Front Cover 3/16/07 8:54 PM Page 1 Cover 2 3/18/07 8:15 PM Page 2 Team Minnesota Sculpts Winning Mathematical Art Stan Wagon Macalester College Stan Wagon Rhapsody in White, Second Place, 2000 Artists’ Choice, People’s Choice Robert Longhurst, designer Stan Wagon A Twist in Time Honorable Mention, 2002 Bathsheba Grossman, designer Stan Wagon Whirled White Web, Second Place, 2003 US Snow Sculpture of the Year Brent Collins and Carlo Séquin, designers Rich Seeley At the 2007 Breckenridge International Snow Sculpture Competition, the Minnesota team of David Chamberlain, Stan Wagon, Dan Schwalbe, Rich Seeley, and Beth Seeley won second place. A spokesperson for the team explained, “the most difficult artistic decision was whether to carve a rectangular box or a perfect cube with a base.” Team Minnesota has a long and successful history of carving beautiful surfaces (see images to the right). Carl Scofield stanwagon.com Stan Wagon teaches mathematics at Macalester College and his web page Cool Jazz, Second Place, 2007 has complete records of the team’s snow sculpting work. David Chamberlain, designer pp.03-04 3/25/07 10:35 PM Page 3 In this Issue Team Minnesota Sculpts Winning Mathematical Art 2 Stan Wagon Math Horizons is for undergraduates and How to sculpt the perfect ice cube. others who are interested in mathematics. Its purpose is to expand both the career and Pythagoras’s Darkest Hour 5 intellectual horizons of students. Colin Adams Truth, betrayal, and hemlock! ARTHUR T. BENJAMIN Harvey Mudd College Hide and Seek 7 JENNIFER J. QUINN Andy Martin Association for Women in Mathematics Drawing a circle around every rational number covers every irrational number, right? Editors When Lions Battle 8 CAROL BAXTER Nicholas Tasaday Managing Editor & Art Director New documents suggest a Newton-Leibniz collaboration. KARA L. KELLER A Dozen Questions About A Dozen Assistant Managing Editor 12 James Tanton DEENA R. BENJAMIN Twelve tricky tributes to the “Tantonian number” twelve. Assistant Editor Mathematical Enquirer 17 Math Horizons (ISSN 1072-4117) is pub- Gary Gordon, Dan Kalman, Liz McMahon, Roger B. Nelsen, and Bruce Reznick lished four times a year; September, Novem- MAA’s first mathematics tabloid. ber, February, and April by the Mathematical Association of America, 1529 Eighteenth If the IRS had discovered the quadratic formula… 21 Street, NW, Washington, DC 20036. April Daniel J. Velleman 2007 Volume XIV, Issue 4. Periodicals Solving a quadratic can be as easy as doing your taxes. postage paid at Washington, DC and addi- tional mailing offices. Annual subscription Book Reviews 22 rates are $29.00 for MAA members, $38.00 Jacob McMillen and Michael Flake for nonmembers, and $49.00 for libraries. Students review The Colossal Book of Short Puzzles and Problems and Mathematical Bulk subscriptions sent to a single address are Fallacies, Flaws, and Flimflam. encouraged. The minimum order is 20 copies ($190.00); additional subscriptions may be Whodunit? 24 ordered in units of 10. To order call (800) Ezra Brown 331-1622. For advertising information call Who proved l’Hôpital’s rule, and other tricky questions. (866) 821-1221. Printed in the United States of America. Copyright ©2007. The Mathe- Statistician’s Blues— 26 matical Association of America. Larry Lesser POSTMASTER: Send address changes to A mean song about statistical love and significant loss. Math Horizons, MAA Service Center, PO Box 91112, Washington, DC 20090-1112. Problem Section 32 Andy Liu THE MATHEMATICAL ASSOCIATION OF AMERICA New Mathematical Definitions and Contest 35 1529 Eighteenth Street, NW Ken Suman Washington, DC 20036 A contest to discover better mathematical definitions. Cover image: Illustration by Greg Nemec. All rights reserved. WWW.MAA.ORG/MATHHORIZONS 3 pp.03-04 3/25/07 10:35 PM Page 4 EDITORS Arthur T. Benjamin Jennifer J. Quinn Harvey Mudd College Association for Women in Mathematics From the Editors EDITORIAL BOARD Dear Readers, sarah-marie belcastro Dan Kalman Smith College American University Nobody reads the Letter from the Editors! So if Ezra Brown Frank Morgan you are reading this, then please stop reading it Virginia Tech Williams College right now. We warned you. Beth Chance Colm Mulcahy Cal Poly, San Luis Obispo Spelman College If this issue of Math Horizons looks a little funny Timothy P. Chartier Karen Saxe to you, it's because it was intended that way. Our Davidson College Macalester College theme is April Fools, and as a result, most of the Carl Cowen Francis Edward Su IUPUI Harvey Mudd College content is satirical, paradoxical, or simply Joe Gallian James Tanton humorous. The centerpiece of this issue is our University of Minnesota Duluth St. Mark’s Institute of first issue of the tabloid Mathematical Enquirer, Sarah J. Greenwald Mathematics where everything is bogus. Appalachian State University So don't believe everything you read—including STUDENT ADVISORY GROUP this letter. You've been warned! Sam Beck Lee Kennard Jackson Preparatory School Kenyon College Steven Byrnes James-Michael Leahy Jenny and Art Harvard University Columbia University Moshe Cohen Greg Leffert Louisiana State University Maggie Walker Governors School Megan Cornman Melissa Mauck Fredonia University Sam Houston State University Diana Davis Jacob McMillen Williams College Emory University Patrick Dixon Sarah Reardon Occidental College Davidson College Michael Flake Ché Lena Smith Davidson College University of North Carolina Victoria Frost Natalya St. Clair Spelman College Scripps College Instructions for Authors Math Horizons is intended primarily for undergraduates interested in mathematics. Thus, while we especially value and desire to publish high quality exposition of beautiful mathematics, we also wish to publish lively articles about the culture of mathematics. We interpret this quite broadly—we welcome stories of mathematical people, the history of an idea or circle of ideas, applications, fiction, folklore, traditions, insti- tutions, humor, puzzles, games, book reviews, student math club activities, and career opportunities and advice. Manuscripts may be submit- ted electronically to Editors Arthur Benjamin, [email protected], and Jennifer Quinn, [email protected]. If submitting by mail, please send two copies to Arthur Benjamin, Math Department, Harvey Mudd College, Claremont, CA, 91711. Subscription Inquiries e-mail: [email protected] Web: www.maa.org Fax: (301) 206-9789 Call: (800) 331-1622 or (301) 617-7800 Write: Math Horizons, MAA Service Center, P. O. Box 91112, Washington, DC 20090-1112 4 APRIL 2007 Adams 3/19/07 6:21 PM Page 5 “Hemlock bartender. Bring me a glass of hemlock.” Pythagoras’s Darkest Hour Colin Adams Williams College hat the Hades is the matter with you?,” Tri- “Oh, I don’t know. It’s just such a pretty equation, that’s all. angulus asked as he leaned over Pythagoras, You can always hope.” “Wwhose face was buried in the wine-soaked “Yes but you probably should have checked at least one sleeve of his toga, atop the stone bar. Pythagoras lifted his head example before announcing the theorem to the Assembly.” groggily. “Well, that’s apparent now. But give me a break. We’re “What do you want? Leave me alone.” living during the birth of mathematics here. Some of these “Oh please, Pythagoras, it doesn’t do you any good to things aren’t so obvious.” drown your sorrows. You are taking this whole thing much too “Well, maybe there’s some way to save it.” hard.” Pythagoras waved him away. “Yeah right.” “Hemlock, bartender. Bring me a glass of hemlock.” “What if you kept the equation but tried it on some other “Don’t be so melodramatic. It’s just one theorem.” object?” “Oh, right. Just one theorem. I have been humiliated in “What do you mean? Like a pyramid? The Egyptians are front of the entirety of Greek civilization. I have been made a smart. Don’t you think that if it applied to a pyramid, they fool. My name will go down in history. They will call it the would have noticed by now?” Pythagorean Folly. Hold it up to young children as an example “True, true, but what about something simpler?” of pure stupidity.” “Like what?” “You are exaggerating. In a week, no one will even “Oh, I don’t know. What about a square or a rectangle?” remember.” “Nah, Triangulus, that’s no good. There are only two side “Right! Like they don’t remember the Trojan Horse. Like lengths and all the angles are identical right angles. And the they forgot Oedipus and his tiny mistake. I think you have a area is just the product of the two side lengths. Nothing works slightly warped view of Greek forbearance. Some poet will there.” write an epic entitled the Pythagoriad, all about what a goat “Well, I guess you need something with three related brain I am. It’ll rise to the top ten and stay there forever.” quantities, maybe all lengths.” Pythagoras began to suck on his wine-soaked sleeve. “You mean like a comb, with three teeth? No, the teeth “Hey, stop that. Remember who you are. You’re the leader would all have the same length. How about a family, with a of the Pythagorean School of Mathematics.” mother and father and child? We could compare their heights.” “Yeah, right. The enrollments have been dropping like the “Seems an unlikely relation to hold, Pythagoras, at least for Athenians in the Persian Wars. In another two days, the most families. How about we stick to more geometric objects, Pythagorean School of Mathematics will have an enrollment like a triangle?” of one, and that’s because you’re my slave. By Zeus, I don’t “Oh, I see where you are going. Make x, y and z the angles have a hope in Hades of getting out of this.