Table Of Content

Index to Volume 69 AUTHORS Drucker, Daniel and Locke, Phil, A Natural Classification of Curves and Surfaces With Reflection Properties, 249-256 Andersen, K.M., A Characterization of Polynomials, 137-142 Englund, Jan-Eric, see Blom, Gunnar Ash, J. Marshall, The Limit of x7” Falkowski, Bernd-Jiirgen, On the Conver- as x gence of Hillam’s Iteration Scheme, 299- Tends to Zero, 207-209 302 Bailey, D.F., Counting Arrangements of 1’s Farris, Frank A., Wheels on Wheels on and —1’s, 128-131 Wheels—Surprising Symmetry, 185-189 Beauregard, Raymond A. and Surya- Fjelstad, Paul, Extending the Pythagorean narayan, E.R., Proof Without Words: Theorem to Other Geometries, 222--223 Parametric Representation of Primitive Francis, Richard L., Math Bite: Recitation Pythagorean Triples, 189 of Large Primes, 260 Bechhoefer, John, The Birth of Period 3, Franzblau, D.S., see Burgiel, H. Revisited, 115-118 Gadbois, Steve, Poker With Wild Cards—A Benjamin, Arthur T., see McCarthy, Clif- Paradox?, 283-285 ford A. Gass, Saul |., On Copying a Compact Beyer, W.A.; Louck, J.D.; and Zeilberger, Disk to Cassette Tape: An _ Integer- D., Math Bite: A Generalization cf a Cu- Programming Approach, 57-61 riosity that Feynman Remembered All His Life, 43-44 Goldfeather, Jack, Using Quadratic Forms to Correct Orientation Errors in Tracking, Blom, Gunnar; Englund, Jan-Eric; and 110-114 Sandell, Dennis, General Russian Roulette, Golomb, Solomon W., A Symmetry Crite- 293-297 rion for Conjugacy in Finite Groups, 373- Bloom, David M., Probabilities of Clumps in 375 a Binary Sequence (and How to Evaluate Gordon, William B., Period Three Trajecto- Them Without Knowing a Lot), 366-372 ries of the Logistic Map, 118-120 Bloy, Greg, Goldbach’s Problem in Matrix Gutschera, K.R., see Burgiel, H. Rings, 136-137 Harold, Sonia, see DeTemple, Duane Bradley, Michael J., Building Home Plate: Field of Dreams or Reality?, 44-45 Harper, James D., The Golden Ratio is Less Than x? /6, 266 Burgiel, H.; Franzblau, D.S.; and Gutschera, K.R., The Mystery of the Hesse, Robert, see Byrne, Philip J. Linked Triangles, 94-102 Hirschhorn, M.D., A Proof in the Spirit of Byrne, Philip J. and Hesse, Robert, A Zeilberger of an Amazing Identity of Ra- Markov Chain Analysis of the Game of Jai manujan, 267-269 Alai, 279-283 Hogan, Guy T., More on the Converse of Chapman, Robin, A Polynomial Taking In- Lagrange’s Theorem, 375-376 teger Values, 121 Hungerbiihler, Norbert, Proof of a Conjec- Chilaka, James O. Proof Without Words: ture of Lewis Carroll, 182-184 Decomposing the Combination (>). 127 Jovanovié, Milan V. and Jungi¢, Veselin M.., —_—, Proof Without Words: )~"_, k(k + Algebraic Set Operations, Multifunctions, 1)(k + 2) = n(n + 1)(n + 2)(n + 3)/4, 63 and Indefinite Integrals, 350-354 Jungi¢, Veselin M., see Jovanovic, Milan V. —__—,, Proof Without Words: An Alternat- ing Series, 355-356 Jungnickel, Dieter, On the Order of a Prod- uct in a Finite Abelian Group, 53-57 DeTemple, Duane and Harold, Sonia, A Round-Up of Square Problems, 15-27 Kaminsky, Kenneth, Professor Fogelfroe, 62, 142, 209, 303, 383 Devlin, Keith, Good-bye Descartes?, 344- 349 Klamkin, Murray S. and Kung, Sidney H., 399 400 MATHEMATICS MAGAZINE Ceva’s and Menelaus’ Theorems and Their Noy, Marc, A Short Solution of a Problem Converses via Centroids, 49-51 in Combinatorial Geometry, 52-53 Knudsen, Finn F. and Skau, Ivar, On the Papanicolaou, Vassilis G., On the Asymp- Asymptotic Solution of a Card-Matching totic Stability of a Class of Linear Differ- Problem, 190-197 ence Equations, 34—43 Pinkernell, Guido M., Identities on Poini- Kortram, R.A., Simple Proofs for > a sts Line Figures in the Euclidean Plane, 377- k= 382 x = x Ramnath, Sarnath and Scully, Daniel, Mov- 6 and snzx = “IT (1-i =), 122- ing Card i to Position) with Perfect Shuf- 125 fles, 361-365 Krause, Eugene F., Maximizing the Product Reiter, Ashley, see Reiter, Harold of Summands; Minimizing the Sum of Fac- Reiter, Harold and Reiter, Ashley, The tors, 270-278 Space of Closed Subsets of a Convergent Krusemeyer, Mark, A Parenthetical Note Sequence, 217-221 (to a Paper of Guy), 257-260 Robertson, John P., Magic Squares of Kung, Sidney H., Proof Without Words: Squares, 289-293 The Difference-Product Identities, 278 Sakmar, I.A., Chanson Sans Paroles, 62 , Proof Without Words: The Sum- Sandefur, James T., The Baseball-Card Col- Product Identities, 269 lector’s Query, 243-248 , see Klamkin, Murray S. Sandell, Dennis, see Blom, Gunnar Lazebnik, Felix, On Systems of Linear Dio- Schaer, J., Parallels on the Sphere, 298 phantine Equations, 261-266 Schaumberger, N., Power Mean for Zero Ledder, Glenn, A Simple Introduction to In- Exponent, 216 tegral Equations, 172-181 Scully, Daniel, see Ramnath, Sarnath Locke, Phil, see Drucker, Daniel Skau, Ivar, see Knudsen, Finn F. Lofaro, Thomas, see Newton, Tyre Spain, P.G., The Fermat Point of a Trian- Louck, J.D., see Beyer, W.A. gle, 131-133 McCarthy, Clifford A. and Benjamin, Stangl, Walter D. Counting Squares in Zp, Arthur T., Determinants of the Tourna- 285-289 ments, 133-135 Staring, Mike, The Pythagorean Proposi- Mackiw, George, Finite Groups of 2 x 2 In- tion: A Proof by Means of Calculus, 45-46 teger Matrices, 356-361 Stein, S.K., Inverse Problems for Central Meijer, A.R., Groups, Factoring, and Cryp- Forces, 83-93 tography, 103-109 Suryanarayan, E.R., see Beauregard, Ray- Meyers, Leroy F., Update on William Wer- mond A. nick’s “Triangle Constructions with Three Tan, Lin, The Group of Rational Points on Located Points,” 46-49 the Unit Circle, 163-171 Meyerson, Mark D., The x* Spindle, 198- Ungar, Abraham A., see Muldoon, Martin 206 , E. Milanfar, Peyman, A Persian Folk Method Wong, Roman W., On the Convergence of of Figuring Interest, 376 the Sequence of Powers of a 2 x 2 Matriz, Mulcahy, Colm, Plotting and Scheming with 210-215 Wavelets, 323-343 Yuefeng, Feng, Proof Without Words: Jor- Muldoon, Martin E. and Ungar, Abraham Ixy dan’s Inequality = < sine < x, 126 A., Beyond Sin and Cos, 3-14 T Zeilberger, D., see Beyer, W.A. Nelsen, Roger B., Proof Without Words: Bernoulli’s Inequality (Two Proofs), 197 , Proof without Words: Five Means— TITLES and Their Means, 64-65 Newton, Tyre and Lofaro, Thomas, On Us- Algebraic Set Operations, Multifunctions, ing Flows to Visualize Functions of a Com- and Indefinite Integrals, by Milan V. Jo- plex Variable, 28-34 VOL. 69, NO. 5, DECEMBE19R9 6 401 vanovié and Veselin M. Jungié, 350-354 Good-bye Descartes?, by Keith Devlin, 344- Asymptotic Solution of a Card-Matching 349 Problem, On the, by Finn F. Knudsen and Group of Rational Points on the Unit Circle, Ivar Skau, 190-197 The, by Lin Tan, 163-171 Asymptotic Stability of a Class of Linear Groups, Factoring, and Cryptography, by Difference, Equations, On the, by Vassilis A.R. Meijer, 103-109 G. Papanicolaou, 34—43 Identities on Point-Line Figures in the Eu- Baseball-Card Collector’s Query, The, by clidean Plane, by Guido M. Pinkernell, James T. Sandefur, 243-248 377-382 Beyond Sin and Cos, by Martin E. Muldoon Inverse Problems for Central Forces, by and Abraham A. Ungar, 3-14 S.K. Stein, 83-93 Birth of Period 3, Revisited, The, by John Limit of x” as x Tends to Zero, The, by Bechhoefer, 115-118 J. Marshall Ash, 207-209 Building Home Plate: Field of Dreams or Magic Squares of Squares, by John Reality?, by Michael J. Bradley, 44-45 P. Robertson, 289-293 Ceva’s and Menelaus’ Theorems and Their Markov Chain Analysis of the Game of Jai Converses via Centroids, by Murray Alai, A, by Philip J. Byrne and Robert S. Klamkin and Sidney H. Kung, 49-51 Hesse, 279-283 Chanson Sans Paroles, by [.A. Sakmar, 62 Math Bite: A Generalization of a Curiosity Characterization of Polynomials, A, by that Feynman Remembered All His Life, K.M. Andersen, 137-142 by W.A. Beyer, J.D. Louck, and D. Zeil- Convergence of Hillam’s Iteration Scheme, berger, 43-44 On the, by Bernd-Jtirgen Falkowski, 299- Math Bite: Recitation of Large Primes, by 302 Richard L. Francis, 260 Convergence of the Sequence of Powers of a Maximizing the Product of Summands; 2x2 Matrix, On the, by Roman W. Wong, Minimizing the Sum of Factors, by Eugene 210-215 F. Krause, 270-278 Converse of Lagrange’s Theorem, More on Moving Card i to Position j with Perfect the, by Guy T. Hogan, 375-376 Shuffles, by Sarnath Ramnath and Daniel Copying a Compact Disk to Cassette Tape: Scully, 361-365 An Integer-Programming Approach, On, Mystery of the Linked Triangles, The, by Saul I. Gass, 57-61 by H. Burgiel, D.S. Franzblau, and Counting Arrangements of 1’s and -1’s, by K.R. Gutschera, 94-102 D.F. Bailey, 128-131 Natural Classification of Curves and Sur- Counting Squares in Zn, by Walter faces With Reflection Properties, A, by D. Stangl, 285-289 Daniel Drucker and Phil Locke, 249-256 Determinants of the Tournaments, by Clif- Order of a Product in a Finite Abelian ford A. McCarthy and Arthur T. Ben- Group, On the, by Dieter Jungnickel, 53- jamin, 133-135 57 Extending the Pythagorean Theorem to Parallels on the Sphere, by J. Schaer, 298 Other Geometries, by Paul Fjelstad, 222- Parenthetical Note (to a Paper of Guy), A, 223 by Mark Krusemeyer, 257-260 Fermat Point of a Triangle, The, by Period Three Trajectories of the Logistic P.G. Spain, 131-133 Map, by William B. Gordon, 118-120 Finite Groups of 2 x 2 Integer Matrices, by Persian Folk Method of Figuring Interest, George Mackiw, 356-361 A, by Peyman Miianfar, 376 General Russian Roulette, by Gunnar Blom, Plotting and Scheming with Wavelets, by Jan-Eric Englund, and Dennis Sandell, Colm Mulcahy, 323-343 293-297 Poker With Wild Cards—A Paradox?, by Goldbach’s Problem in Matrix Rings, by Steve Gadbois, 283-285 Greg Bloy, 136-137 Polynomial Taking Integer Values, A, by Golden Ratio is Less Than x*/6, The, by Robin Chapman, 121 James D. Harper, 266 Power Mean for Zero Exponent, by 402 MATHEMATICS MAGAZINE N. Schaumberger, 216 Groups, A, by Solomon W. Golomb, 373- Probabilities of Clumps in a Binary Se- 375 quence (and How to Evaluate Them With- Systems of Linear Diophantine Equations, out Knowing a Lot), by David M. Bloom, On, by Feliz Lazebnik, 261-266 366-372 Update on Wiiliam Wernick’s “Trian- Professor Fogelfroe, by Kenneth Kaminsky, gle Constructions with Three Located 62, 142, 209, 303, 383 Points,” by Leroy F. Meyers, 46-49 Proof in the Spirit of Zeilberger of an Using Flows to Visualize Functions of a Amazing Identity of Ramanujan, A, by Complex Variable, On, by Tyre Newton M.D. Hirschhorn, 267-269 and Thomas Lofaro, 28-34 Proof of a Conjecture of Lewis Carroll, by Using Quadratic Forms to Correct Orien- Norbert Hungerbihler, 182-184 tation Errors in Tracking, by Jack Gold- Proof Without Words: }°;_, k(k + 1)(k + feather, 110-114 2) = n(n + 1)(n + 2)(n + 3)/4, by James Wheels on Wheels on Wheels—Surprising O. Chilaka, 63 Symmetry, by Frank A. Farris, 185-189 Proof Without Words: An Alternating Se- x” Spindle, The, by Mark D. Meyerson, ries, by James O. Chilaka, 355-356 198-206 Proof Without Words: Bernoulli’s Inequal- ity (two proofs), by Roger B. Nelsen, 197 PROBLEMS Proof Without Words: Decomposing the Combination ey) by James O. Chilaka, 127 The letters P, Q, and § refer to Pro- Proof Without Words: Five Means—and posals, Quickies, and Solutions, respec- Their Means, by Roger B. Nelsen, 64-65 tively; page numbers appear in paren- Proof Without Words: Jordan’s Inequality theses. For example, P1489(66) refers =2. <si. nz < x, by Feng Yuefeng, 126 to Proposal 1489, which appears on page 66. Proof Without Words: Parametric Rep- resentation of Primitive Pythagorean Triples, by Raymond A. Beauregard and February: P1489-1493; Q844-846; S1464- E.R. Suryanarayan, 189 1468 Proof Without Words: The Difference- April: P1494-1498; Q847-849; S1469-1473 Product Identities, by Sidney H. Kung, 278 June: P1499-1503; Q850-852; S$1474-1478 Proof Without Words: The Sum-Product October: P1504-1508; Q853-855; $1479- Identities, by Sidney H. Kung, 269 1483 Pythagorean Proposition, The: A Proof by December: P1509-1513; Q856-858; $1484- Means of Calculus, by Mike Staring, 45-46 1488 Round-Up of Square Problems, A, by Duane DeTemple and Sonia Harold, 15-27 Anchorage Math Solutions Group, Short Solution of a Problem in Combinato- S$1474(225) rial Geometry, A, by Marc Noy, 52-53 Andreoli, Michael H., $1476(227) Simple Introduction to Integral Equations, Bencze, Mihaly, P1497(143) A, by Glenn Ledder, 172-181 ____, Q852(225) x? ‘ Binz, J.C., P1498(144) Simpie Proofs for Bi= = —; and sinz = Byrd, Stan; Smith, Ronald L.; and Muchlis, k=l me 2 Ahmad, $1468(72) || (:= ia), by R.A. Kortram, 122- Callan, David, P1489(66) k=1 , P1509(384) 125 Space of Closed Subsets of a Convergent Se- Chao, Wu Wei, P1491(66) ____, P1499(224) quence, The, by Harold Reiter and Ashley ___, P1506(304) Reiter, 217-221 ____, Q857(385) Symmetry Criterion for Conjugacy in Finite VOL. 69, NO. 5, DECEMBER 1996 Chapman, Robin, $1466(70) Kotkowski, Bogdan, $1482(308) ___, $1472(149) Kutsenok, Victor, $1481(308) Christensen, Jens Peter Reus and Larsen, Kwong, Harris, $1470(146) Mogens Esrom, Q855(305) Larsen, Mogens Esrom, see Christensen, Con Amore Problem Group, see Plank, Jens Peter Reus Donald Larson, Loren C., P1513(385) Deutsch, Emeric, P1502(224) Lau, Kee-Wai, S1488(390) and Gessel, Ira M., P1494(143) Laugwitz, Detlef, P1510(384) Doster, David, $1477(227) Leong, Thomas, see Eggleton, Roger B. Doucette, Robert L., $1475(226) Lord, Nick, P1503(225) Dove, Kevin L., see Sumner, John S. Lossers, O.P., see Yang, Yongzhi Eggleton, Roger B. and Leong, Thomas, Mellendorf, Stephen, see Grossman, Jerrold $1465(69) W. Fischer, Ismor, Q847(144) Morris, Howard, P1507(304) Foster, L.L., $1484(386) Muchlis, Ahmad, see Byrd, Stan Freidkin, Evgenii S., P1490(66) Plank, Donald and Con Amore Problem Fukuta, Jiro, P1493(67) Group, $1478(228) Gessel, Ira M., see Deutsch, Emeric Rembis, F.C., $1480(307) Golomb, Michael, P1492(67) Rousseau, Cecil, C., see Klamkin, Murray Grossman, Jerrold W. and Mellendorf, Ss. Stephen, Q856(385) Sard, Eugene, Q854(305) Harminc, Matus and Sotak, Roman, Schaumberger, Norman, Q844(67) P1501(224) Shapiro, Daniel B., Q849(144) Holshouser, Arthur L. and Klein, Benjamin Sinefakopoulos, Achilleas, P1495(143) G., P1512(385) ____, $1464(68) Horrigan, Frank A., $1486(388) Smith, Ronald L., see Byrd, Stan Hoyt, John P., Q846(68) Sotak, Roman, see Harminc, Matus Just, Erwin, Q844(67) Stahl, Saul, P1500(224) ____, P1504(304) ____, P1508(305) Kandall, Geoffrey A. and Kappus, Hans, Sumner, John S. and Dove, Kevin L., $1469(144) $1485(386) Kappus, Hans, see Kandall, Geoffrey A. Taylor, Catherine, $1483(309) Karmakar, S.B. and Klamkin, Murray S., Wardlaw, William P., Q845(68) Q853(305) Wee, Hoe Teck and Woltermann, Michael, Klamkin, Murray S., P1496(143) $1487(389) _—__, Q851(225) White, Homer, Q848(144) ——, Q853(305) ____, Q850(225) ____., Q858(385) Woltermann, Michael, see Wee, Hoe Teck —_— and Rousseau, Cecil C., P1505(304) Yang, Yongzhi and Lossers, O.P., $1473(149) —_—., see Karmakar, S.B. Klein, Benjamin G. see Holshouser, Arthur Young, Anne L., $1471(148) L. Zhu, David, $1479(305) Koh, Sung Eun, $1467(/1)