100 Great Problems of Elementary Mathematics THEIR HISTORY AND SOLUTION BY HEINRICH DORRIE TRANSLATED BY DAVID ANTIN NEW YORK DOVER PUBLICATIONS, INC Copyright © 1965 by Dover Publications Inc.: originally published in German under the title of Triumph der Mathematik, © 1958 by PhysicaYerlag WUrzburg All rights reserved under Pan American and International Copyright Conventions Published in Canada by General Publishing Company Ltd., 30 Lesmill Road Don Mills Toronto, Ontario Published in the United Kingdom by Constable and Company, Ltd • 10 Orange Street, London WC This Dover edition first published in 1965 is a new translation of the unabridged text of the fifth edition of the work published by the Physica-Yerlag Wiirzburg, Germany in 1958 under the title Triumph der Mathematik: Hundert beruhmte Probleme aw %wei Jahrtawenden mathematischer Kultur This authorized translation is published by special arrangement with the German-language publishers Physica-Yerlag Wiirzburg Standard Book Number: 486-61348-8 Library of Congress Catalog Card Number:65-140JO Manufactured in the United States of America Dover Publications, Inc 180 Yarick Street New York N.Y 10014 Preface A book collecting the celebrated problems of elementary mathematics that would commemorate their origin and, above all, present their solutions briefly, clearly, and comprehensibly has long seemed a necessary and attractive task to the author The restriction to problems of elementary mathematics was considered advisable in view of those readers who have neither the time nor the opportunity to acquaint themselves in any detail with higher mathematics Nevertheless, in spite of this limitation a colorful and compelling picture has emerged, one that gives an idea of the amazing variety of mathematical methods and one that will-I hope-enchant many who are interested in mathematics and who take pleasure in characteristic mathematical thought processes In the present work there are to be found many pearls of mathematical art, problems the solutions of which represent, in the achievements ofa Gauss, an Euler, Steiner, and others, incredible triumphs of the mathematical mind Because the difficult economic situation at the present time barred the publication of a larger work, a limit had to be set to the scope and number of the problems treated Thus, I decided on a round number of one hundred problems Moreover, since many of the problems and solutions require considerable space despite the greatest concision, this had to be compensated for by the inclusion of a number of mathematical miniatures Possibly, however, it may be just these little problems, which are, in their way, true jewels of mathematical miniature work, that will find the readiest readers and win new admirers for the queen of the sciences As we have indicated already, a knowledge of higher analysis is not assumed Consequently, the Taylor expansion could not be used for the treatment of the important infinite series I hope nonetheless that the derivations we have given, particularly the striking derivation of the sine and cosine series, will please and will not be found unattractive even by mathematically sophisticated readers Preface On the other hand, in some of the problems, e.g., the Euler tetrahedron problem and the problem of skew lines, the author believed it necessary not to dispense with the simplest concepts of vector analysis The characteristic advantages of brevity and elegance of the vector method are so obvious, and the time and effort required for mastering it so slight, that the vectorial methods presented here will undoubtedly ~pur many readers on to look into this attractive area For the rest, only the theorems of elementary mathematics are assumed to be known, so that the reading of the book will not entail significant difficulties In this connection the inclusion of the little problems may in fact increase the acceptability of the book, in that it will perhaps lead the mathematically weaker readers, after completion of the simpler problems, to risk the more difficult ones as well So then, let the book go out and its part to awaken and spread the interest and pleasure in mathematical thought Wiesbaden, Fall, 1932 HEINRICH DORRIE Preface to the Second Edition The second edition of the book contains few changes An insufficiency in the proof of the Fermat-Gauss Impossibility Theorem has been eliminated, Problem 94 has been placed in historical perspective and the Problem of the Length of the Polar Night, which in relation to the other problems was of less significance, has been replaced by a problem of a higher level: "Andre's Derivation of the Secant and Tangent Series." Wiesbaden, Spring, 1940 HEINRICH DORRIE Contents ARITHMETICAL PROBLEMS Archimedes' Problema Bovinum The Weight Problem of Bachet de Meziriac Newton's Problem of the Fields and Cows Berwick's Problem of the Seven Sevens Kirkman's Schoolgirl Problem The Bernoulli·Euler Problem of the Misaddressed Letters Euler's Problem of Polygon Division Lucas' Problem of the Married Couples Omar Khayyam's Binomial Expansion 10 Cauchy's Mean Theorem 11 Bernoulli's Power Sum Problem 12 The Euler Number 13 Newton's Exponential Series 14 Nicolaus Mercator's Logarithmic Series 15 Newton's Sine and Cosine Series 16 Andre's Derivation of the Secant and Tangent Series 17 Gregory's Arc Tangent Series 18 Buffon's Needle Problem 19 The Fermat-Euler Prime Number Theorem 20 The Fermat Equation 21 The Fermat-Gauss Impossibility Theorem 22 The Quadratic Reciprocity Law 23 Gauss' Fundamental Theorem of Algebra 24 Sturm's Problem of the Number of Roots 25 Abel's Impossibility Theorem 26 The Hermite-Lindemann Transcendence Theorem page 11 14 19 21 27 34 37 40 44 48 56 59 64 69 73 78 86 96 104 108 112 116 128 lim Contents page PLANIMETRIC PROBLEMS 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 Euler's Straight Line The Feuerbach Circle Castillon's Problem Malfatti's Problem Monge's Problem The Tangency Problem of Apollonius Mascheroni's Compass Problem Steiner's Straight-edge Problem The Delian Cube-doubling Problem Trisection of an Angle The Regular Heptadecagon Archimedes' Determination of the Number 1r Fuss' Problem of the Chord-Tangent Quadrilateral Annex to a Survey Alhazen's Billiard Problem 141 142 144 147 151 154 160 165 170 172 177 184 188 193 197 PROBLEMS CONCERNING CONIC SECTIONS AND CYCLOIDS 42 43 44 45 46 47 48 49 50 51 52 53 54 An Ellipse from Conjugate Radii An Ellipse in a Parallelogram A Parabola from Four Tangents A Parabola from Four Points A Hyperbola from Four Points Van Schooten's Locus Problem Cardan's Spur Wheel Problem Newton's Ellipse Problem The Poncelet-Brianchon Hyperbola Problem A Parabola as Envelope The Astroid Steiner's Three-pointed Hypocycloid The Most Nearly Circular Ellipse Circumscribing a Quadrilateral 55 The Curvature of Conic Sections 56 Archimedes' Squaring of a Parabola 57 Squaring a Hyperbola 203 204 206 208 212 214 216 217 219 220 222 226 231 236 239 242 Contents IX page 58 Rectification of a Parabola 59 Desargues' Homology Theorem (Theorem of Homologous Triangles) 60 Steiner's Double Element Construction 61 Pascal's Hexagon Theorem " 62 Brianchon's Hexagram Theorem 63 Desargues' Involution Theorem 64 A Conic Section from Five Elements 65 A Conic Section and a Straight Line 66 A Conic Section and a Point 247 250 255 257 261 265 273 278 278 STEREOMETRIC PROBLEMS 67 68 69 70 71 72 73 74 75 76 Steiner's Division of Space by Planes Euler's Tetrahedron Problem The Shortest Distance Between Skew Lines The Sphere Circumscribing a Tetrahedron The Five Regular Solids The Square as an Image of a Quadrilateral The Pohlke-Schwarl Theorem Gauss' Fundamental Theorem ofAxonometry Hipparchus' Stereographic Projection The Mercator Projection 283 285 289 292 295 301 303 307 310 314 NAUTICAL AND ASTRONOMICAL PROBLEMS 77 The Problem of the Loxodrome 78 Determining the Position of a Ship at Sea 79 Gauss' Two-Altitude Problem 80 Gauss' Three-Altitude Problem 81 The Kepler Equation ~ S~&~ • 83 The Problem of the Sundial 84 The Shadow Curve " 85 Solar and Lunar Eclipses , 86 Sidereal and Synodic Revolution Periods , 87 Progressive and Retrograde Motion of the Planets 88 Lambert's Comet Problem 319 321 323 327 330 lli 336 340 342 346 349 352 Contents x page EXTREMES 89 90 91 92 93 94 95 96 97 98 99 100 Steiner's Problem Concerning the Euler Number 359 Fagnano's Altitude Base Point Problem " 359 Fermat's Problem for Torricelli 361 Tacking Under a Headwind 363 The Honeybee Cell (Problem by Reaumur) 366 Regiomontanus' Maximum Problem 369 The Maximum Brightness of Venus 371 A Comet Inside the Earth's Orbit 374 The Problem of the Shortest Twilight 375 Steiner's Ellipse Problem 378 Steiner's Circle Problem 381 Steiner's Sphere Problem 384 Index of Names 391 Arithmetical Problems ... spotted, and brown bulls and x, y, z, t to designate the white, black, spotted, and brown cows, we obtain the following seven equations for these eight unknowns: (1 ) (2 ) (3 ) (4 ) (5 ) (6 ) (7 ) X- T=... - T= loZ, Z - T = !~X, x = 172(Y + y), y = to(Z + z), Z = g(T+ t), t=g(X+x) From equations (I), (2 ), (3 ) we obtain 6X - 5Y = 6T, 20Y 9Z = 20T, 42Z - 13X = 42T, and taking these three equations... b"ee'(ab' - ba ') + e"b"(be'a' - b'ea) = e"a"bb'(e' - c) The solution is more easily seen when expressed in the form of determinants If q represents the reciprocal of Q, equations (1 ), (2 ), (3 ) assume