FOCUS August/September 2001 FOCUS is published by the Mathematical Association of America in FOCUS January, February, March, April, May/June, August/September, October, November, and December. August/September 2001 Editor: Fernando Gouvêa, Colby College; [email protected] Volume 21, Number 6 Managing Editor: Carol Baxter, MAA Inside [email protected] Senior Writer: Harry Waldman, MAA 4Orthogonal Latin Garage Doors [email protected] By Gary Gordon and Liz McMahon Please address advertising inquiries to: Doris Griffin; [email protected] 6 Duke University Featured in MAA Exhibit for Congress President: Ann E. Watkins, California State University, Northridge 7MAA Writing Prizes Announced at MathFest 2001 First Vice-President: Barbara L. Osofsky, Second Vice-President: Frank Morgan, 8Four Forums Will Address Mathematics at the Secretary: Martha J. Siegel, Associate Secretary: James J. Tattersall, Treasurer: School/College Transition Gerald J. Porter Executive Director: Tina H. Straley 10 Quantitative Literacy: Everybody’s Orphan By Bernard L. Madison Associate Executive Director and Director of Publications and Electronic Services: Donald J. Albers 12 Travel Grants for ICM 2002, Beijing, China FOCUS Editorial Board: Gerald Alexanderson; Donna Beers; J. Kevin 13 Clay Mathematics Institute Announces Long Term Fellows, Colligan; Ed Dubinsky; Bill Hawkins; Dan Olympiad Scholar, IMO Awards Kalman; Peter Renz; Annie Selden; Jon Scott; Ravi Vakil. 14 MAA National Election Results Letters to the editor should be addressed to Fernando Gouvêa, Colby College, Dept. of Mathematics, Waterville, ME 04901 or by 15 Short Takes email to [email protected]. Subscription and membership questions 17 From the Sections should be directed to the MAA Customer Service Center, 800-331-1622; e-mail: 18 Evaluating Undergraduate Programs: Indicators of [email protected]; (301) 617-7800 (outside U.S. and Canada); fax: (301) 206-9789. Departmental Health By John A. Dossey and Kenneth J. Travers Copyright © 2001 by the Mathematical Association of America (Incorporated). Educational institutions may reproduce 20 A Different Pencil: Use Technology Wisely articles for their own use, but not for sale, By Dick Jardine provided that the following citation is used: “Reprinted with permission of FOCUS, the newsletter of the Mathematical Association of America (Incorporated).” On the cover: Gary Gordon and Liz McMahon's garage doors are painted according to a Periodicals postage paid at Washington, DC mathematical scheme. See page 4 for the article. and additional mailing offices. Postmaster: Send address changes to FOCUS, Mathematical Association of America, P.O. FOCUS Deadlines Box 90973, Washington, DC 20090-0973. November December January ISSN: 0731-2040; Printed in the United States Editorial Copy September 14 October 14 November 15 of America. Display Ads September 24 October 29 November 26 Employment Ads September 24 October 29 November 26 2 August/September 2001 FOCUS USA Mathematical Olympiad Team Ties Russian Team for Second Place T he 2001 USA USA Mathemati- Mathematical Olym- cal Olympiad piad team, composed Team and repre- of six gifted young sent the United mathematicians, States at the IMO. faced off in the Inter- “We are very national Mathemati- proud of their cal Olympiad (IMO) achievement,” against the best high- said Titu An- school students from dreescu, head 82 counties and came coach of the US Team. “It is the re- in second to the team The US IMO team pictured here with Titu Andreescu, Director of the American Math- from China. sult of many years ematics Competitions. Left to right: Dong Shin, Gabriel Carroll, Reid Barton, Tiankai of hard work . All Liu, Oaz Nir and Ian Le. Photograph courtesy of the Clay Mathematics Institute. Mathematical Olym- the students and piad teams of the the coaching staff USA have done very well in International American Mathematics Competitions, an gave their best, and we can see the results. Mathematical Olympiad competitions official program of the Mathematical As- Congratulations to them all.” since first entering in 1974. In twenty- sociation of America presented by the seven years of competition, USA teams Akamai Foundation. The first exam is IMO participants were treated to several have won three times and finished in the taken by more than 350,000 students. The gala events, including the Opening Cer- top five twenty-two times. In 1994, all six USA Mathematical Olympiad, the last of emonies on July 4 presented by the members of the USA team achieved per- the exams, involves only 180 students. Akamai Foundation, and Closing Cer- fect scores, an unprecedented accom- emonies at the Kennedy Center and a lav- plishment by any other team. ish dinner at the National Building Mu- seum, both presented by the Clay Math- During the first two weeks of July at ematics Institute. George Mason University, 475 high- school students from 83 countries com- The Akamai Foundation was the present- peted in the International Mathematical ing sponsor of the IMO 2001 USA, with Olympiad. The students were allotted the National Science Foundation, U.S. nine hours to solve six difficult problems Department of Education, Wolfram Re- that are regarded as challenging by most search, Texas Instruments, and the Na- professional mathematicians. As evi- tional Security Agency all as title spon- dence of the difficulty of the IMO prob- sors. The Clay Mathematics Institute was lems, only four of the 475 students re- asked in May 2000 to participate as a part- ceived perfect scores, Reid Barton and ner by planning and executing the clos- Gabriel Carroll of the USA and Liang Andrew Wiles speaking at the closing ing ceremonies in conjunction with their Xiao and Zhiqiang Zhang of China. ceremonies of the International Math- annual meeting. The University of Ne- ematical Olympiad. Photograph cour- braska-Lincoln and George Mason Uni- Barton, a member of four USA Math- tesy of the Clay Mathematics Institute. versity were academic sponsors. ematical Olympiad teams, distinguished himself by becoming the first IMO con- For additional details on participants, testant to win four gold medals. In addi- The twelve students with the highest results, problems, photos, and history of tion to Barton and Carroll, Ian Le and scores are named winners of the USA the International Mathematical Olym- Tiankai Liu of the USA also won gold Mathematical Olympiad and they are piad, go to the following web sites: http:// medals. Oaz Nir and Dong Shin, the honored at ceremonies held in Washing- www.maa.org, http://imo.wolfram.com, other two members of the USA team, ton, DC hosted by the Mathematical As- http://www.claymath.org/events/ won silver medals. sociation of America, the National Acad- olympiadevent.htm, and http:// emy of Sciences, the National Science www.imo2001.org/. The USA team is selected through a se- Foundation, and the US Department of quence of three examinations of the State. The top six performers make up the 3 FOCUS August/September 2001 Orthogonal Latin Garage Doors Then we tried a more symmetric square, pictured on the left below. Trial and error By Gary Gordon and Liz McMahon produced a square orthogonal to it. (Al- though it is always possible to relabel the squares so that the first row in each square When we moved into our new house on the other to obtain ordered pairs, each is 1, 2, 3, 4, we liked the two squares given in the fall of 1986, the roof was gray, the one of the n2 possible ordered pairs ap- below better.) siding was gray and the two garage doors pears precisely once. (To see these or- were white. In addition, the weather in dered pairs clearly, superimpose one ar- Easton, Pennsylvania is frequently gray, ray on the other and then slide one of 1234 at least during the winter. The four of us the arrays a bit to the left.) (two mathematicians and our two daugh- 2143 ters) came to the same conclusion: the Latin squares are important in all sorts 3412 exterior of our house needed something of applications in experimental design, colorful to cheer up the neighborhood. geometry and graph theory, and they 4321 arise naturally as group multiplication The Project tables. The subject of Latin squares is more than 200 years old; amateurs at- The two wooden garage doors are con- tempting to design whist tournaments 1342 structed in horizontal sections, with four did much of the early development. Al- 4213 sections making up a door. Each section most any text on combinatorics treats this has four raised rectangular panels, so each topic; in fact, a pair of orthogonal 4 x 4 2431 door has 16 panels arranged in a 4 x 4 Latin squares decorates the cover of van 3124 grid. The doors need to be painted every Lint and Wilson’s interesting text A few years, so we soon hit on an obvious Course in Combinatorics. It is possible to solution to our gray lives: Paint the doors find a pair of n x n orthogonal Latin some color that wasn’t white. squares for every n except 2 and 6. Euler It is not hard to show that there can be at conjectured that no orthogonal Latin most three pairwise orthogonal 4 x 4 Latin squares, and, in general, at most n-1 We talked about the particulars of such a squares exist when n is congruent to 2 painting project and came up with a few mod 4, but in 1959, this conjecture was pairwise orthogonal n x n Latin squares. This maximum number is achieved constraints. First, do something math- disproved when a pair of 10 x 10 orthogo- ematical. Since both adults in the house nal Latin squares was found at the Uni- whenever there is a projective plane of are mathematicians and we frequently versity of North Carolina by Parker, Bose order n. All known projective planes have order n = pk for some prime p and some have students over for various activities, and Shrikhande.