THE AUSTRALIAN MATHEMATICAL SOCIETY GAZETTE Volume 31 Number 1 March 2004 1 Editorial 2 President’s column 4 Obituary W.P. Galvin by J. Giles 6 Math matters by P. Hall 12 Mathellaneous by D. Mathews 18 Brain drain by J. Stillwell 21 2004 B.H. Neumann Awards 22 2004 Australia Day honours 23 2004 ANZIAM and Michell medals 25 Comprehension as an assessment tool for mathe- matics by J. Simpson 26 International mathematical olympiad by A. Di Pas- quale 29 Primary schools mathematics competitions by J. Giles 30 The prisoners may be in two minds by A.J. Roberts 34 A random hopscotch problem by M.D. Hirschhorn 36 On the twin prime conjecture by A. van der Poorten and T. Ah-Mah 40 Squares from products of integers by W.D. Banks and A. van der Poorten 43 Modular equations and eta evaluations by W.B. Hart 48 Acknowledgement of priority by P. Schultz 49 Book reviews 58 Obituary D.A. Spence by J.J. O’Connor and E.F. Robert- son 60 News 69 AustMS bulletin board THE AUSTRALIAN MATHEMATICAL SOCIETY Gazette Jan de Gier and S. Ole Warnaar (Editors) E-mail: [email protected] Maaike Wienk (Assistant Editor) Web: www.austms.org.au Department of Mathematics and Statistics Tel: +61-(0)3-834-49682 The University of Melbourne Fax: +61-(0)3-834-44599 VIC 3010 Australia The individual subscription to the Society includes a subscription to the Gazette. Libraries may arrange subscriptions to the Gazette by writing to the Treasurer. The cost for one volume consisting of five issues is AUD 78.10 for Australian customers (includes GST), AUD 86.00 (or USD 69.00) for overseas customers (includes postage no GST applies). The Gazette follows policy established in Vol. 1, No. 1 and reiterated in Vol. 14, No. 1. The Gazette seeks to publish items of the following types. • Mathematical articles of general interest, particularly historical and survey articles • Reviews of books, particularly by Australian authors, or books of wide interest • Classroom notes on presenting mathematics in an elegant way • Items relevant to mathematics education • Letters on relevant topical issues • Information on conferences, particularly those held in Australasia and the region • Information on recent major mathematical achievements • Reports on the business and activities of the Society • Staff changes and visitors in mathematics departments • News of members of the Australian Mathematical Society Local correspondents are asked to submit news items and act as local Society represen- tatives. Material for publication and editorial correspondence should be submitted to the editors. Notes for contributors The editors seek contributions for the Gazette. These can be sent through Local Correspon- dents or directly to the editors. Articles should be fairly short, easily read and of interest to many of our readers. Technical articles are refereed. Authors are encouraged to typeset technical articles in LATEX 2ε, AMS-LATEX or variants. A special Gazette classfile gazette.cls can be downloaded from www.austms.org.au or may be obtained by email from the editors. In exceptional cases other formats may be accepted. Other contributions should preferably also be typeset in LATEX 2ε or variants, but may also be submitted in other editable electronic formats such as plain text or Word documents. 1 Editorial The present issue of the Gazette is the first produced by the new editorial team of Jan de Gier, Ole Warnaar and Maaike Wienk. It is appropriate with a change of guard to begin by thanking the previous editors, Bill Galvin and John Giles, for their dedication and excellent work over the past three years. It is with great sadness, however, that we report that Bill has not been able to witness this latest Gazette. Bill passed away last December, only shortly after editing the final Gazette of 2003. On behalf of the AustMS we extend our deepest sympathies to Bill’s wife and family. An obituary written by John Giles may be found elsewhere in this issue. Maths matters will be a new regular feature in the Gazette. In each issue a promi- nent Australian mathematician will be given the opportunity to voice their opinion on any mathematics issues. We are delighted that Peter Hall has been willing to kick-start the series, airing his view on the ‘decline of the mathematical sciences in Australia.’ Any responses to Peter’s column will be most welcome. We are also pleased to introduce Daniel Mathews. Daniel is an undergraduate stu- dent at The University of Melbourne with a distinct taste for the wild-side. His daring mathematical adventures can be found in Mathellaneous. In this issue he will surprise the reader with the beautiful and mysterious sequence 1,3,4,. An ongoing problem for Australian mathematics (see also Peter Hall’s column) is that many of our top-talents are scooped up by better paying, better resourced — better everything in fact — foreign universities. In the coming issues of the Gazette we hope to let some of our brain-drainees tell their side of the story. John Stillwell, who late last year wrote a letter to The Age about the brain-drain, will be the first to relate his motives for leaving Australia. From the regular technical articles it will become clear that little does not fall under the banner of mathematics. Hopscotch, prisoners, twins, and even eggplant and meatgrinders (page 38) are put under the mathematical microscope. An electronic version of the Gazette may be found at http://www.austms.org.au/ Publ/Gazette. Finally we wish to thank Richard Brak for designing the brainy dingbats. Deadlines for submissions to the Gazette Volume Number Deadline 31 2 13 April 2004 31 3 15 June 2004 31 4 10 August 2004 2 President’s column by Tony Guttmann This issue of the Gazette is the first to be described as disappointing. While long compiled by the editorial team of Jan de on rhetoric, and containing some good Gier and Ole Warnaar, with TeXnical as- recommendations, it is short of mean- sistance from Maaike Wienk. On behalf ingful actions. In particular it steers of the Society I would like to thank Jan, around any unequivocal statement that Ole and Maaike for assuming this respon- there is a substantial shortage of math- sibility. ematics teachers in schools, in all States. A number of significant events have re- As also previously noted, as a result cently occurred. As many of you will be of the Federal Government’s Higher Ed- aware, AMSI has been awarded funding ucation Legislation, a revised Relative to establish an International Centre of Funding Model (RFM) has been imple- Excellence for Education in Mathemat- mented. This new RFM is just as dis- ics (ICE-EM). The Federal Minister for missive of mathematics as its predeces- Education Brendan Nelson made the of- sor, with Government contributions to ficial announcement at the University of mathematics courses being lower than New South Wales on 27 January while those for comparable science and IT dis- visiting AMSI’s second annual Summer ciplines. This probably represents the School, which attracted over 150 partic- greatest threat to our long-term viabil- ipants. The Minister spoke enthusiasti- ity as an academic discipline, so we will cally of the importance of mathematics, be lobbying hard to bring about changes. saying it is “the foundation science for al- It was a major topic of discussion at the most everything that happens in our so- recent Heads of Departments meeting at ciety.” UNSW on February 13th. The International Centre of Excellence Professor Ian Petersen’s appointment for Education in Mathematics will be as the ARC’s Executive Director for managed from AMSI’s offices in Mel- Mathematics, Information and Commu- bourne but activities will be devolved na- nications Sciences will end at the end of tionally. It has funding of $7.8 million February, and he will be succeeded by over four years. ICE-EM will strengthen Professor Ah Chung Tsoi of Wollongong Australia’s capability in education in University. Prof. Tsoi has a background mathematics at all levels, from pre-school in complex systems, in particular the ap- to postgraduate studies. AMSI is cur- plications of neural networks. Professor rently negotiating finer operational de- Michael Cowling of the University of New tails with the Department of Education, South Wales invited him to attend the Science and Training, and hopes to have Heads of Department meeting, but he was a contract signed within a few weeks. unable to attend, and also expressed his As I noted in my column in the pre- reluctance to attend such events prior to vious issue, the Review of Teaching and officially assuming his new role. I also Teacher Education has been released. wrote to Professor Tsoi in my capacity as This six-volume report can be succinctly President, expressing my concerns about President’s column 3 the poor representation of mathematical The AMSI and Heads of Departments scientists on the panel, and about the meetings on February 12th and 13th re- quality of the advice received. We will flected the collaborative spirit currently be discussing these matters with him af- operating in addressing the issues raised ter he assumes office. above. On a more positive note, in the past With the start of the academic year six months AMSI and the ARC Centre upon us, I hope you can fire up the in- of Excellence for Mathematics and Sta- coming generation of undergraduates and tistics of Complex Systems (MASCOS) graduate students with enthusiasm and have worked well together and separately passion for our discipline—and do re- to sponsor a wide range of conferences, member to sign up incoming honours and which I believe have raised our interna- graduate students to a year’s free mem- tional profile.