Table Of Content

VOL. 71, NO. 5, DECEMBER 1998 401 Schaumberger, Norman, Hofstra Univer- Stroyan, Keith, University of Iowa, Iowa sity, Hempstead, NY City, IA Schwenk, Allen, Western Michigan Univer- Tan, Lin, West Chester University of Penn- sity, Kalamazoo, MI sylvania, West Chester, PA Sherman, Gary, Rose-Hulman Institute of Tanis, Elliot, Hope College, Holland, MI Technology, Terre Haute, IN Thompson, Richard, University of Arizona, Shreve, Warren, North Dakota State Uni- Tucson, AZ versity, Fargo, ND Vanden Eynden, Charles, Illinois State Smith, David, Duke University, Durham, University, Normal, IL NC Wagon, Stan, Macalester College, St. Paul, Stein, Sherman, University of California, MN Davis, CA Watkins, John, Colorado College, Colorado Steinbach, Peter, Albuquerque Technical- Springs, CO Vocational Institute, Albuquerque, NM Wetzel, John, University of Illinois, Ur- Stone, Alexander, University of New Mez- bana, IL ico, Albuquerque, NM Williams, Kenneth, Carleton University, Straffin, Jr., Philip, Beloit College, Beloit, Ottawa, Ontario, Canada WI Zeilberger, Doron, Temple University, Strogatz, Steven, Cornell University, Philadelphia, PA Ithaca, NY Index to Volume 71 AUTHORS Bivens, Irl; Simoson, Andrew, Beholding a Rotating Beacon, 83-104 Aczél, Janos; Alsina, Claudi, Trisection of Bloom, David M., A Quadratic Residues Angles, Classical Curves, and Functional Parlor Trick, 201-203 Equations, 182-189 Bridger, Mark; Frampton, John, Bounding Adams, Colin C., The Newest Inductee in Power Series Remainders, 204-207 the Number Hall of Fame, 341-349 Cater, F.S., Uniqueness of the Decomposi- Alsina, Claudi, see Aczél, Janos tion of Finite Abelian Groups: A Simple Anderson, Marlow; Feil, Todd, Turning Proof, 50-52 Lights Out with Linear Algebra, 300-303 Corless, Robert M., Variations on a Theme Armstrong, Gerald, see Lamoreaux, Jack of Newton, 34-41 Ayoub, Ayoub B., Triangles with the Same Dang, Mai, see Martelli, Mario Centroid, 221-224 Darst, Richard; Palagallo, Judith; Price, Bagby, Richard J., A Convergence of Lim- Thomas, Fractal Tilings in the Plane, 12- its, 270-277 23 Bailey, Herbert; DeTemple, Duane, Squares De Angelis, Valerio, see Beslin, Scott J. Inscribed in Angles and Triangles, 278-284 Deiermann, Paul, see Mabry, Rick Bajaj, Prem N., Math Bite: Equality of Dence, Thomas, see Swain, Gordon Limits in Ratio and Root Tests, 299 DeTemple, Duane, see Bailey, Herbert Baney, Douglas J., see Beslin, Scott J. Feigelstock, Shalom, Comparing Sets, 213- Beslin, Scott J.; Baney, Douglas J.; De 216 Angelis, Valerio, Small Denominators: No Feil, Todd, see Anderson, Marlow Small Problem, 132-138 402 MATHEMATICS MAGAZINE Ford, Ben; Franzmeier, Cory; Gayle, The Length of a Polygonal Arch, 377 Richard, Confusing Clocks, 190-195 Manvel, Bennet, see Krier, Nicholas Frampton, John, see Bridger, Mark Martelli, Mario; Dang, Mai; Seph, Tanya, Frantz, Marc, The Telescoping Series in Defining Chaos, 112-122 Perspective, 313-314 McWorter, Jr., William A.; Meyers, Leroy Franzmeier, Cory, see Ford, Ben F., Computing Eigenvalues and Eigenvec- Gau, Yihnan David, Proof Without Words: tors Without Determinants, 24-33 Double Angle Formulas, 385 Meyers, Leroy F., see McWorter, Jr., Gayle, Richard, see Ford, Ben William A. Giblin, Peter; Trout, Matthew, Cusps on Mezzino, Michael; Pinsky, Mark, Letbnitz’s Wheels on Wheels on Wheels, 309-313 Formula, Cauchy Majorants and Linear Differential Equations, 360-368 Gordon, Russell A., When is a Limit Func- tion Continuous ?, 306-308 Moran, D.A.; Stewart, B.M., The Random- ness of Remainders, 139-141 Grundman, H.G., Unity in a Field, 359 Myers, Nadine C., The Crossing Number of Hall, Leon M.; Roe, Robert P., An Unez- pected Mazimum in a Family of Rectangles, Cm x Cn: A Reluctant Induction, 350-359 Nanjundiah, T.S., Math Bite: On n'/" > 285-291 (n+1)'/*) n> 3, 138 Henle, Michael, Will the Real Non- Euclidean Parabola Please Stand Up?, Nelsen, Roger B., Proof Without Words: 369-376 Sums of Integers as Sums of Cubes, 65 Hirschhorn, Michael D., Two Diophantine Norwood, Rick, Math Bite: Why 2 + 2 Equations Studied by Ramanujan, 386-388 Equals 2 x 2, 60 Hungerbihler, Norbert, Proof Without Palagallo, Judith, see Darst, Richard Words: Bijection Between Certain Lattice Pingya, Xu, The Sufficient Condition for Paths, 219 the Differentiability of Functions of Several Kahan, Steven, Perfectly Odd Cubes, 131 Variables, 297-299 Kalman, Dan, A Lambda Slaughter, 122 Pinsky, Mark, see Mezzino, Michael Kedlaya, Kiran S., When is (xy + 1)(yz + Pinter, Klara, Proof Without Words: The 1)(zx +1) a Square?, 61-63 Area of a Right Triangle, 314 Klee, Victor; Reay, John R., A Surprising Pratt, Vaughan, Math Bite: The Volume of but Easily Proved Geometric Decomposi- a Cone, Without Calculus or Square Roots, tion Theorem, 3-11 376 Kleiner, Israel, A Historically Focused Price, Thomas, see Darst, Richard Course in Abstract Algebra, 105-111 Reay, John R., see Klee, Victor Kreminski, Rick, DifferentiatAimnongg In- Ribenboim, Paulo, Galimatias Arithmeti- finite Series, 42-50 cae, 331-340 Krier, Nicholas; Manvel, Bennet, Counting Roe, Robert P., see Hall, Leon M. Integer Triangles, 291-295 Saari, Donald G.; Valognes, Fabrice, Geom- Lamoreaux, Jack; Armstrong, Gerald, The etry, Voting, and Paradozes, 243-259 Fundamental Theorem of Calculus for Sagan, Hans, Math Bite: On the Nowhere Gauge Integrals, 208-212 Differentiability of the Coordinate Func- Lenard, Andrew, What Can Be Learned tions of the Iséke Curve, 296 From n < n!?, 58-60 Sanchis, Gabriela R., Swapping Hats: A Mabry, Rick; Deiermann, Paul, Proof With- Generalization of Montmort’s Problem, out Words: Adding Like Sines, 130 53-57 Machover, Maurice, The Square Root of Schwenk, Allen J., The Steady State Sab- Two is Irrational: Proof by Poem, 131 batical Rate, 195-200 Mallinson, Philip R., Proof Without Words: Segercrantz, Jerry, Inclusion-Ezclusion and Area Under a Polygonal Arch, 141 Characteristic Functions, 216-218 Mallinson, Philip, Proof Without Words: Seph, Tanya, see Martelli, Mario VOL. 71, NO. 5, DECEMBER 1998 403 Simoson, Andrew, see Bivens, Irl Cusps on Wheels on Wheels on Wheels, by Sondow, Jonathan, An Antisymmetric For- Peter Giblin and Matthew Trout, 309-313 mula for Euler’s Constant, 219-220 Defining Chaos, by Mario Martelli, Mai Sprows, David, Digitally Determined Peri- Dang, and Tanya Seph, 112-122 odic Points, 304-305 Differentiating Among Infinite Series, by Stewart, B.M., see Moran, D.A. Rick Kreminski, 42-50 Straffin, Jr., Philip D., Liu Hui and the Digitally Determined Periodic Points, by First Golden Age of Chinese Mathematics, David Sprows, 304-305 163-181 Fractal Tilings in the Plane, by Richard Swain, Gordon; Dence, Thomas, Darst, Judith Palagallo, and Thomas Archimedes’ Quadrature of the Parabola Price, 12-23 Revisited, 123-130 Fundamental Theorem of Calculus for Tan, Lin, Proof Without Words: Eisen- Gauge Integrals, The, by Jack Lamoreaux stein’s Duplication Formula 2csc(20) = and Gerald Armstrong, 208-212 tan 6 + cot 6, 207 Galimatias Arithmeticae, by Paulo Riben- boim, 331-340 Thompson, Richard B., Global Position- ing System: The Mathematics of GPS Re- Geometry, Voting, and Paradozes, by Don- cetvers, 260-269 ald G. Saari and Fabrice Valognes, 243-259 Trout, Matthew, see Giblin, Peter Global Positioning System: The Mathe- Valognes, Fabrice, see Saari, Donald G. matics of GPS Receivers, by Richard B. Thompson, 260-269 Wise, David S., Proof Without Words: A Generalization from Pythagoras, 64 Historically Focused Course in Abstract Al- gebra, A, by Israel Kleiner, 105-111 Zulli, Louis, The Incredibly Knotty Checkerboard Challenge, 378-385 Inclusion-Ezclusion and Characteristic Functions, by Jerry Segercrantz, 216-218 TITLES Incredibly Knotty Checkerboard Challenge, The, by Louis Zulli, 378-385 Antisymmetric Formula for Euler’s Con- Lambda Slaughter, A, by Dan Kalman, 122 stant, An, by Jonathan Sondow, 219-220 Leibnitz’s Formula, Cauchy Majorants and Archimedes’ Quadrature of the Parabola Linear Differential Equations, by Michael Revisited, by Gordon Swain and Thomas Mezzino and Mark Pinsky, 360-368 Dence, 123-130 Liu Hui and the First Golden Age of Chi- Beholding a Rotating Beacon, by Irl Bivens nese Mathematics, by Philip D. Straffin, Jr., 163-181 and Andrew Simoson, 83-104 Bounding Power Series Remainders, by Math Bite: Equality of Limits in Ratio and Mark Bridger and John Frampton, 204- Root Tests, by Prem N. Bajaj, 299 207 Math Bite: Onn'!/” > (n41)'/(+) n> 3, Comparing Sets, by Shalom Feigelstock, by T.S. Nanjundiah, 138 213-216 Math Bite: On the Nowhere Differentiabil- Computing Eigenvalues and Eigenvectors ity of the Coordinate Functions of the Iséke Without Determinants, by William A. Mc- Curve, by Hans Sagan, 296 Worter, Jr. and Leroy F. Meyers, 24-33 Math Bite: The Volume of a Cone, With- Confusing Clocks, by Ben Ford, Cory out Calculus or Square Roots, by Vaughan Franzmeier, and Richard Gayle, 190-195 Pratt, 376 Convergence of Limits, A, by Richard J. Math Bite: Why 2+2 Equals 2 x 2, by Rick Bagby, 270-277 Norwood, 60 Counting Integer Triangles, by Nicholas Newest Inductee in the Number Hall of Krier and Bennet Manvel, 291-295 Fame, The, by Colin C. Adams, 341-349 Crossing Number of Cm x Cn: A Reluctant Perfectly Odd Cubes, by Steven Kahan, 131 Induction, The, by Nadine C. Myers, 350- Proof Without Words: A Generalization 359 from Pythagoras, by David S. Wise, 64 404 MATHEMATICS MAGAZINE Proof Without Words: Adding Like Sines, manujan, by Michael D. Hirschhorn, 386- by Rick Mabry and Paul Deiermann, 130 388 Proof Without Words: Area Under a Polyg- Unexpected Mazimum in a Family of Rect- onal Arch, by Philip R. Mallinson, 141 angles, An, by Leon M. Hall and Robert P. Proof Without Words: Bijection Be- Roe, 285-291 tween Certain Lattice Paths, by Norbert Uniqueness of the Decomposition of Finite Hungerbuhler, 219 Abelian Groups: A Simple Proof, by F.S. Proof Without Words: Double Angle For- Cater, 50-52 mulas, by Yihnan David Gau, 385 Unity in a Field, by H.G. Grundman, 359 Proof Without Words: Eisenstein’s Dupli- Variations on a Theme of Newton, by cation Formula 2csc(260) = tan@ + cot, Robert M. Corless, 34-41 by Lin Tan, 207 What Can Be Learned From n < n!?, by Proof Without Words: Sums of Integers as Andrew Lenard, 58-60 Sums of Cubes, by Roger B. Nelsen, 65 When is (xy+1)(yz+1)(zx+1) a Square?, Proof Without Words: The Area of a Right by Kiran S. Kedlaya, 61-63 Triangle, by Klara Pinter, 314 When is a Limit Function Continuous ?, by Proof Without Words: The Length of a Russell A. Gordon, 306-308 Polygonal Arch, by Philip Mallinson, 377 Will the Real Non-Euclidean Parabola Quadratic Residues Parlor Trick, A, by Please Stand Up?, by Michael Henle, 369- David M. Bloom, 201-203 376 Randomness of Remainders, The, by D.A. Moran and B.M. Stewart, 139-141 PROBLEMS Small Denominators: No Small Problem, by Scott J. Beslin, Douglas J. Baney, and The letters P, Q, and § refer to Pro- Valerio De Angelis, 132-138 posals, Quickies, and Solutions, respec- Square Root of Two is Irrational: Proof by tively; page numbers appear in paren- Poem, The, by Maurice Machover, 131 theses. For example, P1539(66) refers Squares Inscribed in Angles and Triangles, to Proposal 1539, which appears on by Herbert Bailey and Duane DeTemple, page 66. 278-284 February: P1539-1543; Q874-876; S1514- Steady State Sabbatical Rate, The, by Allen 1518 J. Schwenk, 195-200 April: P1544-1548; Q877-879; S1519-1523 Sufficient Condition for the Differentiabil- June: P1549-1553; Q880-882; $1524-1528 ity of Functions of Several Variables, The, by Xu Pingya, 297-299 October: P1554-1558; Q883-884; $1529- 1533 Surprising but Easily Proved Geometric De- composition Theorem, A, by Victor Klee December: P1559-1563; Q885-886; S$1534- and John R. Reay, 3-11 1538 Swapping Hats: A Generalization of Mont- 1996 MathCamp Students of U. of Wash- mort’s Problem, by Gabriela R. Sanchis, ington, P1544(142) 53-57 Abad, Vic, $1526(228) Telescoping Series in Perspective, The, by Akhlaghi, Reza and Sami, Fary, $1515(69) Marc Frantz, 313-314 Alkan, Emre, P1561(389) Triangles with the Same Centroid, by Ay- oub B. Ayoub, 221-224 Bailey, Robert, see White, Homer Trisection of Angles, Classical Curves, and Baker, John and Lawrence, John, Functional Equations, by Janos Aczél and $1521(146) Claudi Alsina, 182-189 Baker, Matt, Q874(67) Turning Lights Out with Linear Algebra, by Beasley, Brian D.; Jones, Lenny; Blount, Marlow Anderson and Todd Feil, 300-303 Karen; Reigle, Dennis; and Stockslager, Beth, $1538(395) Two Diophantine Equations Studied by Ra- VOL. 71, NO. 5, DECEMBER 1998 Beck, Matthias; Tefera, Akalu; and Zeleke, Michael, $1530(317) Melkamu, $1534(390) Ovchinnikov, Sergei, Q881(226) Bencze, Mihaly, P1550(225); P1555(315); Pachter, Lior, $1514(67) Q876(67) Peng, Gao, $1531(319) Beslin, Scott, see Wickner, John Reigle, Dennis, see Beasley, Brian D. Binz, J.C., $1518(72) Reigle, Dennis; Stockslager, Beth; and Blount, Karen, see Beasley, Brian D. Blount, Karen, $1529(316) Blount, Karen, see Reigle, Dennis Rey, Joaquin Gémez, P1559(389) Boase, Mansur, P1558(316) Richardson, Ken, P1548(143) Bracken, Paul, see Donini, Daniele Rosenholtz, Ira, Q880(226) Chao, Wu Wei, P1541(66); P1552(225); Sami, Fary, see Akhlaghi, Reza P1560(389); P1563(390) Sard, Eugene W., Q877(143) de Angelis, Valerio, see Wickner, John Sastry, K.R.S., P1549(225) Deutsch, Emeric, Q884(316) Seiffert, Heinz-Jiirgen, $1528(230) Donini, Daniele and Bracken, Paul, S$1523(149) Shapiro, Lou, see Horrigan, Frank A. Sinefakopoulos, Achilleas, see Kappus, Galperin, Gregory and Gauchman, Hillel, P1556(315) Hans Singer, Nicholas C. and Western Maryland Gauchman, Hillel, see Galperin, Gregory College Problems Group, $1527(228) Golomb, Michael, P1540(66); P1543(67) Stockslager, Beth, see Beasley, Brian D. Grossman, Jerrold W. and Turett, Barry, Stockslager, Beth, see Reigle, Dennis P1542(67) Straffin, Jr., Philip D., $1537(394) Holshouser, Arthur L., see Klein, Benjamin Tefera, Akalu, see Beck, Matthias G. Tian, Zheng-Ping, Q886(390) Horrigan, Frank A. and Shapiro, Lou, $1536(392) Turett, Barry, see Grossman, Jerrold W. Jager, Thomas, $1522(148) Vanden Eynden, Charles, Q878(143) Jones, Lenny, see Beasley, Brian D. Vowe, Michael, see Noltie, Stephen Wardlaw, William P., Q882(226) Just, Erwin, P1545(142) Wee, Hoe Teck, Q875(67) Kappus, Hans and Sinefakopoulos, Achil- leas, $1532(320) Western Maryland College Problems Group, $1533(321) Klamkin, Murray S., Q883(316) Klein, Benjamin G. and Holshouser, Arthur Western Maryland College Problems L., P1546(142) Group, see Singer, Nicholas C. Knuth, Donald, P1539(66) White, Homer and Bailey, Robert, P1547(143) Lakshmanan, Neela, see Young, Robert L. Wickner, John; Beslin, Scott; and de Ange- Lawrence, John, see Baker, John lis, Valerio, P1562(390) Lord, Nick, see Noltie, Stephen Wong, Yan-loi, $1535(391) Mallahi, Keivan, Q885(390) Woo, Peter Y., P1557(316) Miller, G.R., $1517(71) Morris, Howard Cary, P1554(315); Yiu, Paul and Zhu, David, $1520(145) P1551(225); $1519(144) Young, Robert L. and Lakshmanan, Neela, S$1524(226) Mycielski, Jan, Q879(143) Zeleke, Melkamu, see Beck, Matthias Newcomb, William A., $1516(70) Zhu, David, see Yiu, Paul Nieto, José H., $1525(227) Zorn, Paul, P1553(226) Noltie, Stephen; Lord, Nick; and Vowe,