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Mathematical Gazette Index to Volume 92 Numbers 523 to 525 M Cz 2008 www.m-a.org.uk THE MATHEMATICAL ASSOCIATION £1 Index to the Mathematical Gazette e Vol. 92 2008 NO. MONTH PAGES 523 March i= 524 July 193 — 384 525 November 385 — $92 Edrtorials Articles Notes Feedback Teaching Notes Problem Corner . Student Problems Correspondence Reviews - Editors Matter for Debate Articles AUTHOR rITLt ; PAGI - Tom M. Apostol and The method of sweeping tangents 396 Mamikon A. Mnatsakanian Adam Bailey A geometric interpretation of equal sums of cubes 8 C. J. Bradley The Fermat point configuration 214 Rob Eastaway Joined up mathematics (Presidential Address 2008) 386 Michael Fox, Adrian Oldknow, The Soddy spheres of a 4-ball tetrahedron: Part | 205 John Rigby, Christopher Zeeman Michael Fox, Adrian Oldknow, The Soddy spheres of a 4-ball tetrahedron: Part 2 418 John Rigby, Christopher Zeeman H. Brian Griffiths The Divine Proportion, matrices and Fibonacci numbers 14 Martin Griffiths Hypercubes and the normal distribution 223 Jonathan Groves Nice polynomials with three roots l Jonathan Groves Nice symmetric and antisymmetric polynomials 437 Clark Kimberling Twenty-one points on the nine-point circle 29 Barry Lewis More power to Pascal 454 John Mahony The cone-paraboloid intercept 235 Maria Flavia Mammana Quadrilaterals of triangle centres 466 and Biagio Micale ll THE MATHEMATICAL GAZETTE AUTHOR TITLE PAGE Alex Pathan Euler's and Barker's-equations: A geometric derivation ‘ of the time of flight along parabolic trajectories 39 Sandra Pulver Quaternions: The hypercomplex number system 43] Simone Rinaldi and Indecomposability: polyominoes and polyomino tilings 193 D. G. Rogers D. G. Rogers Jacob Bronowski (1908-1974) 476 Norman Routledge Computing Farey Series 55 Norman Routledge Summing Euler's ¢-function 242 Bogdan Suceava, Adrian Vajiac, The power of a point for some real algebraic curves 22 Mihaela Vayiac Robert M. Young \ serendipitous path to a famous inequality 50 Correspondence Michael de Villiers 167 Nick Lord 167 Andrew Jobbings 348 Matter for Debate AUTHOR TITEE PAGI Christopher Ormell Can we understand uncountability? 252 Feedback ——__ —< Been = | March Michael Fox ae ieee . ipp. 165-16 july ~ Mike Hirschhorn Terence Jackson Andrew Jobbings © |pp. 349-355 Nick Lord J. A. Scott Stuart Simons | November Nigel Backhouse ——- Mike Hirschhorn John Mahony ; |—pp. 57(0-572 a = Miscellaneous Thirteenth Annual Mathematical Gazette Writing Awards 222 Acknowledgements 591 INDEX ili Notes March 92.01 — 92.29 July 92.30 — 92.59 November 92.60 — 92.90 AUTHOR NO TITLE Jack Astin 92.39 A discriminant that forms a geometric sequence Michel Bataille 92.75 A curious way to the weighted AM-GM inequality Roger Bilisoly 92 Anasquares: Square anagrams of squares C. J. Bradley 92.5 Generalisation of the Droz-Farny lines P. S. Bullen and 92.11 A remarkable formula R. Vyborny David Cariolaro and 9? - The edge-choosability of the tetrahedron waa = Ww Ko-Wei Lih Stephen Casey 9? The converse of the theorem of Pythagoras 309 Robin Chapman Evaluating Y (a + ndy again 9? Robert J. Clarke Some rules of the triangle 319 Stefano Costa Geometry of the sums of consecutive integer powers 516 P. Dale Application af inversion to touching hyperspheres 128 . Prithwijit De A pair of insoluble Diophantine equations arising 261 from an equilateral triangle Thomas P. Dence 92. Sums of simple rearrangements of the alternating harmonic series Russell Euler and 92. A cubic Diophantine equation re-examined Jawad Sadek Harley Flanders 92.36 A singular integral Colin Foster 92. Avoiding Pythagoras Paul Gailiunas 92. Orthologic triangles and Miquel's theorem Martin Griffiths 92.: Ellipses, ovals and the area of the M25 Martin Griffiths 92. Affine transformations of the graphs of polynomials Martin Griffiths 92. Learning Russian Martin Griffiths 92.8 Partitioning rugby scores Nuran Guzel and 92.7 Proof without words: the sine of a sum Hasan Unal Nguyen Minh Ha and 92.26 An elementary proooff t he generalised Fermat problem Nikolaos Dergiades Michael D. Hirschhorn 92.29 Three scenarios involving elastic collisions Michael D. Hirschhorn 92. oo Pythagoras' Theorem THE MATHEMATIC, AL GAZETTE AUTHOR NO TITLE PAGI Larry Hoehn 92. 49 A generalisation of Pythagoras’ theorem 316 Steve Humble 92. 40 Rendezvous constants Iyoung Michelle Jung 9? 69 A proofo f ¥ 25 * tan(2“x) = cotx — =2 ”| cot(2”x) A. D. Keedwell 92. ~ Euclid's algorithm and the money changing problem Jeremy D. King 9?2 .28 Angles in croquet Adrian R. Kitson 9)2 .42 The prime analogue of the Kepler-Bouwkamp constant Thomas Koshy 92. Pythagorean triples with Pell generators Philip W. Kuchel 9? 10n° + 2 revealed Awani Kumar 9? Magic knight's tours for chess in three dimensions Ron Larham 9? A curved railway platform canopy 278 Teik-Cheng Lim [wo infinite nested radical constants 96 Nick Lord 2.03 A uniform construction of some infinite coprime sequences 66 Nick Lord 9? Maths bite: averaging polygons 134 Nick Lord 9? Maths bite: Pythagoras causes chaos! 290 Nick Lord 9? Euler-type constants for the mid-ordinate rule 300 Nick Lord ‘Look-see’ constructions of a regular pentagon 324 Nick Lord rhe closed form evaluation of ¥ and related $99 Euleriansums Nick Lord On inequalities eqyivalent to the inequality of the means A t © Nick Lord Maths bite: irrational powers of irrational numbers an can be rational , . Nick Lord and 9? 4 generalised algebraic identity bites Pythagoras Paul Glaister Grzegorz Lysik The e-binomial inequality Des MacHale The well-ordering principle for Des MacHale 4 mathematical gem D. MacHale and 9? Some Diophantine tricks C. van den Bosch John Mahony 9? An improved approximation to a well-known integral K. Robin McLean Irisecting angles with ruler and compasses Douglas W. Mitchell The 2:3:4, 3:4:5, 4:5:6 and 3:5:7 triangles W. D. Munn Ellipses circumscribing convex quadrilaterals Robin Murphy Exploring numerical integration on a spreadsheet J. R. Nurcombe On an infinite series INDEX AUTHOR NO TITLE PAGI Jane O'Flynn "92.05 When is the sum of consecutive nth powers an nth power? 7] Michael O'Loughlin 92.65 Graphs of quadratic polynomials 495 Michael O'Loughlin 92.66 The centre of symmetry of a cubic polynomial Thomas J. Osler 92.43 Euler's little summation formula and special values of the zeta function Oren Peles 9? 06 A relation between the roots of a polynomial and its coefficients Poobhalan Pillay 92.64 Proofo f an aphorism from the Vedas Juan Pla 92.6] Linear recurrences which produce Pythagorean triples Angel Plaza and 92.60 \ recurrence relation for Fibonacci sums: Sergio Falcon a combinatorial approach Edgar Rechtschaffen th 7) A Real roots of cubics: explicit formula for quasi-solutions Jennifer Richinick 92.48 The upside-down Pythagorean theorem John Rigby th a we) \ regular (6n | )-gon inscribed in an equilateral triangle 3 Anthony C. Robin Sharing resources as fairly as possible J. A. Scott 92.04 Square-free integers once again J. A. Scott 92.56 Another rectangular hyperbola for the triangle H. A. Shah Ali The numbef of S.P numbers is finite J. P. Shiwalkar and 97? 97 The number of HH's in a coin-tossing experiment and M. N. Deshpande the Fibonacci sequence Stuart Simons 92.08 Summing digits of an arithmetic sequence Stuart Simons 92.14 Some aspects of the behaviour of the graph of f(x, y)J2 (x, y)... Jn(xX, vy) = & Stuart Simons 92.21 Area and perimeter ratios Stuart Simons 92.59 Curious collisions Stuart Simons 92.67 Parametric forms for general polynomial equations Stuart Simoéns 92.86 Some simple geometry Stuart Simons 92.90 \ novel approach to projectile paths Stuart Simons and Stable charge clusters Andrew Tworkowski Paul Stephenson lilings into tilings Paul Stephenson Counting faces on Archimedean solids Paul Stephenson Reconstructing heronian triangles B. Sury Of grand-aunts and Fibonacci B. Sury A generalisation of the irrationality of¢ THE MATHEMATICAL GAZETTE AUTHOR NO. TITLE PAGE Fukuzo Suzuki 92.18 Generalisations of the Napoleon theorems, 115 and triangles circumscribing a given triangle Hung-Ping Tsao 92.09 Explicit polynomial expressions for sums of powers of an arithmetic progression Roger Voles 92.25 Finding lost cousins Tony Ward 92.46 A permutation-ordering algorithm E. E. Wright 92.88 On a quadrilateral with three equal sides I. J. Zucker 92.34 The cubic equation — a new look at the irreducible case Teaching Notes Glyn George A seldom used formula for ODEs 344 Peter Macgregor A rapid method to find a tangent to a circle 160 Des MacHale ‘ The distance from a point to a line Des MacHale Integration without tears — two for the price of one Daniel Timms Proof without words: Infinite sum of a GP David Wells A surprise with parallel lines: an exploration that went wrong, then right INDEX Vii Reviews AUTHOR TITLE REVIEWER PAGE Titu Andreescu, Geometric problems on Michael de Villiers 589 Oleg Mushkarov, and maxima and minima Luchezar Stoyanov Avner Ash and Fearless symmetry, exposing the hidden Anthony C. Robin 584 Robert Gross patterns of numbers Antonio Becchi, Essays on the history of mechanics Michael de Villiers 180 Massimo Corradi, Federico Foce & Orietta Pedemonte, (edrs) Kai Lai Chung and Markov processes, Brownian motion and David Applebaum 186 John B.Walsh time symmetry (second edition) Richard Crandall and Prime numbers — a computational Ron Knott 183 Carl Pomerance perspective (2nd edn) Thibault Damour, Once upon Einstein Tony Crilly 179 (trans. Eric Novak) - : Michael J. de Smith Maths for the mystified Jennie Golding 368 Antonio Duran, Georges The life of numbers - Michael Ward 371 Ifrah and Alberto Manguel Karin Erdmann and Introduction to Lie Algebras Mark Hunacek 382 Mark Wildon Leonhard Euler (edited Elements of algebra Graham Hoare 373 by Christopher James Sangwin) Keith E. Hirst Calculus of one variable Piotr Zarzycki 58 Tom Hull Project origami: activities for Sue Pope 590 exploring mathematics Livia Giacardi (ed.) Da Casati a Gentile. Momenti di storia Ivor Grattan-Guinness ; dell’insegnamento secondario della 374 matematica in Italia Steven G. Krantz Mathematical apocrypha redux — More Graham H. Storr 372 stories and anecdotes of mathematicians and the mathematical Benoit B. Mandelbrot The (mis)behaviour of markets Graham Hoare 187 and Richard L. Hudson E. A. Maxwell Fallacies in mathematics Nick Lord 366 Carlos J. Moreno and Sums of squares of integers Martin Griffiths 377 Samuel S. Wagstaff Paul. J. Nahin. Dr. Euler's fabulous formula: cures S. C. Coutinho 582 many mathematical ills Adrian Oldknow (ed.) ICT and Mathematics: A guide to teaching Michael de Villiers 190 & learning mathematics 11-19 using ICT Vili THE MATHEMATICAL GAZETTE AUTHOR TITLE REVIEWER PAGE Wulf Rossmann Lie groups: an introduction through Mark Hunacek 380 linear groups Stephen C. Roy Complex numbers: lattice simulation and Peter Shiu 375 zeta function applications John Stillwell Yearning for the impossible: the surprising Jennie Golding 37) truths of mathematics Alan D. Taylor Social choice and the mathematics of Paul Hudson 188 manipulation Patrick Taylor Closepacks & quasi-closepacks {nthony C. Robin M. B. W. Tent The prince of mathematics Dennis Morris Carl Friedrich Gauss Graham Upton and Oxford dictionary of statistics (2nd edn.) Peter Saunders lan Cook Daniel J. Vellman How to prove it — a structured John Baylis approach (2nd edn.) Nicolai N. Vorobiev Fibonacci numbers Mic hael de J illie rs Hans Walser 99 points of intersection Gerry Leversha Leonard M..Wapner The pea and the sun: Tony Crilh a mathematical paradox N. J. Wildsberger Divine proportions: rational trigonometry Gerry Leversha to universal geometry Student Problems Numbers 2007-5 & 6 2008 = 1 & 2 2008-3 & 4 2008 — 5 & 6 Statement 91 p. 566 92 p. 176 « 92 p. 363 92 p. 579 Solutions 92 pp. 176-178 92 pp. 363-365 92 p. 580 Problem Corner Numbers SLE, F,G,H 91 1,5,K,1 Statement 91 p. 358 91 p. 561 Solution 92 pp. 171-175 92 pp. 357-362 Numbers 92 A, B,C, D 92 E, F,G, Hi 92 1, J, K, I Statement 92 p. 170 92 p. 356-357 92 pp. 573-574 Solution 92 pp. 574-579 93 93