Maneuvers, and Turnpikes

If you wish to drive from my mother's to move 4 checkers from one corner of a the solution to any maneuvering prob- house in Cleveland, Ohio to my ofice checkerboard to the diagonally opposite lem on the plane obeying a few natural in Claremont, California, here's what corner, you should spend almost all of conditions will spend almost all of its you do. Get on the 1-7 1 freeway South your- time moving the configuration time repeatedly translating one or two to 1-70 West. Go about 1700 miles until efficient configurations. you get to I- 15 (around Beaver, Utah). For more details, see Take 1- 15 South about 700 miles to I- 10 "Graphs, Maneuvers, and Turnpikes," West. Go another 20 miles and you're Arthur Benjamin, ~rn&.r Research, vol. practically there. The point is that you 38, #2 (1990), pp. 202-216. will spend almost all of your time trav- "Optimal Leapfrogging," Arthur Ben- elling on high speed roads. jamin with J. Auslander and D. S. A similar phenomenon takes place Wilkerson, Mathematics Magazine, vol. 66, when maneuvering a collection of objects up and over in two (double-jump) #1 (1993), pp. 14-19. from one location to another. For instance. mol-es. In general, it can be proved that -Arthur Benjamin

Starting positirv? A does a double jump B does a double jump

C does a donble jump D does n dorible jullrp

Math Horizons November 1998 19 Census 2000: Count on Controversy

ne of a statistician's main jobs is Not a Classroom Problem extremely difficult to find and count. to estimate unknown quantities. Many, such as criminals and illegal "Basic training" in the science of Counting the number of residents of the aliens, do not want to be found. estimation occurs in the typical under- United States on April 1,2000, the next More people enter and leave the graduate course in mathematical statis- official census date, is an entirely dif- country than ever before. This in- tics, which focuses on problems of the ferent sort of estimation problem from cludes undocumented workers such following kind: A random sample of size the type described above. First, the as crop pickers who migrate accord- n is drawn from a large population of in- parameter of interest is the unknown size ing to seasonal patterns. dividuals or items. The goal is to esti- of the population itself. And the word mate the value of some important char- census means a complete enumeration \\'ith a projected cost for the 2000 acteristic of the population, referred to of the population-that is, thesample is Census of approximately $4 billion dol- as a parameter. For example, one might the entire population, not a proper sub- lars, the United States Census is the most wish to estimate the mean lifetime of a set of it. Clearly with a full enumera- expensive estimation problem in the certain brand of light bulb, or the pro- tion, there is no sampling error, and in I\-orld.To appreciate the difficulties in portion of bulbs that will fail before the principle no error whatsoever in the obtaining an accurate count, consider time given in the manufacturer's guaran- result. This is why censuses receive scant the follo~singanalogy provided by tee. Values of the variable of interest are mention in most mathematical statistics Tommy \\1-iglit, chief of the Statistical measured for each member of the sample, courses, where the main goal is to quan- Research Di\.ision of the U.S. Bureau then used to form the parameter estimate. tify and minimize the error that arises of the Census [I]: Suppose someone asks The subject of mathematical statistics due to sampling. you to count the number of people in addresses the question of how to effi- The United States Census takes place, attendance at a local high school bas- ciently use sample measurements to es- however, not in the classroom but in the ketball game, and to make the count timate parameters. A sample is repre- real world, and there is indeed a con- during half-time. Many spectators will sented mathematically by n indepen- siderable error in the result. Counting go for refreshments or to the restroom, dent random variables with a common all U.S. residents is a daunting task that u.hile some \\.ill leave the arena alto- probability distribution. This distribu- has become harder with each census. gether. Some fans~\.ills~vitch seats. Play- tion models the variation in a certain Here are some of the reasons: ers. coaches and referees will be in the variable (e.g., lifetimes of light bulbs) locker rooms. Could you just use the At close to 2'70 million people, the ticket count? So,because some persons throughout the population. U.S. population is larger and more As normally only a portion of the are admitted without tickets and some diverse than ever before. The first who purchased tickets do not come to population is sampled, an estimate will census, directed by Thomas almost certainly deviate from the actual the game. .bycounting procedure you Jefferson, found less than 4 million attempt certainlywill miss many people parameter value. The resulting differ- residents. ence is known as the sampling error. By while counting others more than once. analyzing the performance of various es- The Bureau conducts the initial and The difference between the actual timators mathematically it is often pos- primary phase of the census by mail. population and the estimated popula- sible to determine an optimal procedure However, the rate of voluntan re- tion is known as nzearurement error. One that minimizes the likely size of the sam- sponse to the mailed census question- of the Census Bureau's main goals is to pling error. naire (before hrther prompting) has minimize this error for the United States declined significantly, from approxi- as a whole. But there is more: When mately 85% in 1970 to 75% in 1980 Congress establishecl the United States to only 63% in 1990. decennial census in 1790, its main pur- Large numbers of people are home- pose was to provide a basis for a fair MARK SCHILLING is Professor of Mathemat- less or live in rented garages, trail- apportionment of congressional seats to ics at California State University, Northridge. ers, etc. Clearly these individuals are each state in the Union. In this century

20 Math Horizons November 1998 the census has also become used for di- ceding one, despite being the most ex- planned to let census enumerators use viding the enormous amounts of fed- pensive census on record. their own discretion in deciding hot+. eral funds (billions of tax dollars per How might the Bureau reverse this to achieve the 90% quota, but received year) that are distributed to the states. trend? Surprisingly, improving the ac- criticism for the subjectivity and poten- Thus the Bureau attempts to obtain curacy of the census (reducing the mea- tial biases that enumerators might in- counts that are as accurate as possible surement error, i.e., the undercount) troduce into the process.) for the United States as a whole as well may require utilizing a certain amount Finally, the Bureau would make a sta- as for each of the many thousands of of selective sampling (thus introducing tistical estimate of the number of remain- local regions that together comprise the some sampling error). A more accurate ing persons missed, using the informa- C.S.. all while attempting to limit costs. count might be obtained by not trying tion obtained from the units that re- to count everybody. Accordingly, here sponded in the follow-up sample. To- is what the Census Bureau has proposed taling the conventional count \\.ith the Sampling: More From Less? for the 2000 Census: follow-up count and the estimate of those Although the narional cencus is con- The United States population is di- missed by both counts would then gii-e ducted only once men- ren vears. the vided into approximately seven million the overall estimated population of each Census Bureau conrinu.xl'\ cnnducts sur- CP~S~ISblocks-essentially, neighbor- tract. The Bureau would then estimate veys and analyses for w!?er purposes. hoods, each containing perhaps 13 to the total U.S. population b!- sumnling This makes it possible toc..hrzin an idea 30 housing units. A tract is a contigu- over all of the tracts in the L-nited States. of the measurement emrin a ~-ende- ous collection of such blocks. The Bu- Census Bureau officials have also decennial census. For the Census. reau has formulated a plan to employ vised a strategy for using sampling to pm the official count !\-as 21\.7F'P.P73 conventional counting methods in the vide a quality check on the accuraq of people, while evidence from n!her cur- first phase, then use sampling to bring the estimated counts. \\hat's more. this veys and demographic analvcec indi- the count up to at least 90% of the total plan offers the potential to actually i?n- cated a true population ofappm~irnareh- population (as estimated by prior sur- pro-ire the original count: Entirely inde- 253,000,000.The 1990 Census therefore veys by the Bureau and other agencies pendently of the main census count, the produced anundercount ofarouncl -!mil- and by demographic projections) in Bureau proposed selecting a random lion people. Each census since at least each tract. That is, within each tract sample of approximately 23,000 blocks 1940, and possibly all of them. have mi- whose initial response rate is below 90%, nationwide. Census workers would visit dently resulted in a sizable number of a random sample of the nonresponding each housing unit \\.ithin these blocks persons being missed. However. the 1 !Bl units ~\rouldundergo follow-up efforts to and attempt to dete~minethe populations Census I\-asthe first to have an estimated contact enough individuals to reach the of these blocks. .As in the main census, undercount larger than that of the pre- 90% quota. (Initially, the Bureau had some people will again be missed.

Math Horizons November 1998 21 The Bureau would then havetwo num- product of the two census counts de- enumerate missing individuals. Con- bers for each region of the country and scribed above divided by the number of versely, Republicans almost universally for the country as awhole, both expected people found in both counts. The reli- oppose sampling. to be wrong-specifically, too small. How ability of these new figures will depend Although political disputes over the can this dormation be used to create one on assumptions analogous to those in- census are not new-George Washing- potentially very good count? volving the likelihood of tagged perch ton issued the first presidential veto in The answer lies in a time-tested sta- being recaptured. Specifically, it is cer- disagreement with Congress' interpre- tistical procedure known as thecnptlc~e- tainly plausible that people who were tation of the 1790 census figures-the recapture method. Imagine that you wish hard to find by the mail portion of the contemporary level of controversy over to estimate the number of perch in a survey and the follow-up to achieve 90% how the census should be conducted is certain lake. By fishing simultaneously coverage will also be hard to find by the unprecedented. Numerous lawsuits were in several areas carefully distributed door-to-door survey of 25,000 blocks. heard over the last several years regard- within the lake your research team ing the 1990 Census. The issue was catches, tags and releases 40 perch (cap- Will They or Won't They? whether to use the original census count ture). Now you send your team out to or a different number that attempts to fish again. This time they capture 50 The Census Bureau believes that sam- correct for the undercount. The Su- perch, ofwhich 8 are tagged. Since one- pling has the potential to produce a preme Court ultimately ruled in favor fifth (8 out of 40) of the tagged fish more accurate census at lower cost. The of the original, unadjusted count [2]. caught are tagged (recapture),one could National Academy of Sciences, the The 2000 Census is embroiled in a conclude that approximately one-fifth American Statistical Association, and similar controversy. On August 24th of ofall of the perch in the lake were caught several other independent agencies sup- this year, a special panel of the federal this time. Thus the total number of port that view. Yet it now looks as if sam- appeals court ruled that the Census Bu- perch in the lake is estimated as 30 x 5 pling may not be used at all for Census reau may not use statistical sampling in = 230. (This assumes, of course, that 2000. The United States Census, unfor- the next census. TheJustice Department tagged fish were neithermow likely nor tunately, has become a political football. has appealed this decision to the Su- less likely to be caught than untagged The reason is that the level of preme Court, which will hear arguments fish. Conceivably, tagged fish could be undercounting varies dramatically by at the end of this month. A final ruling more gullible than other perch in the income status and ethnicity, being much is expected no later than March 1999. lake. Conversely, they could be more higher for poor communities and for cautious than untagged fish during the people of color. Partly as a result, the References recapture phase, having been caught undercount also tends to be greater for once already.) large urban centers. Potential voters in 1. Tommy Wright, "Sampling and Census The capture-recaptureestimate above these categories are more likely to be 2000: The Concepts", American Scientist, May-June 1998, 495-524. can be written as (40 x 50) + 8. Simi- Democratic than Republican. Thus it is 2. Margo Anderson and Stephen E. not surprising that the President and larly, then, the Census Bureau's capture- Fienberg, "An Adjusted Census in 1990: recapture estimate of the human popu- most Democratic governors and legis- The Supreme Court Finally Decides", lation of a given region would be the lators support the use of sampling to Chance, Vol. 9, No. 3, 1996, 4-9.

Bowling Green State University 1 Department of Mathematics and Statistics

The mathematics and statistics program at BGSU is nationally ranked and one of the strongest in Ohio. We are large enough to offer a wide array of courses and research opportunities, excellent library and computing facilities, and yet small enough to provide our students individual attention in a liberal arts atmosphere. We have 30 faculty and over 60 fully supported graduate students. Strong research areas include Algebra, Analysis, Applied Mathematics, Probability, and Statistics. We offer the Master of Arts in Mathematics and Statistics, the Master of Arts in Teaching Mathematics, the Master of Science in Applied Statistics, and the Doctoral degree in Mathematics and Statistics. Please consider visiting our campus. For further information, contact:

Graduate Secretary (419) 372-2637 Department of Mathematics and Statistics FAX (419) 372-6092 Bowling Green State University [email protected] Bowling Green, OH 43403-0221 http://www.bgsu.edu/departments/math~

22 Math Horizons November 1998 JORDAN ELLENBERG Exploring Moduli Spaces

There's more to math than just numbers and equations. In 2. elements of S that are close to each other correspond to fact, much of mathematics concerns space-not outer space, points of X that are close to each other, and vice versa. but mental representations of space and spatial relationships. While many people think of Descartes as the first person to describe points in a plane by means of a pair of coordi- The Coordinate Plane nates, the idea of coordinatizing planar space was known to Just as a picture of the United States is a map of the set of the Hipparchus in 150 B.C. And by 1640, Pierre de Fermat had states, the plane is a map of the set of pairs of real numbers. already developed the idea of expressing a curve in the plane If S is the set of ordered pairs (s,~)of real numbers, a moduli as the solution set of an equation in x andy. RenC Descartes, space for S would be a plane, which we'll call X. however, was the first to systemically use the coordinate plane How does X satisfi our definition of moduli spaces? The to turn problems of geometry into problems of algebra, and one-to-one correspondence required bv the first part of the vice versa. Today, at the very center of modern research in definition sends an ordered pair (s.~)to the point (x,y) of number theory and geometry lies a much-generalized form the plane. The second part of the definition is satisfied be- of Descartes's idea-that of moduli spaces. cause pairs of real numbers that are vel-\.close to each other, such as (1,2) and (1.OO 1.1.999), map to points of X that are What is a Moduli Space? very close to each other: likewise, nearby points of X correspond to nearby pairs of real numbers. You might think of a moduli space as a map or a picture of a So Descartes's insights about the coordinate plane can be certain set. A map of the United States is, loosely speaking, summed up in the terminolop of moduli spaces: A plane is a moduli space for the set of states. Ifyou had a list of the 50 a moduli space for the set of pairs of real numbers. states and their sizes and shapes, and another list detailing which states bordered on which other states, you might still Challenge 1. \\'hat ~~ouldbe a moduli space for the set of have only a fuzzy idea of what the United States looks like unordered pairs of real numbers (x, y)? and how it is put together. But with a map, you would per- ceive this information clearly. In math, you typically work with sets of objects, such as numbers, points, or lines. Butjust as with the states, you can Distorting Space often have a clearer understanding of a set if you have a You can probably guess that a line is a moduli space for the "map" of it. A moduli space for a mathematical set might be set of all real numbers. a plane, a line, a curve, a cylinder, a hockey-puck-shaped entity, or even a wild hyperamoeba in 26-dimensional space. More specifically, a moduli space for a set S is a geometric object X such that A moduli space for the real nzrmbers 1. the points of Xare in one-to-one correspondence with the elements ofS; and But it is not the only one. We can distort the line in as devi- ous a fashion as we like, taking care not to break it or let it cross itself (which would violate the second and first parts of our definition, respectively)-and the resulting curve will still be a moduli space for the real numbers. JORDAN ELLENBERG received his doctorate in mathematics from Thls example enables us to expand our definition of moduli Harvard in 1998. He is now an instructor of mathematics at spaces a bit: If X is a moduli space for a set S, any distortion of Princeton. X is also a moduli space for S.

24 Math Horizons November 1998 Challenge 2. What if S is the set of ordered pairs (r,O), where r is any real number? - - - - _--- Challenge 3. What ifS is the set of ordered pairs of angles O*)?

Another moduli space for the real nzonhers Playing Around with Correspondences Suppose S is the set of all lines through the origin of the xy Circles, Cylinders, and More Moduli plane. What is a moduli space for S? Spaces A line through the origin can be defined by the angle0 it makes with the x-axis. So we might guess that the moduli What if S is the set of all angles? We know that an angle is space for S is the same as the moduli space for angles: a determined by its measure, which (in radians) is a real num- circle. But as we consider the problem more carehlly, we hit ber between 0 and 2n. But the moduli space for S is not a a snag. The natural correspondence between lines and angles line segment from 0 to 2n because those points, which are is not one-to-one, but two-to-one: the angles x12 and 3nl2 far away from one another in a line, correspond to angles both correspond to the vertical line. How can we address that are identical to each other, violating our definition. To this problem? make a moduli space for S, we need to join the ends of the One good way is to use adiffermt correspondence between line segment to form a circle. Of course, a distorted circle lines and angles: Instead of associating line L to the angle6, will also do. we could associate it to the angle 26. That simple redefini- tion transforms our two-to-one correspondence to a one-to- one correspondence. Using this new correspondence, we can see that the moduli space for S is indeed a circle.

Challenge 4. What if S is the set of all lines in the xy plane?

What I like about problems like this one is that they don't E[IO,r~oduli cp.1~ f~ thr srt of angles require much, or any, pencil-and-paper calculation. You can work on them in the shower or while riding the bus. Better Now let S be the set of ordered pairs (r,O), where r is a yet, the problem admits endless variations, each with its own positive real number and 9 is an angle. If you know about interesting features: What is the moduli space of circles in a polar coordinates. vou knm- [hat the plane with the origin plane? Circles through the origin? Lines through the origin removed is a moduli space for S: (r,O) corresponds to the in three-dimensional space? The possibilities are limited only point \\.hose (s.v) coordinarm are (rcos0, rsin6). This moduli by your imagination (and, if you are working in the shower, space is usuall!' called -[he punctured plane." by the hot water supply). Enjoy! We can also think of [he moduli space for this set in another way. IVe know thar a cirtle is a moduli space for the set Reprinted with permission of The Johns Hopkins University Press of angles. And I\-ecan see that a rav is a moduli space for the from Imagine, MarcWApril 1998, p. 12-1 3. set of positive real numben. So a moduli space for S can be constructed by starting \siirh a circle and attaching a ray to each point. Of course, we don't have to plot r in the same plane as 6. If r is plotted perpendicular to thexy plane, the moduli space now looks like a n-clinder, infinite in one direction. Note that this moduli space isjust a distortion of the punctured plane. .d!~lssn!qoN snourej ay1 s! x :p aBual~eq=> ".sruo~,,e lie3 sue!3~etnay3euraqms e -1nuyBnop e jo a~pnsaw ayq padeys s! X:S aSual~y3 ~rapu!lhayugu! u-e s! x :z aBual1eq3 .{IG 7 x :(X6x))Xq paugap aue~d-j[eyaq 'a~durexa JOJ :aue~d-jeye s! x :I a8ualley3

From punctured pione to cone to qlinde~

Math Horizons November 1998 25 JAMES TANTON A Dozen Questions About a Donut

s a topologist I have long held great respect for the humble donut (torus).The surface is suficiently com- A plicated to be interesting mathematically, exhibiting regions of positive and negative curvature for example, yet the shape is so familiar to us all that discussion about this surface is accessible and meaningful to most every audience. Here are some of my favorite toroidal puzzles resulting from my recent musings upon the donut. I hope you will enjoy thinking about them as much as I did.

1. Care Please! Surprisingly, for houses and companies situated on a plane this problem is never solvable. (Tnit for yourself.) Even tak- The Jordan Curve Theorem, stated carelessly, says that any ing into account the spherical structure of the earth and al- simple closed curve divides whatever it is drawn on into two lowing the possibiliy of pipes circumnavigating the entire distinct pieces, an inside and an outside. globe the problem is still unsolvable (Why?)But maybe the situation ~vouldbe different if the earth were the shape of a torus. Could the problem ever be solved on a toroidal planet?

3. Hidden Symmetries One can form a torus from a square piece of paper simply by gluing the top edge to the bottom edge to form a cylinder and then bending this cylinder to glue the left edge to the Show that this "utterly obvious" theorem is in fact false for right edge. (live topologists say that the opposite edges have some curves drawn on the surface of a torus. been idenr$ed.)

2. Housing Issues Here's a classic problem from graph theory: Three houses are each to be connected to three utility companies (electricity, water and gas) in such a way that no lines or mains cross. Is this ever possible ? '4 But suppose we first apply a clock~\.iserotation of 90 degrees to the square before identi5ing the edges and thereby interchange the roles of edges A and B. \\hat effect does this have on the formation of the torus? JAMES TANTON is Assistant Professor of Mathematics at St. Mary's A square has, in fact, four rotational symmetries in all, as College of ~Maryland. well as four reflection symmetries. How does the application

26 Math Horizons November 1998 4-7 5. Inside Out Turning a punctured sphere inside out yields another sphere.

A JI of each of these symmetries affect the final presentation of the torus? One can also form a torus starting., with a hexagon- and gluing opposite edges in the analogouswa~.(Try it!).Ahexa- What do vou obtain ifvou turn a ~uncturedtorus inside out? gon has far more symmetry than a square, so where does this i I 1 extra 6-fold symmetn lie hidden in the fully formed torus?

4. The Search for Irrationality Now that we kno\\-that a torus can be formed from a square with opposite edge idencifications, let's try running the fol- 6- Cartographer's Wisdom lowing experiment on a unit square. Cartographers have known for centuries thatjust four colors Given a dra'r a line of 'lope starting from the are for coloring any map on the globe, the stipula- bottom left comer of rhe square. When this line hits the top tion being that no rwoneighboring regions are assigned the edge have it reappear (asa "wrap around" effect) at the same same color. corresponding point on the bottom edge, to continue on at the same slopem. Similarl\-for when the line hits a point on the right edge.

But if the earth were the shape of a torus four colors would not suffice. Can you produce a map on the torus that requires a minimum ofJive colors to complete? How about one (In doing this we hare. in effect. tiled the entire first quad- that requiressur colors? yet another that requires a mini- rant of the Cartesian plane with unit squares and superim- mum of seven! (Hint: Noting question 3, try drawing maps posed the pictures from each square ontojust one.) on a hexagon.) Show that this cun-e wilF rerurn to the bottom left vertex if, and only if, the slope ic nrinnal. Thus, if you enter an irra- tional slope m, the line wi!;continue on forever, never re- 7. A Torus with a Twist turning to its starting poinr. Take a band of paper and glue the top edge to the bottom all the way round. This forms a torus.

Suppose however this band was given a half twist in its construction. Do we still obtain a torus by gluing the top edge to Here's a hncomputer project: \\iite a program that would the bottom edge? visuall!; do this for you (either br actually drawing this line for you or by plotting a specified number of points on this ;vf~)/,.i,J , line) and input various numbers for m. Ilhich value of m -...... __...., + ...../.,//; .... -+ ?? seems to fill up the screen the fastest?

Math Horizons November 1998 27 8. Trouser Trouble 10. Tricky Triangulation A pair of trousers right side in is physically equivalent to a Notice that it is possible to completel! cover a torus with pair right side out. One can transform one into the other warped triangular regions in such a \\.a! that any two neigh- simply by turning the item in hand inside out. boring triangles touch either in asi~~ple\.ertes or alongjust one entire edge. In this picture II've drawn a trianLgulationthat uses more than 62 triangles. Can you do better? Using mark- ers on a bagel can you produce a triangulation that requires fewer than, say, 15 triangles? What is the minimum number of triangles required to triangulate the torus?

But suppose in each case we sew the leg openings to each other to form punctured tori.

11. Is the Universe a Donut? Columbus, by sailing west and hoping to return from the sew sew east, apparently believed that the earth was not flat. The mod- Are these two structures still equivalent? Is there a physical ern world guessed it was a sphere but it could have turned manipulation that would convert one to the other? If so, out to be a torus (or even a multi-torus). how? If not, why not? We've seen that a two-dimensional torus sitting in three- space comes from a square piece of paper by gluing together the top and bottom edges and then the left and right edges. 9. Triangular Surprise The next higher dimensional analog would be to start with a Normally we think of a triangle as a region, finite in area, solid cube and glue the top and bottom faces together, then bounded by three distinct straight lines. Thus a triangle has the left and right walls together, and finally the back and three edges connecting three distinct vertices. front walls. This would give a three-dimensional torus sitting On a surface, a "straight" line connecting two points is in four-space. the path a piece of stretched elastic would take between the two points. Hence on a sphere, for example, the "straight lines" are arcs of great circles. On the plane, "straight lines" coincide with our usual notion. On the torus the straight lines are less obvious, but they follow the same principle. Thus, the notion of a "triangle" naturally extends to surfaces as any region on the surface, finite in area, bounded by three distinct straight lines and having three distinct vertices. Now, one learns in school, thanks to Euclid, that the sum We've heard from physicists that "space is curved". Could of the angles in any triangle is 180 degrees. But one has to it be that the universe is the shape of a three-dimensional be careful with such a statement-it is patently not true in torus? How would we know? general. For example, on a sphere we can easily find tri- angleswhose angles add to greater than 180 degrees. In fact this is the case forevery triangle on a sphere. 12. Jordan Returns Euclid's rule also fails on a torus. Show that it is possible An ordinary two-dimensional tonls al\\.a!.s divides space into to construct triangles on a torus whose angles sum to greater two regions-an inside and an outside. Right? than 180 degrees. It is also possible to construct triangles Wrong! Show that there is a two-dimensional tonls sitting whose angles sum to less than 180 degrees. Can you see how? inside a three-dimensional torus that fails to divide the space it is sitting in into two distinct pieces.

28 Math Horizons November 1998 Comments, Answers and Further Questions by dragging circleA around B. Reflecting the square across a horizontal or vertical line turns the square over, preserving 1. Curves that circumnavigate the torus, in either direction, the locations of edgesA and B. Upon folding, this creates a or both, fail to divide the surface into two distinct regions. new donut, but the other side of the paper is now on the These curves, as it were, have no inside nor outside. outside of the torus. Reflection across a diagonal combines these two effects. For a hexagon the three pairs of opposite edges correspond to three line segments within the hlly formed torus. We can think of two of these strands situated to form a circle and the third located so as to dictate a twisted path along which to drag this circle.

TheJordan Curie Theorem is, however, valid for curves drawn on the surface of a plane or a sphere. The surface on which the curves are drawn needs to be mentioned in its statement. This sticky point also explains why this seemingly obvious theorem is so difficult to prove in mathematics. Itsvalidity is saying something civificant about the geometry and structure ofsurfaces. ?his needs to be understood fully before one can begin provinz the "obvious." 2. Indeed there is a toroidal solution.

In a similar way, the rotational symmetries of a hexagon correspond to interchanging the roles of each of the three strands in the formation of the torus. Reflections again yield the combined effect of this and turning the torus inside out. 4. A line y = mx in the plane has rational slope if, and only if, it passes through a point with integer coordinates. As For a proof that rhrre iq no planar solution see [2]. If there each such point is identified as the bottom left corner of the were a solution on rEie curface of a sphere, simply puncturing unit square the claim follows. the sphere and stretchinc the surface flat would yield a planar solution. .& no cuch planar solution exists, there can be 5. One obtains another torus. - no spherical solurinn. -1ibnq this reasoning one step hrther, we can sirnilarl\-dec!uce chat there is no solution on the surface of a qlinder. Hmmer. here is a solution if this cylinder is given a half nr-ict in it< fnrmation to create a Mobius band. Can you find it'

+ 3. Consider first the 4-fold F\-mmetnof a square. The two ~h~ next ,-hallenge is to draw a sequence ofdiaganls that pairs of opposite edges of the ~ecorre~~~~~show even everting multi-tori gives the same surfaces back in the fully formed torus. \\'e can thus think of the torus as again. canyou do it? being formed by dragging circle R around circleA. -

6. Regarding a torus as a hexagon with opposite edge iden- Rotating the square 90 degrees before folding switches tifications we can easily design a map that requires seven the positions ofA and B and hence results in a torus formed distinct colors to complete.

Math Horizons November 1998 29 As it is impossible to convert one set of bands into the other (tn; it!),the two ptu~cturedtori are in fact physically inequivalent. 9. Any triangle dra~t-non the outer face of a tol-us has angles summing to greater than 180 degrees. On the inner face all triangles have angle sums less than 180 degrees. This can easily be verified with three pieces of suing and a protractor on an inner tube or a bagel. Although the "Four Color Theorem" is extremely difficult to prove, relying on extensive computer operations to verify thousands of special cases, a "Seven Color Theorem" for a torus is quite easy to establish. For a proof that seven colors will suffice to color any map on a torus see [I]. 7. This half twist has changed the topology of the problem significantly. Ifyou attempt this feat in practice (and I suggest you usep1r)~doughifyou do) you will soon discover completing the procedure is impossible-the two ends of the tube being 10. Any triangulation of the torus must involve at least 14 created end up lying on opposite sides of the band. triangles. This is easily proven using the Euler equation for a torus. If t is the number of triangles used in a given triangulation, e the total number of edges andv the total number of vertices then v-e +t = 0 (see [3]). Using this with the fact that 3t = 2e (every triangle Completing the process would somehow require pushing has three edges and each edge is counted twice) and one end of the tube through itself so as to link up with the other. Doing this would create a fanciful surface perhaps fa- ..(;)="i."2 miliar to you already-the Klein Bottle. n (each edge connects a pair of vertices) it follows that e r 14 (Incidentally, the formula 3t = 2e shows that it is impossible to cover a donut with an odd number of triangles!) Regard- ing a torus as a square with opposite edge identifications we can demonstrate a triangulation that uses precisely 14 triangles. Of course the construction of such a surface is impossible in our three-dimensional world. However it has been shown that the Klein Bottle is a perfectly valid mathematical object if situated in four-dimensional space. It is the curious non- orientable (one sided) cousin to the torus. 8. Regard the torus as being formed from a square. Cutting a small hole in the torus corresponds to cutting a small hole in the center of the square. Cutting a large hole, or equivalently, 11. Ifwe ever sent a brave astronaut off into space and many extending this small one, leaves a thin border of material along years later she returned from the opposite direction we'd most the edges of the square. Upon identification this corresponds likely conclude that the universe is not flat. However since we to two thin strips of material connected to form two bands. cannot stand back and look at the universe from a distance we'd have no immediate way of figuring out exactly what shape it is. We'd have to rely on more subtle geometric properties. The way to visualize what life would be like in a three- dimensional torus is to imagine walking into a room completely covered with mirrors, walls, floor. and ceiling. If you look to the left you will see the right all and the same with Doing this for the two tori pictured in the question corre- the back and the front. (This analoq. is not quite correct sponds to two different sets of double bands. however. If you are standing in the center of the room and look up you would see the top of your head. However, in a three-dimensional torus you would see the bottom of your feet). It is possible that our universe is shaped like this. It's not very old so maybe light hasn't travelled far enough yet to be "reflected" back to us along some "wall."

30 Math Horizons November 1998 The study ofcurvature is the study of the geometrical stmc- References ture of shapes and surfaces.As we have seen, for example, on the surface ofa (two-dimensional) sphere all triangles have 1. Arnold, B.H.; Intuitive Concepts in Elementary Topology, Prentice- angles which sum to greater than 180 degrees-a sphere has Hall, 1962. 2. Chartrand, G.; Introductoly Graph Theory, Dover 1977. positivecurvature' On a torus there are regions of positive and 3. Massey, WS.;A Bmic Course h,41gebaic T@&gy, Springer.,"erlag negative curvature. These localproperties give inhabitants of 1991. these surfaces as the the surface On 4. Thurston, W.B.; Three-Dimensional Geometry and Topology, Vol 1, which they live. Local properties may give us some clues as princeton University Press, 1997. to the shape of the universe in which we live. 5. Week, J. R.; The Shape of Space, Marcel-Dekker, Inc., New York, 1985. 12. Regard a (two-dimensional)torus as a square with opposite edge identifications. A circle that fails to divide this region into two distinct pieces is shown as follows:

9R@jD ~$~q2.,+itCr.,. Towing Icebergs, Trigonometric Delights iy?&*.'iy $$$&$$ Falling Dominoes, Eli Maor :":?p:$Lt<+LJP b, It's no secret that trigonometry can By analop, a n\-o-dimensionaltoms G:,s.ps.i1t?>~:t4r.:4 And Other Adventures in Applied &$$$$ Mathematics be boring-but not in the hands of that fails to di\ide a three-dimensional Eli Maor! This isn't a formulaic look at $&$$$! Robert B. Banks one into nvo distinct pieces is shown sines, cosines, and tan~ents." 1t.s - a~ collection of stories, curiosities and as follo~t~s: Robert Banks insights, presents a wide illustrated range of mathemat- with over ical musings, both 100 fasci- practical and enter- nating taining, that have drawings. intrigued him and others: How tall can "Maor one grow?Why do clearly has we get stuck in traf- a great love fic? Can California of trig- water shortages be onometry alleviated by towing formulas icebergs from and all, and Antarctica? his enthusiasm shines through .... If you always Banks shows how math and simple wanted to know where trigonometry reasoning together may produce ele- came from, and what it's good for, Final Comments gant models that explain everything you'll find plenty here to enlighten from the federal debt to the proper you."-Ian Stewart, New Scientist Question 12 is an extension of ques- technique for ski-jumping. Cloth $24.95 ISBN 0-691 -05754-0 tion 1 to higher dimensional tori. More than 100 illustrations and tables. To what extent do the remaining Cloth $29.95 ISBN 0-691-05948-9 Due December questions generalize to hieher di- mensions? For example. can one turn a three-dimensional torus inside-ut An Imaginary Tale New in paperback within four-space (question 5 I' Is it The Story of fl e: The Story of a Number possible to "triangulate" a three- Paul J. Nahin Eli Maor dimensional torus with tetrahedra? What is the minimal number of tet- Paul Nahin tells the 2000-year-old "Maor has succeeded in writing a history of one of mathematics' most short, readable mathematical story. He rahedra required (question 10)' elusive numbers, the square root of has interspersed a variety of anec- Do other solid polygonal figures minus one, also known as i, re-creating dotes, excursions, and essays to light- with face identifications yield alter- the baffling mathematical problems en the flow.. . [The book] is like the that conjured it up and the colorful native universes (question 3, see [-I])? voyages of Columbus as told by the characters who tried to solve them. first mate."-Peter Borwein. Science .be there spheres that sit inside a 48 illurtrations. Paper $14.95 ISBN 0-691-05854-7 three-dimensional torus that fail to Cloth 824.95 ISBN 0-691-02795-1 divide that space into two distinct pieces (question 12)? Good luck with these and other generalizations. Have fun!

Math Horizons November 1998 31 Problem Section

Murray Klamkin University ofAlbej-to

All problems andlor solutions should be submitted in dupli- "S" designated problems are problems set particularly for cate in easily legible form (preferably printed) on separate secondary school students and/or undergraduates. Problems sheets containing the contributor's name (if one is a student with a dagger f indicate they are not original. "Quickie" prob- this should be so noted). mailing address, school affiliation, lems, which are not indicated as such, have their immediate and sent to the editor, Slathematics Department, University solutions at the end of the section and solutions should not be of Alberta, Edmonton. Alberta T6G 2G1, Canada. If an ac- submitted for them unless the solution is more elegant or is a kno~4edgementis desired an e-mail address or a stamped self- worthwhile new generalization. For problem submission infor- addressed postcal-d should be included (no stamp necessary for mation, see the Februaiy or April, 1996 issues. outside Canada and the US).

To be considered for publication, solutions to the following problems should be received by February 5, 1999.

S-20. Proposed by E. 11. Kaye, Vancouver, BC. An infi- 109. Proposed by K. S. Murray, Brooklyn, IVY. Deter- nite arithmetic progression of positive integers contains mine the maximum value of the terms 1, UP, bQ, and cr. Prove it also contains a term of the form kPQr (k > 1). where a, b, c are constants. S-21. Proposed by Mircea Ghita, Flushing, NY. Deter- mine all the solutions of the sim~~ltaneousequations: 110. Proposed by Peter Y. Woo, Biola University. Let Lo, L1, L2 be three given parallel lines, A a variable on x5 y5 = 211(x + y). sj - yj = 55(x - y). + Lo, and 0 a given point not on them. Let C1 be a circle through 0 and A and tangent to L1 and intersecting Lo S-22. Proposed by V5clav Iionetnf, Ferris State Univer- again at P.Let C2 be a circle through 0 and A tangent to sity and the Problem Editor. Determine all real a, b, c, Lz and intersecting Lo again at P' so that A lies between d, and e, such that the equation P and PI. Prove that there is some fixed line L3 parallel to Lo that is always tangent to the circle OPP'. is solvable in terms of radicals. 111. Proposed by the Problem Editor. Solve the difference equation 108. Proposed by Mansur Boase, student, Trinity Col- 16a,+l = 1 4a, d- where a1 = a. lege, Cambridge, UK. Let S be the set of all points in a + + plane whose rectangular coordinates are (a. b) where a is a prime number and b is a square number f 0 and let C 112. Proposed by K. M. Seymour, Toronto, ON. Eval- be the union of circular disks centered at the points of uate S with radii 0.32. Prove that there exists a line y = ma " lnxdx (m > 0) which does not intersect C.

32 Math Horizons November 1998 Solutions

$14. A Digit Problem Solzttion by E. M. Kaye, kincouver, BC. Let the equation of the circle be x2+y2 = 1, e be the x-axis and the vertices The number 1998 has exactly three kinds of digits. If eloL.where QX-= Q+2nk/n, k = 1,2,.. . , n. It now follows each of the positive integers x and y has exactly two that kinds of digits (in base lo), at most how many kinds of digits can x + y have?

Solution by Andy Liu, University of Alberta. The sum can have every kind of digit as given by

The example here uses the minimum of digits for x, y, .-ilso .$oliled bj D.Anderson, Mansur Boase, S. Smith, and and x+y combined to achieve the optimal result. Clearly, th~P~O/IOSPI.. x + y has at least 10 digits, and we may assume that it has exactly 10. If x also has 10 digits but y only 5, then the first 5 digits of x + y will not be distinct. If x has 9 Problem 98. Sorted Numbers digits while y has 6, then the first 4 digits of x + y much be 1000. .&I integer in base b is called "sorted" if its digits are The only other solution getting 9 digzts was a joint one bj nondecreasing from left to right, e.g., 36667 is sorted Ben Andrews and Brad Etters (students). in base 10. Characterize those bases b such that there are infinitely many base b numbers N such that N and .Y' are sorted. That b can be 10 is given by the num- S-16. Condition for an Identity bers 33.. .3366. . ,667 and this appears in Problem 1234 .\la!h. .\lag. 60( 1987)46. The equation 2(xn+ yn + 1)-n~y(x~-~ + y"-3 + 1) = 0 is satisfied identically when n = 5 for y = -x - 1. Are there Solufron b~the proposers. We first show that if the base any other positive integers n 3 for which the equation > b = a:'~l(z > I), then there are arbitrarily large sorted is also satisfied identically for y = -x - l?This problenl integers .\- and N2. One example is the n digit number was suggested by Problem 1538 Math. Mag., Dec. 1997, .Y consisting of (n- 1) m's followed by m + 1 where which was to find all integer solutions for the case n = 5. n1 = a:. For a proof, first note that a and m are valid digits in base b and m2 = a(b- 1). Also, q(bi - @)/(b- 1) Solution by I? Wagner, Chicago, IL. Let Sn = xn + yn zn (in base b) consists of (i - j) q's followed by j zeros. Since with S1 = 0 and let x, y, z be roots of the cubic t3 + bt - c = 0 where here b = yz + zx + n.y and c = xyz. Then multiplying the equation by t, t', t3. . . .and summing over the roots, we successively obtain it follo~vsthat S3 = 3c, S4 = 2b2, S5 = -5bc, Ss = 3c2 - 26", S; = 7h'c. Ss = 2b4 - 8bc2, S9 = 3c3 - 8bc2, . . . . The given equation can be made symmetrical by replacing the 1 by :. giv- ing 2Sn = ncSnPs.It now follows that n = 7 is a solution since 2S7 = 7cS4 = 14cb2. Apparently, there are no other and the first term of which consists of n a's folloa\led by solutions. n O's, the second term is n 0's followed by n a(2z - l)'s, the third term is merely 1. Finally, Also solved by D. Anderson, S. Smith, Z. Ilhrlp. and fhe proposer.

S-17. A Constant Sum and there is no "carry" from the last term. Editorial note. The proposers also have rather lengthy Let AlA2.. .An denote a regular n-gon inscribed in a proofsof the reverse assertion that if a base b has ar- circle C and !a given line. If lines through .-li and parallel bitrarily large N such that N and N2 are sorted then to E intersect C again in B;, i = 1.2.. . . . n, respectively, b = az2 + 1 and the extension that given a base number prove that the sum b with largest digit c, then c is of the form qpn where p, n > 1 if and only if there are infinitely many N such that N,N2,. . . , Nn are all sorted. It would be of interest is independent oft. to find more compact proofs. Math Horizons November 1998 33 Problem 99. On Numerical Integration Solution by Michael Woltermann, Washington and Jefferson College. The length of any chord through P which ter- A recent calculus text gives the impression that the er- minates on the xz and yz planes is at least as long as ror in approximating a definite integral by the midpoint its orthogonal projection onto the plane z = e. In this rule is about half that for the trapezoidal rule and of plane, the problem of finding the shortest chord through opposite sign. Given a number n of subintervals of [O,l], P is equivalent to solving the well-known "ladder in the find a continuously differentiable function f (x) for which corridor" problem found in many calculus books and is the midpoint and trapezoidal rule approximations to the given by (h2l3+ k2/3)3/2and this is not greater than the area under the curve y = f (x)on [O,l] are the same and minimum length chords which terminate in the other the error in the approximation is nonzero. two pairs of coordinate planes. Editorial note. For a non-calculus proof of the 2- Solution by Michael Woltermann, Washington and Jefferson dimensional problem we can take the equation of the line College. Let F(x) = x + 1 - 4 cos 4n7rx, x E [O,l]. Then through (h,k) as sla ylk = 1 where hla klb = 1. We the area under the curve = 312. For a regular partition + + of [O,1] into n subintervals, xk = kln, f(xk) = xr, for then want to minimize dm.By Holder's inequality 0,1,.. . , n. It is now immediate that both approximations give the area under the curve y = x and which is 112. Also solved by John Christqher, Nitin Goil (student), Hence, Patm'ck M. Lang, Patrick D. Shanahan, Z. Wang, and the proposer.

Problem 101. Nested Polygons $20. (Quickie) An A.P. with all Powers Starting with a unit circle, inscribe a square in it. In- scribe a second circle in the latter square, then a regular More generally, any arithmetic progression of the form octagon in the second circle, etc., each time inscribing A, = 1 + nd, n = 1: 2,. . . where d is a positive integer a circle in the last regular polygon and then inscribing contains every power an infinite number of times. To a regular polygon of twice the number of sides of the obtain terms of any power s, just let previous one in the previous circle. Determine the limit (md + 1)' - 1 n = of the resulting sequence of radii. d Then Solution by Mansur Boase (student), Trinity College, Cam- bridge, UK. It follows easily that the ratio of the inradius -4, = (md + 1)" to the circumradius of a regular n-gon is cos~ln.Thus the radius r of the limiting circle is Problem 111. (Quickie) An Easy Recurrence

Letting b, = Jm,the recurrence reduces to 2b,+1 = b, + 3. Hence b, = 3 + a/2,. Since Multiplying top and bottom by sin~/2~and using bl = a = 2bl - 6. sin x cos x = 2 sin 22 repeatedly, we get a, Finally, 1 2 r = lim -sin ($) = -. k-rn 2k-1 71-

Also solved by D. Anderson, Joshua Hughes (student),John Comment. This problem with a1 = 1 is due to Arthur Kacher (student), Gregory Miller (student), Adam C. Rhodes Engel who solved it by mathematical induction. More (student) and Dan Saracino, Iftim,ie Simion, The Abelian - generally, the recurrence an+^ = a + ban + JRwill Group, 2. Wang, and the proposer. reduce to a linear one as above by letting bn = d- provided d2 = 4b(ad + c - bc). Problem 102. Minimum Length Chord

Determine the minimum length chord in the first or- Solutions not previously acknowledged. thant of a rectangular coordinate system which passes S-11. David E. Manes; S-12.The Abelian Group (CPSU), through the point (h, k,e) where 0 < h 5 k 5 !and David E. Manes; Problem 95. David E. Manes, Randy K. terminating on two of the coordinate planes. Schwarz. 34 Math Horizons November 1998