A Review of The Art of Mathematics: Coffee Time in Memphis by Béla Bollobás. xvi + 359 pp. Cambridge University Press, 2006.

BY JAMES PROPP Here is one of my favorites: This proof exhibits economy Thanks For the Suppose 10 chairs are arranged (the chief idea is contained in Tangents he mathematician Tangents in a circle, half of them occupied the five-word hint), surprise Contributions! News from the Department of Mathematical Sciences Fall 2007 T and puzzle-connois- University of Massachusetts Lowell seur Peter Winkler by students. Show that there (who would think to bring ran- once joked, with a exists some whole number n domness and probability into Our thanks to all who have nod to Isaac Newton, between 1 and 9 such that if solving a problem like this?) contributed to the Department “If I have seen far- each of the 5 students moves n and fecundity (the probabilistic of Mathematical Sciences over ther than others, it is because chairs clockwise in the circle, or method has been an enormously the past few years. Your more of them will end up sitting powerful tool in the hands of I have stood on the shoulders generosity has allowed us in a previously occupied chair. Erdös and others). Bollobás’s 2007 Alumni-Awards Banquet A review of of Hungarians.” One of these to make purchases, award Hungarians is the late Paul This is not how Bollobás book is full of tasty little morsels scholarships, and engage in Next Banquet set for April 25, 2008 God Created Erdös, famous within mathemat- actually poses the problem in his like this one, puzzles whose ics for his contributions to num- book; in problem 3 (in a series solution requires attacking them activities that would otherwise the Integers of 157 problems), he asks the from some unexpected angle. have been impossible. he 2007 Alumni- _Stephen ber theory and combinatorics Awards Banquet at Giardini and famed more broadly for his reader to consider a more gener- The ability to come up with Many of you have T the Brewery by Stephen Hawking, 2005, unique lifestyle and lingo (chil- al situation. But the key idea that creative approaches to problems responded generously to • Shapiro Scholar 1162 pages, $29.95 Exchange on April dren are “epsilons,” God is the solves Bollobás’ problem can be can be cultivated, but it cannot UML phonathon and other (outstanding discovered by thinking about the be taught; it is more of an art 20 was well attend- math major):_ “Supreme Fascist”, God’s collec- fundraising contacts. These special case I’ve described—and than a craft. Hence the first half ed by the college Jennifer Cole BY GUNTRAM MUELLER tion of the best mathematical requests can benefit the proofs is “The Book”, and so by following the clue that of the book’s title. dean, current and emeritus * Outstanding o you mean that Department of Mathematical forth). Many of Erdös’ collabora- Bollobás helpfully provides in a I prefer Bollobás’s original faculty, adjuncts and stu- Graduate D Stephen Hawking? tors and successors are also separate section devoted to title, Coffee Time in Memphis Sciences directly if you specify dents. Some forty people Student:_Drew Yes, that Stephen Hungarian, and others have hints. (If you want to think (which his editor convinced him that you wish to have your in all came to wine, dine Lazar Hawking! What about this problem on your to lengthen); it bears more of the adopted what might be called gift directed to Mathematics. and congratulate our 2007 This year was business is it of a own, now would be a good time stamp of Bollobás’s personal Prof. Tom Kudzma and Stephen Giardini. “the Hungarian style,” with an Otherwise it will provide scholarship winners, each the first time the physicist to edit a to put aside this book review!) style. This is a book that emphasis on snappy problems valuable assistance to the of whom was given a book Shapiro Scholarship was and once again our team per- book of mathematics, select- and clever solutions. I can think Bollobás’s clue is a short ques- developed in the author’s mind prize and certificate in formed extremely well, earn- ing at his personal whim tion that at first seems like a non through the course of conversa- University at the College level. awarded and we are very of no better way to get acquaint- addition to any stipend. ing a Meritorious ranking what he considers to be the sequitur: “What about a random tions with students and col- grateful to Bernie and Yana ed with these people and their The awardees were Shapiro for their dedication which put them in the top brightest jewels of mathemat- work than to spend a few rotation?” leagues, sometimes in offices or to the department and gener- 15% of the 800 or so partici- ical thought through these months periodically dipping into If we choose n randomly, classrooms but just as often in • Arthur S. Zamankos then each student has a 4-out- departmental lounges or cafes. ous support of our students. pants worldwide. A copy of last 2500 years? The nerve! Béla Bollobás’s new collection of Scholarship (for outstand- of-9 chance of ending up in a Mathematicians love to find Thank you both. this year’s problems and other The gall! Where’s the griev- mathematical puzzles, titled The ing juniors and information can be found at ance committee? Well for Art of Mathematics: Coffee Time previously occupied chair. So on elegant solutions to their The Department of seniors):_Alan Bartlett, http://www.comap.com and one thing, he is the Lucasian in Memphis. average, the number of students research problems, but they can’t Mathematical Sciences Keith McElroy, Michael their final report is at Professor of Mathematics at Bollobás (the name is pro- sitting in previously occupied always be sure that the chal- entered a team in the Williams, Michelle Polys Cambridge, Newton’s old gig, nounced “bowl o’ bosh”) is one chairs will be 4/9 + 4/9 + 4/9 + lenges they set for themselves Mathematical Competition in http://faculty.uml.edu/ of the most ardent keepers of the 4/9 + 4/9, or 20/9, which is have sweet answers. It can be a • The Richardson Bedell Modeling hosted by the jgrahameagle/mcm2007.doc. and for another, he did just Erdös flame. Since Erdös died - slightly greater than 2. Now great relief from these uncertain- Scholarship (for outstand- Consortium for Mathematics A book prize was awarded to an absolutely excellent job of or, as Erdös would say, “left”—in comes the punch line: The only ties to work on a problem secure ing juniors and and its Applications. The the three team members it. Did his interest in physics 1996, Bollobás has organized a way the average value of an inte- in the knowledge that it has a seniors):_Ryan Beaven department has a strong tra- 2007 class valedictorian color his choice of pieces to conference in his honor every ger-valued quantity can exceed 2 pleasing solution, which the • The Mary Hall Prize (for dition in this competition Chris Evans, Jason Percival include? Well, the selection year at the University of is if it sometimes is 3 or greater. Continued on page 3 the outstanding freshman): stretching back over a decade and Michael Williams. is very strong in analysis, Memphis. (Full disclosure: I Congratulations to the team weak in algebra except for spoke at the 2006 conference.) members and their coach, Gauss, nonexistent in topolo- Bollobás, a professor who Tangents Non-Profit Org Kiwi. gy, but it includes some divides his time between Trinity Department of Mathematical Sciences U.S. Postage The Alumni-Awards 20th century foundations University of Massachusetts Lowell PAID College (Cambridge) and the Olney Hall Lowell, MA Banquet is an annual event and logic. University of Memphis, has UML North Permit No. 69 and we would be delighted Each of these pieces or worked for decades in functional Lowell, MA 01854 to see more people there. excerpts is preceded by an analysis, combinatorics and If you have not been interesting biographical graph theory. In the course of informed of this event in sketch and several pages of years of teaching and research, the past then please contact mathematical mis-en-scene, he has devised (or learned of) Prof Ken Levasseur explaining the context and many easily stated problems ([email protected]) the main results of that par- whose solutions possess one or to make sure you are on our ticular entry. Let’s face it: more of the hallmarks that mailing list. Continued on page 3 mathematicians prize, such as Drew Lazar and Prof. Charlie Byrne. economy, surprise and fecundity. Fall 2007 ➝ Tangents What Are You Up To? Steve’s Want to keep your classmates up to date on what you’re doing and where you are? Take a few moments to tell us Continued from back page Continued from page 1 ? Fall 2007—From the Chair where you are, and whatever else you might like to share. We’ll add it to the UML Math Alumni page, A review of God Created the Integers http://www.uml.edu/dept/math/alumni.htm. A Review of The Art of Mathematics: Editorial Note: Although we are happy that Steve Pennell has agreed to We can be contacted by mail at Department of Coffee Time in Memphis reading all these introduc- What holds the planets in Mathematical Sciences, North Campus, UMass Lowell, serve as chair, his name doesn’t lend it to as catchy a column title as problem-poser will, if pressed, Who are the right readers tions provides an excellent place? Does God do it? “Kiwi’s Korner.” We’ve wrestled with ideas for Steve’s but have decided to Lowell MA 01854. Telephone: (978) 934-2410. Email: [email protected] reveal. Bollobás has included in of this book? Mathematicians, contextualization of a lot Newton thought so. make it a contest. Winner gets a mathematics comic book. Send your entries of mathematics. This is Laplace, 100 years later, You might also wish to contact our Office of the book the sorts of problems certainly—especially younger to [email protected] and Steve will choose his favorite entry. Alumni Relations, Southwick Hall 250, UML North, with which he loves to tease ones who are still building up something that is sorely in his 5-volume Celestial Lowell, MA 01854-3629. Toll free telephone: his colleagues and students, their mental toolkits. lacking in the usual mode Mechanics, showed that iwi recently finished his second term as math department chair. (877) UML-ALUM. Email: [email protected] with pleasure on both sides. Corporate recruiters in Silicon of taking first this course, Newton’s three laws K Despite our pleas, promises of good behavior, and attempted bribes, Fans of books on recre- Valley will probably find some then that, with a strong implied exactly the he decided not to run for a third term. I was elected to succeed him. ational mathematics should be of these problems to be good emphasis on being able to observed phenomena! The ceremonial change of offices took place on July 2. The Tangents Problem warned that this one is not for ways of assessing the mental solve various kinds of Napoleon Bonaparte, Kiwi will be a tough act to follow. His low-key leadership style, the faint-hearted or the mathe- athleticism of potential hires; problems, but with little looking through Laplace’s his good judgment, his sense of humor, and his aversion to depart- S tart with threetart with line segments,three line segments,A, B, and C.A, ConnectB, matically unprepared. For in fact, I wouldn’t be surprised time available for seeing work, remarked that he ment meetings were greatly appreciated by all his colleagues during his six years an endS of A andto an C. end Connect of B and an theend other of A to an instance, the first sentence of to learn that some of these where these mathematical saw no mention of God. as department chair. Thank you, Kiwi. Good on ya. end of B to anend end of Bof and C. Usingthe other this endsystem of B to the solution of problem 84 puzzles are already being used issues come from, nor the Laplace replied: “Sire, reads, “This assertion is consid- in this way. On the other side Thank you also to undergraduate coordinator Shelley Rasmussen and graduate of line segmentsan end on of a C.plane, Using how this can system you of significance that these there is no need of that erably trickier than the usual of the interview desk, job seek- coordinator Charlie Byrne, who keep our degree programs running. They spend enclose maximumline segments area using on a the plane, system how and can results have in the big pic- hypothesis.” many hours advising our students and helping them navigate the sometimes rocky you enclose maximum area using run-of-the-mill limit questions ers might want to practice a given line, as shown in the figure? ture. This book provides For the record, here road to the completion of their degrees. the system and a given line, as shown in the based on subadditivity or sub- delivering the solutions to some of that larger picture. are the authors included: Congratulations to Dan Klain, who was promoted from figure? multiplicativity; here we have some of the more accessible What are some of the Euclid, Archimedes, to be a little more careful.” puzzles, adding some pauses Associate Professor to Professor effective Fall 2007. nuggets to look for? Firstly, Diophantus, Descartes, Bollobás assumes his reader is and brief false turns to make Three of our students gave presentations at the section on Archimedes, Newton, Laplace, Fourier, already acquainted with (and the whole thing sound unre- conferences this summer. Graduate student Vincent Durante perhaps a bit jaded by) large hearsed. Likewise, mathematics the account of the Gauss, Cauchy, Boole, presented his work on “Development of a 2D RBF image chunks of the advanced mathe- professors administering oral astounding devices that he Riemann, Weierstrass, reconstruction toolbox for sharp edged images” at the matics curriculum. If you do examinations to Ph.D. stu- built, his work on the esti- Dedekind, Cantor, International Congress on Industrial and Applied not come equipped with a dents, and Ph.D. students seek- mation of pi—look Ma, no Lebesgue, Gödel, and Mathematics in Zurich, Switzerland. June graduates knowledge of principles like ing to pass those examinations, algebraic notation! Also Turing. In its 1162 pages, Jason Percival and Michael Williams gave presentations continuity, compactness, con- might want to go through this significant is the fact that this book covers a lot of in San Jose, CA at the Pi Mu Epsilon National Conference, UpUp to to four four correct corr ectsolutions solutions from amongfrom among all that allare thatsubmitted he had an estimate of the territory, including some Steve Pennell tractibility and countability of book. In addition, students at part of the Mathematical Association of America’s summer byar eFebruary submitted 15, 2008by Februar will earny the15, solver 2008 a will“Math earn the rational numbers (to men- all levels (high school on up) radius of the earth, and pieces translated for the Mathfest. Jason spoke on “Calculating the area of a spherical triangle,” and Challenge”the solver T-shirt. a “Math You Challenge” may submit yourT-shirt. solution You to may tion just the ones that start may find in the notes a source talked about the theory of first time for this volume. Michael spoke on “Continued fractions with Gosper grids.” Congratulations to [email protected] your solution or to mail [email protected] it to Ken Levasseur, Department with “c”), you will not find all of attractive unsolved research the earth going around the I must say that reading of Mathematical Sciences, North Campus/Olney Vinnie, Jason, and Michael. or mail it to Ken Levasseur, Department of of these problems to be fair problems. sun, 1800 years before some of these pieces is not Hall, UMass Lowell, Lowell MA 01854. We sadly say goodbye to three faculty members. As mentioned in the last Mathematical Sciences, North Campus/Olney challenges; you might want to Erdös’s collaborator Paul Galileo got into that little easy, neither the ancient issue of Tangents, Guntram Mueller retired after the spring 2007 semester. Hall, UMass Lowell, Lowell MA 01854. read this book with a partner Turán once remarked that a dustup with the church. pieces nor the recent ones Mark Yin also retired after the spring 2007 semester, and Alina Stancu resigned and take turns being the one to mathematician is a machine for Secondly, I recommend —I keep bumping my to take a position at Concordia University in Montreal. Best wishes to all three. Solution to the Spring 2007 problem peek at the sections containing turning coffee into theorems. If the section on Laplace. head on Gödel’s work. But the hints and solutions. On the so, then many of the theorems We will miss them. We didn’t get a complete solution to the All of the known planets at the very least, even the other hand, some of the puz- in Bollobás’s book represent In July, Vinnie Durante and I joined over 3,000 other applied mathematicians Spring 2007 problem. revolved about the sun in introductions alone are at the Sixth International Congress on Industrial & Applied Mathematics in zles require nothing more than some extremely potent espres- the same direction and in excellent and well worth Suppose x and y are two integers with elementary arithmetic and the so. It would be a mistake to Zurich. I am happy to report that applied math is thriving. New theoretical 1 < x < y and x + y″ 100. Jill knows their almost the same plane. any student’s, or profes- developments combined with rapid increases in computational power have led to right perspective on the prob- gulp them down too quickly. The observed small pertur- sor’s, time and effort. sum x + y and Paul knows their product xy. lem. Some of my own favorites If enjoyed at a deliberate rate, exciting advances in many areas. To give just one example, Emmanuel Candès bations of their orbits The following conversation now takes place: in this latter vein are problems alone or in conversation, these from Cal Tech gave a talk titled “Compressive sampling.” According to his caused some concern. Do Rating: (out of 4) 7, 13, 20, 21, 24, 31, 34, 40, problems should have a stimu- ≡ ≡ ≡ ≡ abstract, “One of the central tenets of signal processing and data acquisition is Paul: I do not know the two numbers. 48, 55, 87, 102 and 119. lating effect on the prepared they build on each other? the Shannon–Nyquist sampling theory: the number of samples needed to capture Jill: I knew that already. Paul: Now I know the two numbers. Readers who find these more reader who takes the time to a signal is dictated by its bandwidth. Here we introduce a novel sampling or Steve Pennell, Chair Jill: Now I know them also. accessible problems fun would savor them. sensing theory which goes against this conventional wisdom. This theory, now probably enjoy Peter Winkler’s Writers: Ken Levasseur, Tangents is produced biannually known as Compressed Sensing or Compressive Sampling, allows the faithful What are the numbers? two collections of brain-teasers This book review appeared Raj Prasad, Ann Marie Hurley, by the Publications Office for the recovery of signals and images from… far fewer measurements or data bits than Dan Klain, Guntram Mueller, Department of Mathematical (“Mathematical Puzzles: A in The American Scientist in Marv Stick and Kiwi Graham-Eagle Sciences. Your comments are used by traditional methods.” It was an exciting talk, especially now when we are All is not lost, however. Ornella Bascunan, Connoisseur’s Collection” and Fall 2007. welcome. confronted with a flood of data from so many sources. Despite all the excitement, one of our alumni who teaches at Lowell High “Mathematical Mind-Benders”) rioting by math hooligans was kept to a minimum. School, gave the problem to her AP calculus as well as “Proofs from THE Please keep in touch, and stop in to visit the next time you are in the area. class and one of her students, Mark Trout, BOOK” by Martin Aigner and http://faculty.uml.edu/math Remember to check out our web site, http://faculty.uml.edu/math, for items successfully makes the case for x = 4 and Günter Ziegler. of interest. y = 13. All that’s missing was the logic to rule out other cases. Visit our web page for Mark’s submission. ➝ 2 ➝ Tangents Fall 2007 Fall 2007 Tangents ➝ 3