## BH Neumann Awards

The BH Neumann Award honours the influence of Professor Bernhard H Neumann AC (1909–2002) who, after his arrival in Australia in 1962, provided outstanding leadership, support and encouragement for mathematics and the teaching of mathematics at all levels.

### BH Neumann Award Winners

#### 2016 Winners

**Lim Chong Keang
**

**Greg Gamble**

**Andrew Kepert**

#### 2015 Winners

**Kumudini Dharmadasa
Norman Do
Daniel Mathews**

#### 2014 Winners

#### 2013 Winners

**Jan Cavanagh**

Mike Clapper

Mike Clapper

**Karen Diehl**

**Jim Green**

Jillian Neale

Jillian Neale

Jacqui Ramagge

Jacqui Ramagge

Gregory Taylor

Gregory Taylor

Anthony Telford

Anthony Telford

#### 2012 Winners

**Janine McIntosh**

Graham Meiklejohn

Graham Meiklejohn

Sun Wen-Hsien

Sun Wen-Hsien

#### 2011 Winners

**Jane Breidahl**

Gary Carter

Gary Carter

Simon L Chua

Simon L Chua

#### 2010 Winners

**John Dowsey**

Cheryl Praeger

Cheryl Praeger

#### 2009 Winners

**Katrina Sims**

Latchezar Stoyanov

Latchezar Stoyanov

Elena Stoyanova

Elena Stoyanova

#### 2008 Winners

**Benjamin Andrew Burton**

Philip Swedosh

Philip Swedosh

Steve Thornton

Steve Thornton

#### 2007 Winners

**Angelo Di Pasquale**

Anne Hastings

Anne Hastings

William Palmer

William Palmer

#### 2006 Winners

**Brendan Case**

Teo Mui Hong

Teo Mui Hong

Rod Watson

Rod Watson

#### 2005 Winners

**Russell W Coad**

George Harvey

George Harvey

Andrei Storozhev

Andrei Storozhev

#### 2004 Winners

**Anthony Guttmann**

Anna Nakos

Anna Nakos

Jamie Simpson

Jamie Simpson

#### 2003 Winners

#### 2002 Winners

**Roger Curnow**

Sheila Williams

Sheila Williams

#### 2001 Winners

**Andy Edwards**

Michael Evans

Michael Evans

Alice Thomas

Alice Thomas

#### 2000 Winners

**Malcolm Brooks**

Maurice Starck

Maurice Starck

#### 1999 Winners

#### 1998 Winners

#### 1997 Winners

#### 1996 Winners

**Peter O’Halloran**

**David Clark**

**Judith Downes**

**Josephine Edwards**

**Neville Grace**

**David Pederson**

**Robin Thornely**

**Giovanna Vardaro**

#### 1995 Winners

**Neville de Mestre**

Kevin L McAvaney

Kevin L McAvaney

#### 1994 Winners

**Fred Bishop**

**Peter Brown**

Anne Street

Anne Street

Marta Sved

Marta Sved

Robert Thompson

Robert Thompson

#### 1993 Winners

**Warren Atkins**

**Robert Bryce**

**John Carty**

**Terence Gagen**

**Christopher Harman**

**Declan King**

**LG (Laci) Kovács**

Pierre-Olivier Legrand

Pierre-Olivier Legrand

**John Munro**

**MF (Mike) Newman**

Ashley Plank

Ashley Plank

Esther Klein Szekeres

Esther Klein Szekeres

George Szekeres

George Szekeres

**Peter Taylor**

#### 1992 Winners

**William T Akhurst**

**Geoffrey R Ball**

**AL (Larry) Blakers**

**John C Burns**

Neville J (Gus) Gale

Neville J (Gus) Gale

David H Haimes

David H Haimes

Bruce Henry

Bruce Henry

Asaeli Raass

Asaeli Raass

Ann Richards

Ann Richards

**Phillip Schultz**

**James L Williams**

**Neil Williams**

## Obituaries

#### BALL Geoff (1934–2016)

Geoff was born in 1934 in St Mary’s NSW, the only son of the local baker. Through his school days, though he was very competent academically, study took second place to sport, particularly his passion for rugby, which continued throughout his life. When he enrolled at Sydney Teacher’s College, he initially intended to become a Geography teacher, but his interest in mathematics was ignited by Jim Williams and he soon switched his focus. It was there also that he met Dawn Baker, whom he later married. After teaching mathematics in a number of schools, including Sydney Boys High, Geoff was appointed as Head of Mathematics at Fairfield Girls High School in 1964 and in 1967, he was appointed as a senior tutor in the Department of Mathematics at The University of Sydney, where Jim Williams was the Director of First and Second Year Mathematics. He remained at The University of Sydney for the rest of his career, becoming Director of First Year Studies in 1991 and, from the beginning, seeing it as a part of his brief to be involved actively in the dissemination of mathematical ideas to the wider community.

In 1968, he was elected to the Executive of MANSW and held various offices, including Editor of the journal *Reflections *and Chair of the Program Committee for the Biennial AAMT Conference. Having worked with John Mack and Jim Williams on the NSW Mathematical Olympiad program for many years from the early ’70s, Geoff took over as convenor of the committee in 1982, developing MATHSEARCH, a project-based investigative and research program which has stimulated an interest in mathematics in generations of high school students over more than 20 years. This also led to his involvement with IMO selection and in 1983, he became the NSW AMOC Director, developing correspondence programs for students training for Olympiad selection. He carried out this role with great energy until Bill Palmer took over in 1996, though he has continued to assist Bill until recent ill health made that impossible.

In 1983, he succeeded George Szekeres as the deputy leader of the IMO team, a position which he held for seven years, three of them with a youthful Terry Tao under his care! During that period, he was also Director of Training. He was a member of the Executive Board which organised the 1988 IMO in Canberra.

Geoff was involved with the AMC problems committee in the early days of the competition, particularly in setting the Senior Paper. He also contributed heavily to the development of the Mathematics Challenge for Young Australians Enrichment series, including as a writer. Specifically, he was co-editor of the Polya Enrichment Development Team (1992 to 1995 with Keith Hamann), and he was also a member of the Gauss Enrichment Development Team (1993 to 1995).

Geoff received a BH Neumann award in 1992, the very first year of the awards, and was awarded an OAM in 2005 for ‘promotion of the study of mathematics and service to the professional associations’.

Geoff will be greatly missed by his many friends from AMT and our condolences are extended to his widow Dawn and his family.

#### BILCHEV Svetoslav (Slavy) Jordanov (1946-2010)

#### Born Rouse 30 September 1946

#### Died Rouse 24 March 2010

When the first International Mathematical Olympiad (IMO) was organized in Romania in 1959 there was a student from Rouse region (Todor Penev Todorov) in the Bulgarian team. In the years to come, among authorities and citizens in Rouse, this fact sparked a considerable interest toward mathematics and mathematics education. A short announcement inviting school students to a “math circle” and “problem solving” , posted at a few places in the town, gathered over 800 (instead of the expected tens) in the backyard of the Evening Gymnasium.

This was the beginning of the famous Rouse Students’ Mathematical Circle (RSMC) guided by Docho Todorov Dochev. Slavy Bilchev was one of its most active participants. The creative atmosphere and the desire to learn how to solve difficult mathematical problems produced very interesting and stimulating results. Soon the Bulgarian Mathematical Olympiad was heavily dominated by winners from Rouse. Year after year school students from Rouse made their (not easy!) way to the Bulgarian Team for IMO. In 1964, 1968, 1969 and 1988 there were actually three Rouse students in each of the teams. Slavy Bilchev was a part of this success and contributed to it not only by helping others. He participated in IMO in Wroczlaw, Poland, 1963, and in Moscow, USSR, 1964. Both times he returned home with bronze medal. His rapid advancement encouraged other students from Rouse and motivated them to work hard.

On the basis of his successful participation in IMO Slavy Bilchev was given a state stipend to study at the famous Mechanics-Mathematics Department (better known as “Mech-Math”) of the Moscow State University. He graduated from that institution in 1969 and immediately entered the staff of Rouse University where he worked till his death on March 24th, 2010. Slavy Bilchev earned his PhD in 1975 . In 1979 he became Associate Professor. Less than a week after his death he was elected a Full Professor in a finalization of a procedure that had started a year earlier. Over the years he occupied different positions related to the administration of research and education in his university: Chairman of Department, Vice Director and Director of the Center of Mathematics, Dean of the Pedagogical Faculty (1994 – 2003) and Chairman of the Department of Algebra and Geometry ( 2004 – 2010).

Slavy Bilchev’s major research topics included (but were not limited to) Geometry, Ordinary and Partial Differential Equations, Differential Games, Mathematical Modeling in Economics, Mathematics Education, Mathematical Competitions and Olympiads. His results were published in more than 130 articles and 10 books. Unlike many others who profited enormously from the Bulgarian system for identification and support of mathematical talent and never returned “to pay back the debt” afterwards, Slavy Bilchev was a passionate supporter and promoter of the work with students of higher mathematical abilities during all his life. At least 25 of the school students from Rouse and Rouse region who took part in IMO participated in the regular class and beyond-class activities provided by him. Many of his former school students are now PhD holders, Associate Professors and Full Professors of Mathematics and/or Informatics in Bulgaria, USA, Canada, England, France.

As a lecturer at the university he also enjoyed highest appreciation on the side of students. Here is what an anonymous former student of him wrote in an Internet Forum immediately after learning about his death:

*I know it’s weird. I do not know where to start from. He is, perhaps, one of the most prominent citizens of Rouse. At least in my opinion. I did not have too many touching points with him. During my first year at the university he was lecturing math. He was a great guy and a lecturer. Unique as a person. He irradiated sincere bonhomie and cordiality. Something that I had rarely seen before…. I will never forget how we shook hands after the exam. … It is strange how a man you had only a few contacts with could leave such deep impressions on you. But he did. Can not explain exactly why… The coolest teacher in this university. … Rest in peace, Prof. Bilchev!*

Slavy Bilchev had great communication skills. He easily established fruitful contacts, liked to travel and to participate in conferences, developed numerous friendships all over the world. These abilities made him a desired partner in an impressive number of projects financed by the European Union. Meetings and discussions with colleagues were an essential part of his life. Yet, not too many knew that he was a lyrical poet as well (one of the 10 books mentioned above is a love poetry). He was a loving father, esteemed colleague and an honest and helpful friend. It was a pleasure to be around him. His smile rarely left his face. He was always ready to tell an encouraging story or a joke fitted well to the discussed topic. He kept his good mood and intensive life style even after the severe heart operation he had several years ago and despite the cancer he was fighting with till his last days…

Professor Slavy Bilchev was survived by his wife Ginka Bilcheva, daughter Marina and grand-daughter Anastasia.

Petar S. Kenderov

August 2010, Sofia,

Bulgaria

#### DAVIDOV Ljubomir (1913–1996)

Australian mathematicians and friends of Ljubomir Davidov (photographed above, right, in Pravets, 1994 with his compatriot Petar Kenderov) were saddened to hear of his death on 7 February 1998. His funeral in Sofia on 10 February attracted about 1000 fellow mathematicians and friends.

This is his obituary, written by Jenny Sendova.

### About Ljubo — Smiling Through Tears

* What matters is how you live, not how long. … –Seneca *

`Now here’s a beautiful problem for you …’ — how often have we, Ljubomir Davidov’s friends and colleagues, heard these words from his mouth. So have his students, many of whom are Olympic champions of the past or the present. These words spontaneously followed even the first declaration of his feelings, and his chosen one took that as the highest guarantee of his sincerity.

Ljubo’s keen sense for æsthetics was manifested in a particularly powerful way in the choosing of problems for training mathematical talents — an audience which appreciates the apt setting of the problem and the elegance of the solution more than any other. But he was also incredibly skilled at transmitting his enthusiasm for the beauty of mathematics to the less well-versed in this art.

Fate so decreed that that the last thing he saw on the computer screen was the cover of this book. `*Problems from the Putnam Contest* will reach the Bulgarian fans of mathematics at last’, his smiling face seemed to say. The feeling of deep satisfaction was perfectly natural, because into the preparation of this book Ljubomir Davidov had invested the zeal of a collector and a researcher, the high professional skill of an expert in mathematical contests and a technological competence which is still rarely seen among the practitioners of *pure* mathematics.

`Ljubo was such a perfect born gentleman. I have been wondering what his family background was’, George Berzsényi, a well-known American mathematician, writes in his letter of condolence. This may also be of interest to the students, the colleagues and the friends of the author of this book.

Ljubomir Davidov was born on 2 June 1947 in Sofia in the family of one of the capital city’s most prominent lawyers, Ivan Davidov. His maternal grandfather was Georgi Manev, professor of physics and founder of the Chair of Theoretical Physics, at various times Minister of Education and Convener of the University of Sofia. He was the first to introduce his young grandson to the realms of mathematics and physics. His exceptional skill of a storyteller Ljubo probably inherited from his father.

Following his graduation in mathematics at the Department of Mathematics and Mechanics (subsequently Department of Mathematics and Computer Science, DMCS) of the Kliment Okhridski University of Sofia in 1970, Ljubomir Davidov takes up a job at the Institute for Mathematics and Mechanics. In 1978 he defends a doctoral thesis in mathematics in the Moldavian Academy of Sciences. In 1986 he receives an associate professorship in algebra at the DMCS, where he works until 1990. Since then he is an associate professor at the Department of Education in Mathematics and Computer Science at the Institute for Mathematics and Computer Science (IMCS), Bulgarian Academy of Sciences. As an acknowledgement of his great organising skills, he is elected President of the Union of Bulgarian Mathematicians (1992-1993) and Deputy Director of the IMCS (1997).

His scientific interests are mainly in algebra, more specifically in the structural theory of associative rings and algebras.

In 1980 he becomes actively involved in the problems of teaching mathematics in the high school.

Though a wizard with a chalk in front of a blackboard, Ljubo was enthusiastic about exploring the potential of the new information technologies for enriching maths classes. His broad interests and knowledge made him an indispensable adviser in all manner of thorny situations — from solving a problem from a contest or competition to taming a new piece of software or elucidating the theory of electoral systems and mechanisms. His activity in the development of teaching materials and methods for integrating information technologies in the teaching of mathematics earned wide international recognition. In 1989 Ljubomir Davidov was invited to lecture at the University of California for a month, and in 1990 he was a Fullbright Professor at the Stevens Institute of Technology in New Jersey, USA.

From 1989 to 1993 Ljubomir Davidov was the leader of the Bulgarian team at international Olympiads in mathematics and contributed significantly to their success. Let it be reminded that the Bulgarian team led by him won the third place (after China and Germany), with 2 gold and 4 silver medals, at the International Mathematical Olympiad in Turkey, 1993. His prestige as a person with a great part in the formation of the traditions of mathematical contests was further strengthened at the Second International Conference of the World Federation of National Mathematics Competitions (Pravetz, 1994), where he was one of the principal organisers. During the Eighth World Congress of Mathematical Education (ICME’8, Sevilla 1996) Ljubomir Davidov was elected regional representative of the WFNMC for Europe. At the most recent instalment of the International Mathematical Olympiad (Argentina, 1997) he was invited by the organisers to supervise the coordination.

With his works (over 40 scholarly publications and 25 books), his brilliant lectures in linear algebra and higher algebra, his course of contemporary school algebra and the electives he read at the University of Sofia, the Burgas Free University and the New Bulgarian University, Ljubomir Davidov made a lasting mark in Bulgarian mathematics and ignited hundreds of yound minds with his admiration of the beauty of this science.

His dedication, his deep perception of problems, gift for organising, analytical mind, personal refinement and sense of humour fill our hearts with gratitude for the time spent with him. …

Ljubomir Davidov’s cause, and his name with it, will have the recognition of all lovers of mathematics young and old.

*–Jenny Sendova*

#### ERDÖS Paul (1913-1996)

Paul Erdös (or, in the original Hungarian way of writing his name: Erdös Pál) died on the 20th of September 1996 in Warsaw (Poland) of a heart attack while participating at a combinatorics conference. The world has lost one of the greatest, most prolific, most original and most loveworthy mathematicians of all time.

I shall try to elucidate on all four adjectives of the previous sentence because they are all meant in all earnesty. Erdös pursued an incredibly wide range of mathematics and had outstanding results in a dozen or so different fields, some of them being created by himself. He started as a number theorist and some proofs from his early youth are still amazing to this day for their exceptional mathematical beauty. Later he turned to combinatorics, in particular to graph theory (continuing to work in number theory, too, of course), then after having discovered so-called `probabilistic methods’ (both in combinatorics and in number theory) he became a leading scholar in probability theory itself. He also became one of the top authorities in classical set theory and founded quite new branches of it, like e.g. “partition theory’. But he also founded combinatorial geometry, transfinite combinatorics, and many more — often with co-authors (see below). He has literally thousands of theorems which count as key results in their respective disciplines.

It is commonly agreed that Erdös was the second most prolific mathematician of all times, being superseded only by Euler. The number of his published papers is around 1500 and another 50 or more are still to be published after the death of their author. Erdös undoubtedly had the greatest number of co-authors among all the mathematicians of all times — the number of his co-authors is about 500. It is not by chance that the mathematicians of the world introduced the concept of the “Erdös number”. Someone has Erdös number 1 if he/she has written a common paper with Erdös, someone having a common paper with someone who has an Erdös number 1 (but not with Erdös himself) has Erdös number 2, etc. A huge number of today’s mathematicians have a very small Erdös number. Erdös himself sometimes jokingly mentioned fractional Erdös numbers: someone having n common papers with himself has Erdös number 1/n. Two people (A. Hajnal and A. Sárközy) have Erdös number less than 1/50 and the number of people having Erdös number less than 1/10 is close to 30. But he was exceptionally prolific in other ways too. As it is well-known he travelled widely and unceasingly, and wherever he went he gave talks. These were not only the usual scholarly type of talks about new and earlier results of himself and others — he very often gave talks on unsolved problems in geometry, number theory, combinatorics, etc. He had a few more favourite topics, among them “The problems I would most like to see solved” and “Child prodigies”. He also wrote quite a few books, usually with co-authors. I would like to mention just one pearl among them, the `Topics from the Theory of Numbers’ written with J. Surányi [4], an English translation of which is due to appear soon.

Erdös was highly original both in the mathematical and in the everyday sense. I have mentioned already a few aspects of his mathematics, but it is important to emphasize the overwhelming weight *good questions* had in his thinking of mathematics. Many mathematicians consider Erdös the greatest problem poser of all times. For him building a new theory — a primary ambition of many colleagues of him — was never an aim. He just asked the right questions and the theory grew out by itself like a plant. And he also had a superb ability to know which question to ask from whom, quite often simultaneously. It was not an uncommon sight to see three or four people sitting in different corners of a room, Erdös walking from one to the other, making significant progress whith each of them on problems *belonging to quite different areas of mathematics* at the same time. It is true (but should be interpreted correctly — see the next paragraph) that the whole life of Erdös was mathematics; he was doing it all the time. Once I was sitting with him at a concert; as soon as the concert began, Erdös pulled out a notebook from his briefcase and started solving problems. After about half an hour he turned to me and asked: “What is this noise?” But this again should be interpreted correctly: Erdös had his own vocabulary and in this vocabulary `noise’ just stood for music. Actually he was very fond of music and he knew perfectly well that he was sitting at a concert, just it was normal for him to listen to the music *and* to do mathematics at the same time. This personal vocabulary of Erdös was well-known to his many friends. To give just a few sample examples: `Joe’ stood for the Soviet Union (derived from Stalin’s first name) and `Sam’ stood for the USA. `Bosses’ meant women (in particular wives) and `slaves’ men (in particular husbands). And `epsilon’ stood for any child. (I vividly recall my first encounter with Erdös. I was 15, attending high school and we paid a visit to Erdös at the Mathematical Institute in Budapest together with my class-mate L. Lovász. “Here are the epsilons!”—exclaimed Erdös and he started pouring mathematical problems on us as if we were professional mathematicians.) One more example should be mentioned because it sheds light on Erdös’s thinking about mathematics. `The Book’ meant the collection of the best, simplest, most brilliant proofs of all mathematical theorems, a book possessed by God only (which is not going to say that Erdös himself was religious in the traditional sense of the word). He himself surely produced quite a few proofs coming from The Book.

Erdös was loveworthy in a number of different and remarkable ways. To start with, when he asked a father of four: “How are the epsilons?”, it did not mean that he did not remember the names of the children. Just on the contrary — he perfectly knew the names, ages, past illnesses, etc. of those (and a few thousand other) children and he was genuinely interested in how they were. Incidentally, he had an incredible memory. When I first heard him `preaching’ (Erdös’s phrase for `giving a talk’) in 1962, I was shocked by passages like: “This was first asked from me by Kakutani in the summer of 1937 at Rome airport. I could not answer his question that time, but when we next met on a boat trip in May 1942, I told him … We published a joint paper on that in the June 1943 issue of the Bulletin of the American Mathematical Society.” And so on with almost every problem mentioned during his talk (and there were many dozens of them). I mentioned above that Erdös was doing mathematics all the time. I warned you that this should be interpreted correctly, because Erdös was doing other things, *too*, during almost all the time. He had an exceptionally wide range of interests and gladly discussed anything belonging to that range any time. I recall one discussion in the middle of problem-solving with some fellow-mathematicians, where the principal topics was whether ball-pens are cheaper in a certain Asian country than in a certain South-American one. These childish-sounding topics led to an extremely fascinating discussion covering many aspects of the economical, social, etc. life of those two countries, which was made so fruitful precisely by the presence of Erdös, who visited almost every country several times and due in no small way to his excellent memory and vivid interest in almost everything had a lot to say in way of comparison, evaluation and so on. But these evaluations were not those of an economist or sociologist, but those of a common-sense man. The style can be exemplified by `Sam wants this and this, but Joe doesn’t want it.’ You could learn a lot from such discussions.

But Erdös was also exceptionally generous, too. Practically all the money he earned from royalties, etc., he gave to friends in need, young mathematicians having financial troubles, for charity purposes, or just to anybody who needed it. Of the $50,000 which he received with the Wolf prize in 1984, he kept only $720 for himself, the rest went to friends, relatives, colleagues and a large part of it to endow a postdoctoral fellowship at the Technion (Haifa) to commemorate his mother. During his lifetime Erdös had many close friends and faithful and cherished disciples, but he had by far the closest emotional contact with *Anyuka* (`mom’ in Hungarian). *Anyuka* started to accompany Erdös on his incessant travels when she was 84(!) and continued to do so till her death at the age of 91 in 1971. The death of *Anyuka* was an incredible blow to Erdös from which he never fully recovered. After some time he found remedy in doing even more mathematics than before.

Erdös’s money did not go only to needy people and organizations. As it is widely known he offered money prizes for problems he could not solve for a while, the amount being anywhere between $10 and $10,000 depending on the estimated difficulty of the problem. Many mathematicians received those cash prizes in due time. There are also many legends about Erdös’s lifestyle; most of them not true literally, although resembling the truth. One such legend says that Erdös had no home for himself. This is not true: he owned an apartment in Budapest, but he did not use it after the death of *Anyuka*, even when staying in Budapest — he loaned it to colleagues in trouble or visiting foreign mathematicians (free of charge, of course).

It is only too fitting in this journal to say a few words about Erdös’s relationship to mathematics competitions and talented young people in general. Erd\H os, like practically every great Hungarian mathematician of the twentieth century started his mathematical career by solving problems published in * Középiskolai Matematikai Lapok*, a monthly mathematical journal for high-school students, founded in 1894. The best solutions sent in by students were published a few months after the publications of the problems themselves and the photos of the most successful problem solvers were published at the end of each school year. Erdös’s photo appeared in each of his high school years: 1927–1930. He remained faithful to the journal: he often published articles or problems in it.

But his greatest contribution to talent nurturing was his incessant search of and help for mathematically gifted young people. I mentioned earlier Erdös’s great affection to *epsilon*s in general, but he felt particularly at home with epsilons who showed signs of serious mathematical talent. He treated such youngsters as his peers, gave them suitable (often unsolved!) problems and paid close attention to their mathematical progress. He gave his own account of several of these young disciples under the title “Child Prodigies” (see item 71.01 in the bibliography of [1], vol. 1.), and he later retold the story several times at lectures all over the world, the list always expanding, of course.

For those who want to get a detailed account of Erdös’s life I recommend the masterly written paper of L. Babai [3], which also contains a wealth of background material. [2] is a collection of the most important and influential papers of Erdös, containing many classic gems. From the two volumes of [1] you can have a glimpse of the state of knowledge in the various areas of mathematics where Erdös’s influence has been decisive. They also contain the most up-to-date bibliography of the publications of Erdös.

#### References

[1] *Combinatorics*, Paul Erdös is Eighty (eds.: D. Miklós, V. T. Sós, T. Szönyi), Bolyai Society Mathematical Studies, Budapest, vol. 1.: 1993., vol. 2.: 1996.

[2] P. Erdös: *The Art of Counting (Selected Writings)* (ed.: J. Spencer), M.I.T. Press, 1973.

[3] L. Babai: *In and Out of Hungary: Paul Erd\H os, His Friends and Times*, in: [1], vol.~ 2., pp. 7–95.

[4] Erdös Pál — Surányi János: *Válogatott fejezetek a számelméletböl* (Topics from the Theory of Numbers, in Hungarian) 1st ed.: Tankönyvkiadó, Budapest, 1960., 2nd, revised ed.: Polygon, Szeged, 1996. (An English translation of the second edition is to appear at Springer Verlag)

József Pelikán,

Dept. of Algebra and Number Theory,

Eötvös Loránd University,

Múzeum körút 6–8.,

H–1088, Budapest,

HUNGARY

e-mail: pelikan@cs.elte.hu

(This obituary appeared in Mathematics Competitions, **9**, 2, 1996, pp15-20.)

#### GREITZER Samuel L (1905–1988)

For fifteen years, competitors and mathematicians interested in high-level compteitions found it near-impossible to think of the prestigious USA Mathematical Olympiad without having in mind Dr Samuel L Greitzer, founding chairman of the committee in charge of the USAMO. Even the abbreviation USAMO centred on his name Sam, as his friends and admirers called him. Throughout his tenure and for years afterwards (through the journal *Arbelos*, which he prepared more or less single-handedly). Sam’s concern with the mathematical well-being of highly talented students was similarly centralized. Only death could terminate his devotion, on February 22, 1988, at age 82.

Sam Greitzer emigrated to the United States from Russia in 1906, graduated from the City College of New York in 1927, and earned his PhD from Yeshiva University some years later. For over 25 years he was teaching at the secondary level, and then at Yeshiva University, the Polytechnic Institute of Brooklyn, Columbia University and Rutgers University. He published numerous pedagogical and research articles as well as several books, of which *Geometry Revisited* (co-authored with Coxeter) and *International Mathematical Olympiads 1959-1977* are probably the best known in competition circles. He was the leader of the US team to the International Mathematical Olympiads during the first ten years of US participation, and took part in training the teams prior to the IMOs. He traveled extensively to help other countries with their efforts also, and he was well known in Australia, in Europe and South America as well.

The initial impetus for starting the USAMO was due to Professor Emeritus Nura Turner, whose intense work with the participants of this competition is still in progress. Sam fully recognized the importance of the USAMO, and soon became its most ardent supporter. His expectations from the students were high, his leadership of the committee was firm, and his administration of the competition was flawless. So it is far from co-incidental that SAM’s name and guiding spirit will remain central to the USAMO as long as this superb competion is in existence.

G Berzsenyi and W Mientka

1988

This obituary was published in * Mathematics Competitions*,

**1**, 1, 1988.

#### NEUMANN Bernhard Hermann (1909–2002)

Emeritus Professor Bernhard Hermann Neumann, who provided the greatest inspirational influence in mathematics in Australia over a 40-year period, died in Canberra on 21 October 2002 not long after happily celebrating his 93rd birthday.

He first visited Australia for three months in 1959, during sabbatical leave, and fell in love with the country. So when, late in 1960, he was invited to found a Department of Mathematics in the research-focussed Australian National University, he was receptive to the idea. Within days of his permanent arrival on 2 October 1962, he also became involved in activities supporting the teaching of mathematics in schools.

Bernhard had a great influence in the founding and administration of the Australian Mathematics Trust. He became a mentor and source of inspiration to Peter O’Halloran (1931-1994) who, while on the staff of the Canberra College of Advanced Education (later the University of Canberra) during the period of the early 1970s to the early 1990s, is acknowledged as the Founder of the Trust. Peter gained great strength from Bernhard’s encouragement, not only while Bernhard held his position as head of mathematics in the Institute of Advanced Studies at the Australian National University, but also after Bernhard’s retirement.

Bernhard took an active personal part in the Trust’s activities. He was the Inaugural Chairman of the Australian Mathematical Olympiad Committee, a position he held from 1980 to 1986. He was also the representative of the Canberra Mathematical Association (a sponsor of the Australian Mathematics Competition (AMC)) on first the AMC Governing Board, and then on the Advisory Committee of the Trust. He was an active member of the Advisory Committee until his death.

Bernhard was also the principal inspiration behind the founding of other mathematics organisations. For example, he was the founding President of the Canberra Mathematical Association and the Australian Association of Mathematics Teachers (AAMT).

Bernhard’s continual and effective promotion of mathematics among students of all ages, combined with his interest in and support for mathematics education at all levels, was significant and together profound in its impact upon all aspects of mathematics in Australia.

Early in 1998 the Australian Academy of Science arranged an interview with Bernhard as part of its Video Histories of Australian Scientists program. An edited transcript is available on-line here. This contains a clear and detailed account of his personal life and career in Mathematics up to that time.

Very briefly, he was born on 15 October 1909 in Berlin and grew up there. He studied at Freiburg and Berlin and earned his doctorate in 1931, making him one of the youngest ever to receive this award in mathematics from a German university. The difficult economic and political situation, especially related to race, caused him to leave for England in 1933, where he undertook a PhD at Cambridge completed in 1935. The employment situation was difficult and it was late 1937 before he began a university career at Cardiff. He was then, with the onset of World War 2, interned as an alien and later recruited into the British military, first in the Pioneers and finally in Intelligence. He recommenced his university career in Hull in 1946 and then in 1948 moved to Manchester where later he became a Reader. He earned various honours including the Adams Prize (1951) and election to the Royal Society of London (1959).

In 1938 he married Hanna von Cämmerer after a long secret engagement. They raised five children under the sometimes difficult domestic arrangements associated with two careers. The children have all had successful careers; one of them, Peter, became a Fellow of The Queen’s College, University of Oxford and Chairman of the United Kingdom Mathematics Trust (a similar body to the Australian Mathematics Trust). At the same time as Bernhard’s Canberra appointment, Hanna was offered a Readership at ANU. She arrived in August 1963 and within a year became the foundation Professor of Pure Mathematics, where her primary responsibility was undergraduate teaching. She rapidly involved herself with mathematics teaching in schools and with the AAMT.

Bernhard was the author of over a hundred research papers. Most of his work was in group theory. He also worked in other branches of algebra and sometimes in geometry. One notable example of the latter is the Douglas-Neumann Theorem, an extension of Napoleon’s theorem discovered independently by Bernhard and the Fields Medallist, Jesse Douglas.

Bernhard’s department raised the visibility of Australia in the mathematics community and mathematics in Australia. It provided strong graduate training, producing over 50 PhDs by the time he retired from ANU in 1974. Many of these graduates went on to senior positions and some to very active roles in mathematics education.

He was one of the few people who had attended all ICME conferences from ICME-1 (1969) in Lyon, ICME-2 (1972) in Exeter, ICME-3 (1976) (Karlsruhe) up to and including ICME-7 (1992) in Quebec.

Bernhard was elected to the Australian Academy of Science in 1964. In that context he made significant contributions to mathematics education, primarily through vitalising the Australian Subcommission of the International Commission on Mathematical Instruction (ICMI), serving as chairman and National Representative on ICMI. His visibility at the early International Congresses on Mathematical Education meant that he and other AAMT members were able to respond with enthusiasm to a suggestion (at Karlsruhe) that Australia host a forthcoming ICME. Intensive work, in which many AAMT members played decisive roles, led to the successful bid to hold the 1984 Congress in Adelaide and this conference was remarkably successful.

In addition he was active in getting the Academy involved in providing materials for schools; after a long gestation period, six volumes of *Mathematics at Work* appeared in 1980-1.

The main award of Australian Mathematics Trust recognises, in Bernhard’s name, people who have made significant contributions over many years to the enrichment of mathematics learning in Australia and its region. Until his death, Bernhard took an active interest in this award, travelling around Australia where necessary to present the awards (now restricted by quota to no more than three per year), always protecting the polish on the silver tray with his trademark white gloves.

The Australian Mathematics Trust also has another award in Bernhard’s name. Bernhard had a fascination for the ability of students who could achieve a perfect score in the Australian Mathematics Competition for the Westpac Awards, and such an achievement now merits a BH Neumann Certificate.

His involvement in Olympiad activities was wider than may have appeared on the surface. He became a member of the Executive Committee of ICMI (1979-82) and, chairing a Site Committee, ensured better structure and operation of the International Mathematical Olympiads. The holding of the 1988 IMO in Australia, in our bicentenary year, had much to thank him for.

He was also active in The Australian Mathematical Society; it awards prizes in his name, on the basis of research student presentations at its annual meetings.

In 1994 he was made a Companion of the Order of Australia (AC) for service to the advancement of research and teaching in mathematics.

These important achievements are only to be weighed beside the many individual acts of advice and assistance that Bernhard, initially with Hanna and then with the complete support of his second wife, Dorothea, gave to all who sought this from him. And they should be viewed in the light of many other activities including cycling, cello-playing, chess and exploring the countryside.

May the memory of him as a warm and influential person, and of his many contributions to mathematics, to mathematics education, and the organisations activities he helped found, serve to stimulate us all!

An interview with Bernhard Neumann, taken about the time of his 90th birthday, is available on the Academy’s website and can be found by clicking here.

John Mack, Mike Newman and Peter Taylor

#### O'HALLORAN Peter Joseph (1931–1994)

The death occurred on 25 September 1994 of Peter O’Halloran, founder of the Australian Mathematics Competition and a range of national and international mathematics enrichment activities.

Peter was born in Sydney on 27 April, 1931, the youngest of a family of four boys and three girls. His father died when he was four years old, leaving the family in poor circumstances. His mother took in boarders to make ends meet and put several of the children through University.

Peter attended Marist Brothers School in Kogarah, Sydney, and it was there that he developed his life-long fascination with mathematics. He went on to the University of Sydney, on a government teaching scholarship, where he graduated with a Bachelor of Science and Diploma of Education.

Peter married Marjorie in 1955 and taught mathematics in several high schools in Sydney and the country regions. At Manly Boys High School, which had a noted discipline problem at the time, Peter also coached the Rugby Union and Rugby League teams, both with high degrees of success.

In 1965 Peter became head of mathematics at the Royal Australian Navy Academy at Jervis Bay. It was there that he completed a Master of Science, specialising in oceanography.

In 1970, now with four children, Peter and Marjorie moved to Canberra. Peter was one of the original appointments at the Canberra College of Advanced Education (later to become the University of Canberra) and taught in the first official semester of the CCAE. At the CCAE Peter developed other interests, including operations research and what was to become his main interest, discrete mathematics.

In 1972/3 he was the first CCAE academic to take study leave. Part of this was taken at the University of Waterloo, Canada, where he gained the idea of a broadly based mathematics competition for high school students. On his return he often enthused to his colleagues about the potential value of such a competition in Australia.

In 1976, while President of the Canberra Mathematical Association, he established a committee to run a mathematics competition in Canberra. This was so successful that the competition became national by 1978 as the Australian Mathematics Competition, sponsored by the Bank of New South Wales (now Westpac Banking Corporation). It is now well-known that this competition has grown to over 500,000 entries annually, and is probably the biggest mass-participation event in the country.

In 1979 he established the Australian Mathematical Olympiad Committee.The activities of this committee have grown to a complex web of competition and enrichment activities, at the highest level culminating each year in Australia’s participation in the International Mathematical Olympiad (IMO). Australia has participated internationally with distinction, for example this year being placed 12th out of 69 countries.

In 1983 Peter founded University of Canberra Mathematics Day, based on a similar event he had seen in America. This is now well-established and supports similar days in other parts of Australia. It has in turn become a model for other Mathematics Days throughout the world.

In 1984 Peter founded the World Federation of National Mathematics Competitions. For several years the main activity of the WFNMC was the production of a Journal, which acted as a vital line of communication for people trying to set up similar activities in other countries. In recent years its activities have expanded to include an international conference and a set of international awards (the Hilbert and Erdos Awards, to recognise mathematicians prominent in enriching mathematics education. Most recently, the WFNMC has become a Special Interest Group of the International Commission on Mathematical Instruction (ICMI).

One of the highlights of Peter’s career was hosting the 1988 IMO in Canberra, which attracted a record number of countries at the time and set new standards in many aspects of organisation.

In 1989 Peter established the Asia Pacific Mathematics Olympiad, providing a regional Olympiad for countries in the dynamic Pacific rim area.

In 1990 Peter completed the spectrum of competitions and enrichment activities in Australia, with the range of Challenge and Enrichment activities now organised by the Australian Mathematical Olympiad Committee.

Perhaps the most significant event in Peter’s career in the last two or three years was his role in the establishment of the Australian Mathematics Trust, which is an umbrella body administering all the activities with which he has been associated and which are referred to above.

Peter’s last main duty was to preside at the WFNMC conference in Bulgaria in July 1994. It was obvious to all who were there that Peter was ill. It was generally thought that he was experiencing another bout of pleuro-pneumonia, from which he had suffered in 1993. On his return home however further tests revealed that Peter’s condition was much more serious, and cancer was diagnosed. He spent most of his last month at home.

On 31 August he was presented with the David Hilbert Award, which he had declined to accept earlier in the year while still president of the WFNMC. A small party of 30 to 40 of Peter’s relatives and local colleagues were in attendance at his home. The David Hilbert Award is the highest international award of the WFNMC and in Peter’s case was awarded for “his significant contribution to the enrichment of mathematics learning at an international level”.

On 19 September he was awarded the World Cultural Council’s “Jose Vasconcelos” World Award for Education at a ceremony at Chambery, France. This award “is granted to a renowned educator, an authority in the field of teaching or to a legislator of education policies who has a significant influence on the advancement in the scope of culture for mankind”. The qualifying jury is formed by several members of the interdisciplinary Committee (of the World Cultural Council) and a group of distinguished educators.

Due to his illness Peter was unable to travel. Instead his eldest daughter Genevieve and son Anthony travelled to Chambery to receive the Award on his behalf. Fortunately Peter was still alive on 23 September on their return to Australia and was able to receive the award in person.

Peter was also recognised in many other ways throughout his career. In 1983 he was awarded the Medal of the Order of Australia (OAM), in 1991 he was awarded a Doctor of Science (honoris causa) from Deakin University and in 1994 he was promoted to Professor in his own University.

Many mathematicians have made significant individual contributions to the subject itself. Peter’s influence was much more direct, bringing mathematics to the world. With his driving energy and the institutions he created he has significantly increased people’s awareness of mathematics and what it can do, throughout the world.

One of Peter’s main concern in life was to assist the disadvantaged. He saw the AMC as being able to bring mathematics to children in remote places. In the earliest times of the AMC he travelled the Pacific and introduced the competition to a large number of island nations, such as Fiji, Tonga, Western Samoa, French Polynesia (for whom the paper was made available in French) and many smaller countries, some of which only had radio or occasional steamer contact with the outside world.

The Australian Government recognised these efforts and funded this project as one of the few cultural links between Australia and its Pacific neighbours.

Peter was proud of the fact that more than 50\% of the AMC entrants were girls, but concerned that they did not get the same representation among the medals. He was also proud of the many letters he received from country schools, thankful for the opportunity to participate in the same event as their city cousins.

Peter saw the main advantage to be derived from the WFNMC was the help it could give to mathematics education in developing countries. I was seated next to him in a debate on the value of competitions at the 1992 International Conference on Mathematical Education in Quebec where he was payed the ultimate compliment to which he would have aspired. One delegate gave a well-planned attack on competitions, based on the usual lines, that competitions encouraged elitism, etc. In response, a delegate from the small African country of Malawi, unknown to Peter, responded with an emotional thank you to Peter and the people of the Australian Mathematics Competition for what they had made possible in her country. This was a most moving experience.

I made these observations to Peter’s eldest brother Ted after Peter’s funeral. Ted had become senior partner in one of Australia’s largest law firms before going into semi-retirement and was both a father-figure and inspiration to Peter. Ted reflected on these comments and noted the similarity between Peter and another brother Michael (now deceased). Michael had gone into the Marist order and had shown the same concerns for the disadvantaged. He had become responsible in the order for the distribution of international aid. It was probably no coincidence that Peter’s and Michael’s paths once crossed. Several years ago they found themselves together on the same remote Pacific island, each going about their own separate tasks.

Peter, of course will be irreplaceable. Fortunately, however, he had the foresight to establish institutions in such a way that they all have the resources, particularly human resources, to ensure that the good work will continue.

Peter is survived by his wife Marjorie, four children and six grandchildren.

Peter Taylor

Canberra

10 October 1994

The above obituary was also published in the January 1995 edition of the Bulletin of the Institute of Combinatorics and its Applications, where it was preceded by the following personal tribute by Professor Ralph Stanton. Professor Stanton was the first head of mathematics at the University of Waterloo, and as a result was responsible for the appointment of excellent teachers to the faculty and the establishment there of what became the Canadian national mathematics contests.

**PETER O’HALLORAN
A PERSONAL TRIBUTE**

**R.G. Stanton, University of Manitoba**

Early in September, I was talking on the telephone, from the University of Leeds, with Peter O’Halloran at his home; he was his usual cheerful self, but it was evident, from his coughing, that he was extremely unwell. When Peter Taylor sent me an email later in the month that Peter had died, I felt a deep sadness because the mathematical world has lost a man who could still have contributed a great deal. But we can all be very proud of the enormous impact that he has had on Australian mathematics and, indeed, on world mathematics.

My own feeling is that Peter O’Halloran contributed more to the development of mathematics in the past 25 years than any other single person. Those of us who write research papers typically publish results that are read by a small circle of researchers interested in similar ideas. This is true even of “important” papers. But Peter influenced millions of students. He took the idea of a Mathematics Competition, as he had seen it operate at the University of Waterloo, and developed it to an extent and scope that would have been impossible to anyone who did not possess his driving energy, his boundless enthusiasm, and his love for the subject. The Australian Mathematics Competition alone attracts more than half a million entries a year, and its importance is underlined by the fact that His Royal Highness, The Prince Philip, K.G., K. T., graciously consented to be Patron of the Competition. And the other activities that Peter initiated and promoted have likewise possessed a widely ranging impact.

Many people over-emphasize the importance of mathematical research. Research is important, but it is only one facet of mathematics. As mathematicians, we must also be alive to the importance of the preservation of knowledge, the transmission of knowledge, and the formation and nurturing of the next generation of mathematicians. Peter’s great service to mathematics was his phenomenal contribution to the stimulation of mathematical interest in young people around the world. Through his Competition, and the other activities now preserved though the Australian Mathematics Trust, he has had a tremendous influence that no contemporary mathematician can match.

We are fortunate that Peter’s colleague, Peter Taylor, has contributed an account of Peter’s life and work; this follows immediately after these few words of recognition.

#### PAGET David (1943–1997)

Dr David Paget, photographed above with his wife Leone and Professor BH Neumann after receiving an Australian Mathematics Trust BH Neumann Award in Hobart in February 1997, died after a long illness on 30 November 1997.

He was the mainstay of the Olympiads in Tasmania and State Director for 8 years, 1988 to 1995.

David worked on many fronts tapping emerging mathematical talents throughout Tasmania. To a large extent this was single-handed; without him there would have been no Tasmanian activity or representation.

The Tasmanian “Friday night group” was one of David’s initiatives, including arranging access for students in the north of the state. He looked after all the student needs, including pastoral needs, often handling problems via parents.

Hobart on two occasions had the strongest Australian or New Zealand score in the International Mathematics Tournament of Towns, a direct result of the very strong group of students David had developed. Two members of these groups, including one from a remote location 80km west of Launceston, gained Australian selection at IMO, winning silver medals.

David soon became active on the national scene. In 1990 he became Director of Training of the Australian Mathematical Olympiad Committee, a position he held for six years. He developed and professionalised the position to the point at which it became one of AMOC’s three substantive senior positions. He followed on a personal basis the fortunes of the elite students throughout Australia and coordinated their attendance at training schools.

On the international scene he was Deputy Leader of the Australian Team in 1990 and Team Leader in 1991, through to 1995.

David also took on administrative roles in the Trust, becoming an alternative member of the Trust’s Board and a Director of AMTOS Pty Limited, the company which administered the Trust’s activities until 1998.

David was also active promoting mathematics in general, organising camps for a wide number of students and obtaining sponsorship.

There are many areas in which David contributed to mathematics outside the Trust.

It is appropriate to conclude by attaching a profile about himself written for the Australian Mathematics Trust’s in-house journal *The Globe*, written in March 1995 while he was still active with the Trust’s work and about to lead Australia’s team to the IMO in Toronto.

Peter Taylor

January 1998

**David’s Profile (written by himself in 1995)**

I was born in 1943 at Twickenham, near London, just a stone’s throw from the home of rugby football. My earliest memories are of the street parties held to celebrate the end of World War II; there was so much food!

One of my three older sisters taught me the basics of reading and arithmetic before I started school (she became a teacher!). The Infant and Primary teachers of that era demanded higher standards than would be educationally acceptable today, so, before I left Primary school, I had been introduced to literature in Robert Louis Stevenson and Charles Dickens, and also to the elements of Algebra. One particular teacher is responsible for developing my life-long interest in sport. He encouraged us to attend live sports meetings all over London. At eleven years old I frequently cycled 15 miles or more through London traffic to attend an athletics meeting, cricket match or football game.

Passing the Eleven-Plus Examination gave me entry to the local Grammar School, where I was introduced to Euclid, and from there my love of mathematics just grew and grew. I had the good fortune to develop friendships with a group of students with similar academic leanings. We worked extraordinarily hard, not because we were studious but because we were intrigued; we needed to understand how and why science worked. Our enthusiasm was well managed by our teachers. For our Applied Mathematics course we were required to produce 30 solved problems per week. These solutions were checked against the official solutions in the teacher’s cupboard. One day I solved a problem for which there was no official solution. The teacher was delighted and asked me to write out my solution on an official card, saying that he had been unable to do it. My friends and I noted all the missing solutions and we formed a Friday night club to produce them. Each time we produced a solution the teacher was overjoyed. Four years later, visiting that same classroom, I discovered that all our solutions had been removed and a small band of students was working enthusiastically on those problems the teacher couldn’t solve!

In 1961 I was awarded a State Scholarship to Southampton University where I read Mathematics. That’s right, nothing else, just Mathematics for three years. At that time there were 32 full-time members of the Mathematics Department and about 400 Mathematics students in each of the three years. I understand that staff increases have almost kept pace with student increases right up to the present time. After the initial shock of tertiary mathematics I settled down to a diet of mathematics and rugby. For the first two terms the University 1st XV played twice a week and trained three times a week. This left a lot of mathematics to be done in vacations and the Summer term.

From the age of 16 onwards I spent my summer holidays in Europe and Israel, hitch hiking and occasionally working. On returning to London after these trips abroad I increasingly felt that there must be better places to live than England. Consequently in my final year at Southampton I sought advice from the University Careers Guidance Officer, expressing a wish to go overseas. Unfortunately he was of an era in which one received a gold watch after 40 years service and retired on a pension. His only suggestion was for me to join the British Army! I looked elsewhere for advice. Eventually I settled on a position at an Anglican boarding school in Hamilton, New Zealand. At that time the University of Waikato was being established and I was able to keep up with some Mathematics by attending occasional seminars. I was also able to keep up with my rugby; indeed my first game in New Zealand was against an Auckland side captained by the then All Black captain, Wilson Whineray. The fitness, toughness and total commitment of NZ rugby players was something of a shock to an Englishman who would fall on the ball as a defensive tactic. Such an action in NZ could only be considered suicidal.

In December 1966 I married Leone and in January 1967 we moved to Hobart where I was to take up a Research Assistantship under Professor David Elliott with the proviso that I play for the University Associates Rugby Club. Tasmania was practically unknown to New Zealanders. Leone thought she was coming to a whaling port whilst the New Zealand travel agent, unaware of Hobart Airport, booked us a journey by plane, ferry and bus which took two days.

My research interests have been in Numerical Analysis and Approximation Theory. For ten years I was Chief Examiner of Mathematics for the Schools Board of Tasmania. I resigned when the Board failed to support the maintenance of high standards. In 1988 I became Tasmanian Director for the AMOC and joined the Coordination Team (Markers) for the 29th International Mathematical Olympiad in Canberra. Here at last was the sort of mathematical enthusiasm I could identify with. I was pleased to deepen my involvement. It was truly rewarding to work alongside and learn from people of the calibre of Geoff Ball and David Hunt. Most of all I am grateful to the late Peter O’Halloran who saw the need for the AMOC programme, had the vision and enthusiasm to assemble a committee of mathematicians to carry it out, and had the communication skills to make everybody feel they had a vital role to play.

With my heavy involvement in the AMOC programme, these days I have very little time for research. Like many others in the AMOC, I believe the work we do with talented youngsters is of enormous benefit. Unfortunately, university values are based almost entirely on research output.

To finish on a personal note, Leone and I have four children. The youngest is still at high school whilst the other three are at various universities, ranging from 2nd year Science to 4th year PhD.

All those who have spent time at any of the training schools will know that I have a hobby. I gain much pleasure from perusing and cataloguing my cigarette card collection which now weighs in at close to 25,000 cards.

#### POTTS Renfrey Burnard (1925–2005)

#### Born Adelaide 04 October 1925

#### Died Adelaide 09 August 2005

Ren Potts, one of Australia’s most inspirational mathematicians in recent decades, was educated at Rose Park Primary School and Prince Alfred College before studying at the University of Adelaide from 1943 to 1947, when he graduated with first class honours in mathematics.

In 1948 he was the Rhodes Scholar for South Australia and went to Oxford University, where he was at Queen’s College and graduated with a D Phil, researching Mathematical Physics. This earlier work included the development of what became the Potts Model, a significant cornerstone of statistical mechanics. However Ren soon moved his interests to being one of the real founders of Operations Research, and the Potts Model’s citations, etc were to come only 20 years later, well after he had changed fields. His contribution to Operations research, and developing it is a significant mathematical discipline were significant, not only with research, but also in considerable application in improving traffic flow in cities in the US and Australia, and many related applications.

By 1959 Ren was appointed as a Professor of Applied mathematics at the University of Adelaide, holding one of the University’s four named Elder Chairs until his retirement in 1990. These were truly golden years for mathematics at Adelaide, with Adelaide having consistently one of the highest outputs of graduate students in Australia during this time.

The atmosphere in the department during this time was highly conducive to quality research and the morale was very high. Ren’s research group was certainly one of the largest with over 15 doctoral students at times. Ren took not only a personal interest in the progress of all students in the department, but also monitored all the undergraduate classes, visiting them from time to time to extol the virtues of continuing their studies and the excellent job prospects.

During most of his life Ren was an active sportsman. Not only had he been a fine athlete during his undergraduate days but through his working life he pursued a number of sports vigorously, usually until injusries took their toll. These sports included hockey, squash, badminton, marathon running and swimming. He also took a leading role in running the Adelaide University Sports Association and was often a successive active lobbysist in improving the facilities available to students.

Ren played an active role in organising ICME-5 in Adelaide, which attracted several thousand mathematics educators from around the world, and his plenary speech at that conference was a memorable example of his considerable oratory skills.

In Canberra in 1988, Australia hosted for the first time the International Mathematical Olympiad, the largest held to that time, with many Asian countries participating for the first time. Ren chaired the steering committee and also the Jury, a highly challenging job which he handled with considerable skill.

Ren recieved a number of national honours including Officer of the Order of Australia (AO) in 1991 for “service to Australian society and science in operations research”. He was made a Fellow of the Australian Academy of Sciences (FAA) in 1975 and Fellow of the Australian Academy of Technological Sciences and Engineering (FTSE) in 1983, one of the few to be Fellow of both Academies.

He was active in founding ANZIAM (the Applied Mathematics division of the Australian Mathematical Society) and the Australian Computer Society and won the inaugural medal of ANZIAM in 1995.

In recent years he became active in Keith Hamann’s Saturday morning enrichment group in Adelaide, improving the problem solving skills of many of Adelaide’s most talented students. During this time this group has produced a number of IMO medallists, including Justin Ghan, Ross Atkins and Konrad Pilch. Justin and Ross continued also while living in Adelaide to work with the students and Konrad is still eligible while at school for another year for further IMO selection.

Ren is survived by his wife Barbara (Dr Barbara Kidman, a Computer Scientist), whom he married in Oxford in 1950, and two daughters Linda and Rebecca.

Peter Taylor

September 2005

(With acknowledgement to material published in the Adelaide Advertiser for some details.)

#### PRESTON Gordon Bamford (1925–2015)

http://www-history.mcs.st-andrews.ac.uk/Biographies/Preston.html

#### STRZELECKI Emanuel (1921–2015)

Dr Emanuel Strzelecki of Monash University was an inspiring teacher of mathematics. He lead our IMO team in 1987 and 1988, trained many secondary students, and was a member of AMOC Senior Problems committee and a Challenge moderator.

#### SZEKERES George (1911–2005) SZEKERES Esther Klein (1910–2005)

### George Szekeres

#### Born Budapest 1911

#### Died Adelaide 28 August 2005

### Esther Klein Szekeres

#### Born Budapest 1910

#### Died Adelaide 28 August 2005

Other than Bernhard Neumann, George and Esther Szekeres were the other two great nonagerian Australian mathematicians in recent years, all three having established their mathematical traditions in Europe before having to make their way out during the 1930s and eventually finding themselves in Australia.

George and Esther first met in a mathematics club in Budapest in 1933 and were married in 1936. They escaped the nazi persecution just in time and arrived at a safe refuge in Shanghai where they stayed during the war. After the war George was offered a lectureship at the University of Adelaide, which he took up in 1948. In 1963 he moved to Sydney to take up a chair at the University of New South Wales. He remained in this position until retirement in 1976.

George remained active in research, still working in an office in UNSW until after his 90th birthday, where he continued to produce significant results. Only in 2004 did they return to live in Adelaide, where both of their children lived and also Esther’s close life long friend Marta Sved (who has in turn since died in Adelaide later in 2005).

Esther and Marta were in the same class at a high school in Budapest and both loved mathematics. Because of quotas only one of them could participate in the national mathematics competition, while the other in the equivalent physics competition. It became history that Marta ended up writing the mathematics competition and finished in the honour list in third place. Number one student in that year, 1928, was George Sved, who later became Marta’s husband.

After retirement George became very active in helping set up Australia’s Mathematics Olympiad structure, particular with the training. George was Deputy Leader of the Australian team in its first years and did much of the training, especially for the students in Sydney. George’s Sunday nights at this time were always reserved at home for students who visited him to work through training problems.

While at UNSW George was the strongest influence in establishing the school mathematics magazine *Parabola*, modelled on similar ideas he had brought from Hungary. Parabola is still published today, via an editorial team from UNSW, but now this magazine is published by the Australian Mathematics Trust, and the equivalent Monash magazine *Function* has been merged in with it.

Esther, like her friend Marta, had a strong teaching commitment. She joined the staff of Macquarie University and was one of the best known academics there during her career.

Known in Hungary as Esther Klein (her maiden name), Esther was featured in the recent film on the life of Paul Erdös, the most famous Hungarian mathematician, who was a friend of both George and Esther, which resulted in Erdös often visiting Australia.

Esther was active in other enrichment activities and was one of those teachers who most frequently worked with the Chatswood enrichment program organised by Terry Gagen.

Both George and Esther had many other interests, including music and art. George in fact played viola in the Ku-rin-gai Philharmonic Orchestra, and both enjoyed bushwalking.

Significantly, both George and Esther have composed separate problems which were set in International Mathematical Olympiad papers.

George was made a Member of the Order of Australia (AM) in 2002.

They are survived by son Peter, who is an mathematical physicist at the University of Adelaide, and daughter Judy.

Peter Taylor

December 2005

(with acknowledgement to information published in the Sydney Telegraph and the Adelaide Advertiser for some details.)

#### VASSILIEV Nikolai Borisovich (1940–1998)

As a problem solver, he was second to none in his generation. Even when he was in his forties and fifties, the speed, depth and elegance of his solutions placed him at least at an equal level with the best performers in the next generations — Dima Fomin of Saint Petersburg, Sergei Konyagin and Sasha Razborov of Moscow, Maxim Kontsevich of Moscow and Paris, Andy Liu of Edmonton. Yet, strangely enough, Kolya Vassiliev had no official achievements in math problem solving contests; in fact, he never actually participated in any. His interest in problem solving (and indeed more generally in mathematics) arose rather late and, in a sense, accidentally, when he was about to graduate from high school, a specialized school for future musicians. Until then, his main interest had been music, and his intention was to enter the Moscow Conservatory in order to study the piano and/or the theory of music. However, he had medical problems with the joints of his fingers, which hampered the prospects of a career as a concert pianist. He began to lose interest in the theory of music because of “the huge amount of boring details” it involved, and so Kolya finally abandoned the idea of a career in music just before graduation. He then began preparing for the entrance exams to Moscow University, which included written and oral tests in mathematics, a subject that he had previously neglected. It was his good fortune to be advised by an excellent mathematics teacher, I.Kh. Sivashinski, who discerned Kolya’s exceptional mathematical talent and convinced him to apply for the Mechanics and Mathematics Faculty of Moscow University, where his talent blossomed.

Kolya’s love of music was never suppressed by his mathematical activities: he was an excellent amateur pianist, an assiduous concert goer, and, most important, the artistic and aesthetic aspect of his personality permeated his work as a composer, compiler, and style editor of mathematics olympiad problems. There is no doubt that *Kvant* magazine and the Tournament of Towns were extremely fortunate to profit, for decades, from the fact that their problem sections were headed by a person combining the depth of a high class research mathematician with the brilliant aesthetic character of a composer of music. For Kolya himself, the most rewarding period of his activity as a chooser and editor of problems was 1967–1979 (which may be called the golden era of Russian olympiads), when he presided the Methodological Commission of the All-Union Olympiad of the USSR, either de jure or de facto.

The choice of problems and of their final formulation, both in *Kvant* and in the Tournament of Towns, was always the result of considerable (sometimes heated) discussion. We feel that Nikolai Vassiliev’s credo in these matters was best expressed in one of these discussions: when one of us asked him why he was so vehemently against the inclusion of a certain problem, Kolya answered: “How does a composer decide whether to make a new melody available to the public or to throw it into the waste basquet? He assesses if the melody will be true gift, if it will give joy to the listener. If not, it is best to forget about it. I choose mathematics problems in the same way.”

Kolya Vassiliev was always a mild, soft-spoken, unassuming person, never raising his voice in anger or irritation, never complaining about the vicissitudes of the life of an underpaid scientist in Moscow. To be in his company in or around the university, or to visit his large bedroom-living room, crowded with overflowing bookshelves, two pianos and ancient furniture, with its striking view of the Kremlin, was always a pleasure and a meaningful intellectual experience.

On all important issues, he was a person who stood firmly for high ethical principles, whether this was to his personal advantage or not. In his work ethics concerning mathematics problems, he was inflexible, and would not rest until the best problems were chosen and their best formulations and solutions had been worked out. In this activity he in fact created a specific style in contest problem formulation: the problems must be natural and attractive, they should involve a minimum of professional mathematical jargon and logical formalization, and yet be unambiguous and rigorously stated. This style has now become the prevailing tradition in many places, but we should not forget that Nikolai Vassiliev, more than anyone else, was its creator.

Besides the style, there is also the substance, what a person leaves behind after he is gone from this world. In that respect, Nikolai Vassiliev’s legacy is exceptional: besides the numerous problems authored and solved personally, there are those that he selected and edited for *Kvant* magazine (over a thousand!), for the Tournament of Towns, for the Moscow Math Olympiad and for the All-Union Olympiad; he is the co-author of five olympiad problem books. Kolya was the person who convinced us that the Tournament of Towns should become international and played a leading role in implementing this idea, not an easy task when Soviet Russia was still behind the iron curtain. He is also, in a sense, the main author of the subject-matter of the three previous volumes in the Tournament of Towns series, as well as of the present volume.

A.A. Egorov, N.N. Konstantinov, A.B. Sossinsky

Moscow

September 1998

#### WILLIAMS James Lewis (Jim) (1917–1993)

#### Born East Newcastle 1917

#### Died Sydney 31 January 1993

Jim Williams was born and raised in Newcastle, attending Newcastle Boys High School. In Newcastle he was a keen surfer and played grade cricket and baseball. (He had a strong love of sport and was also in later life an avid follower of Australian teams in cricket, rugby league and rugby union.)

He attended the University of Sydney and graduated in 1938 with first class honours in physics and first class honours in mathematics. In 1939 he graduated from Sydney Teachers College, winning the Jones Medal. In the same year he set out as a secondary school teacher with the NSW Department of Education.

Jim served with the Royal Australian Air Force during World War 2, first meeting his future wife Dorothy in Melbourne while training, then serving in the Solomon Islands until they were captured by the Japanese in 1942. He then saw further action while variously based in New Hebrides, New Caledonia, Hawaii and Milne Bay. He married Dorothy in Melbourne in 1944 and reached the rank of Squadron Leader.

In 1946 he resumed his role as a teacher. In 1946 and 1947 he became a Lecturer in Mathematics at the Sydney Teachers College and in 1947 was awarded the Gowrie Research Travelling Scholarship, which enabled him to attend King’s College Cambridge. Between 1947 and 1949 he graduated BA with honours in Parts 2 and 3 of the Mathematical Tripos. During this period he also served on the Council of the Mathematical Association (UK).

He returned to Australia, and between 1950 and 1959 he was Head of the Department of Mathematics at Sydney Teachers College. During this time he was a member of all mathematics syllabus committees of the NSW Board of Secondary Studies, was assistant examiner, examiner and chief examiner of mathematics at the NSW Leaving Certificate examinations, member of the primary mathematics syllabus committees of the NSW Department of Education and had become a part-time Lecturer in Mathematics at the University of Sydney.

From 1950 to 1955 Jim was Editor of the *Australian Mathematics Teacher*.

In 1952 he graduated MSc at the University of Sydney and in 1954 MA at Cambridge University.

From 1959 to 1964 Jim was Senior Lecturer in Mathematics at the University of Sydney, and from 1964 to 1975 he was Director of First and Second Year Studies in Pure Mathematics there. During this time he co-authored, with Alistair McMullen, the very successful series of secondary school mathematics texts *On Course Mathemtics* published by Macmillan Co. of Australia.

From 1950 to 1975 Jim was a member of the Executive of the Mathematical Association of NSW (MANSW). In 1967 and 1968 he was its President. Through this time he was also a member of the National Council for Teachers of Mathematics (US). He was also a regular contributor of articles to the Australian Mathematics Teacher and to the Mathematics Bulletin of the NSW Department of Education, and Lecturer at Mathematics Seminars organised by the NSW Inservice Training Department. In 1975 Jim was made an Honorary Life Member of MANSW.

In 1964 he was a member of the organising committee for the UNESCO Mathematics Conference held in Sydney.

In 1969 Jim was awarded a Carnegie Travelling Scholarship and spent a year in the US investigating mathematics education at secondary and tertiary levels, attending NCTM Conferences, and taking part in Summer Schools.

From 1973 to 1976 he was Convenor of a Committee which prepares published solutions to the Higher School Certificate examination papers for MANSW. In 1975 he inaugurated a NSW Mathematical Olympiad for Year 11 students based on the USA Mathematical Olympiad. This was later renamed by MANSW as the JL Williams Competition.

During the late 1970s this experience enabled Jim to become one the the main lobbyists for Australia to becaome a participating country at the International Mathematical Olympiad (IMO) and from 1979 to 1986 he was the National Director of Training for the Australian program and Team Leader for Australia. The last of these events was the first of the IMOs attended by Terry Tao. In this time Jim was also a member of the Problems Committee of the Australian Mathematics Competition, which had just begun.

Jim working with colleagues Bob Bryce (ANU, left) and John Mack (Sydney) at a meeting of the Australian Mathematics Competition Problems Committee in 1979.

Jim with the 1981 Australian IMO team, Australia’s first.

Jim, centre front, as member for the 1983 Problems Committee of the Australian Mathematics Competition.

Andrew Hassell, Australia’s first IMO Gold Medallist, being congratulated in Helsinki, 1985, by Team Leader Jim Williams after learning the result.

Jim as Leader and Geoff Ball as Deputy,in Warsaw, with the 1986 IMO team which includes a 10-year-old Terry Tao.

In 1992 Jim was honoured with the presentation of a BH Neumann Award of the Australian Mathematics trust.

In his MANSW obituary in 1993 for Jim, Jim’s friend Geoff Ball noted that Jim was a *character*. Jim was certainly that. In his presence Jim was larger than life, always positive, cheerful, a great sense of humour, strongly batting for the presence of geometry at all occasions. For people like Geoff, who worked with him for many years, he was a major inspiration. Even for others of us who only knew Jim in his later career, he also was. Jim, more than anyone, epitomised a great and colourful period of mathematics and mathematics teaching, in New South Wales and beyond.

Peter Taylor

December 2012

(based to a large extent on information supplied by his family)

## Historical Biographies

#### DIRICHLET Peter Gustav Lejeune (1805-1859)

Peter Gustav Lejeune Dirichlet was born in Düren, then in the French Empire, but now in western Germany, on 13 February 1805 and was educated at the University of Göttingen, where Carl Friedrich Gauss was one of his mentors. He was fluent in both French and German and as such was often involved in communicating ideas between French and Geman mathematicians.

He made major contributions in the fields of number theory, analysis and mechanics, and taught in the Universities of Breslau (1827) and Berlin (1828-1855) before succeeding Gauss at the University of Göttingen.

It was Dirichlet who proposed (in 1837) the Theorem in his name which states the exisence of an infinite number of primes in any arithmetic series *a*+*b*, *2a*+*b*, *3a*+*b*, …, *na*+*b*, in which neither of *a* nor *b* are divisible by the other. For example, 5, 11, 17, 23 and 29 are among the primes of the form 6*n*+5.

Independently, he and Legendre independently proved Fermat’s Last Theorem for the case *n*=5, reportedly using an idea suggested by Sophie Germain. Actually, Dirichlet’s proof was published in 1825 and reportedly had an error which was corrected by Legendre.

He developed the theory of units in algebraic number theory and made major contributions to the theory of ideals.

In 1837 he introduced the modern concept of a function with notation *y*=*f*(*x*) in which *y* is uniquely determined by the value of *x*. This work was inspired by significant contributions he had made to the understanding on Fourier Series, particularly with respect to conditions of convergence.

In mechanics he investigated the solutions of boundary problems with partial differential equations in the interior of a region. If the unknown is prescribed on the boundary of the region the problem is known as a Dirichlet problem. The minimisation principle with respect to certain classes of boundary value problems is oftern known as Dirichlet’s minimisation principle.

It was Dirichlet who formulated the Pigeonhole Principle, often known as Dirichlet’s Principle, which states

*If there are p pigeons placed in h holes and p>h then there must be at least one pigeonhole containing at least 2 pigeons.*

It was he who underlined the use of its systematical application as a powerful tool for creating mathematical proofs. Dirichlet’s Principle enables short and elegant solutions of problems, previously attacked by complicated constructive methods.

Dirichlet died on 5 May 1859, in Göttingen.

Written by Peter Taylor, January 2001.

A number of historical references were used in composing this short biography. They included

*Men of Mathematics*, ET Bell, Simon and Schuster, New York, 1937, 1965.*An Introduction to the History of Mathematics*, Howard Eves, 4th ed,, Holt, Rinehart and Winston, New York, 1976.

Dirichlet is the subject of the Australian Mathematics Trust T Shirt in 2001.

This is available for sale through AMT Publishing

#### EULER Leonhard (1707-1783)

Leonhard Euler was the most published mathematician of all time. There is probably not a single branch of mathematics known during his lifetime which he did not influence. If a difficult problem arose, Euler was generally consulted, and could often solve it.

Euler was born near Basel, Switzerland and raised in the village of Riehen. His father was a Protestant Minister, and his mother was also from a clerical family. He was expected to follow his father into the clergy. He was an able student, mastering languages and mathematics and a memory for matters of detail.

He entered the University of Basel at the age of 14. A Professor of mathematics there was Johann Bernoulli (1667-1748), arguably the world’s greatest active mathemtician. Euler became a good friend of Bernoulli, who became his mentor. Both men appeared to have inspired each other greatly during their regular meetings. He obtained a Bachelor of Arts and Master of Philosophy Degree from Basel University.

He did afterwards enter divinity school but found the call of mathematics to be greater. Bernoulli’s son Daniel (1700-1782) moved to Russia in 1725 to take up a position at the newly formed St Petersburg Academy. In the following year Euler was invited to join him and he arrived in the year 1727. Living at the same home as Daniel Bernoulli Euler was able to discuss and collaborate with him extensively.

At about this time his work on exponential functions led him to introduce the constant *e*, the symbol for the important transcendental number 2.71828… . He also discovered the result

linking *e*, *π*, and *i*, the symbol he developed for the square root of -1.

In 1733 Bernoulli moved to a Chair in Switzerland. This enabled Euler to move from a post in Physics to take up Bernoulli’s Chair in Mathematics. He married Katherine Gsell (d. 1773) and they had 13 children, only five of whom reached adolescence and three of whom survived him.

This was a period during which Euler did much consulting work for the Russian Government and publishing many results, including the solution to the much debated Basel Problem in 1735 (see below).

In 1736 Euler solved the Königsberg Bridges Problem, which is described below. This solution established the branch of mathematics now known as Graph Theory, and which is the basis of the understanding of networks, including computer networks.

Whereas Euler’s research continued at an astonishing pace, there were some problems encountered during the next period, including the death of Catherine I, a subsequent backlash against the foreigners who dominated the Academy, and in 1738 the first signs of failing eyesight, with the loss of sight from his right eye.

During this time he still produced ground-breaking works, including work on shp-building, acoustics, music, Classical Number Theory in collaboration with Christian Goldbach (1690-1764), Analytic Number Theory, and a text *Mechanica* presenting Newtonian mechanics in a framework of Calculus.

In 1741, while still in the employ of the St Petersburg Academy, Euler and his family moved to Berlin at the invitation of Prussia’s Frederick the Great (1712-1786) to join the revitalised Berlin Academy. He was to stay in Berlin until 1766.

In Berlin he published his most widely read book, *Letters to a German Princess*, which contains over 200 “letters” inspired by the instruction he was required to give to the Princess of Anhalt Dessau. The letters cover a wide range of topics in mathematics and physics, including the explanation of commonly observed phenomena. It is a classical example of excellent writing to explain science to the masses.

During his time in Berlin, Euler kept in excellent contact with the St Petersburg Academy, which was still paying him, and fell out gradually with Frederick the Great. While in Berlin he also fell out with the other leading identity Voltaire (1694-1778) who was more in favour with the King and was rather disdainful of Euler, who had not learned philosophy. While absent the St Petersburg Academy had also been revitalised under the influence of Catherine the Great (1729-1796) and in 1766 he returned to St Petersburg for the remainder of his life.

Euler’s work in St Petersburg continued at a breathtaking pace despite the death of his wife (he later married her half-sister) and the substantial loss of sight in his good eye, forcing him to dictate all of his writings to scribes. He died of a massive hemhorrhage on the afternoon of 18 September 1783, a day on which he had still been working at his normal pace. The St Petersburg Academy Journal had a massive backlog of his work to publish, a task which took a further 48 years to complete.

The complete works of Euler, *Omnia Opera*, was only published in the latter part of the twentieth century after a commitment by the Swiss Academy of Science in 1909. It is very expensive and can only be found in major research libraries. It comprises 29 volumes on mathematics, 31 on mechanics and astronomy, 12 on physics and other topics, 8 on correspondence. Further volumes on manuscripts is still to appear.

Euler’s work took him into virtually every branch of mathemtics and physics known during his life. Here we briefly discuss some problems for which he became famous. The individual problems discussed below indicate the flavour of Euler’s work and do not indicate his massive contribution to what we now call applied mathematics.

#### The Königsberg Bridges Problem

Königsberg (now the Russian city of Kaliningrad, on the Baltic Sea) was a city in East Prussia laid out on the River Pregel, which had split into two courses forming two islands. The various regions of the city were connected by bridges.

The left hand diagram shows the layout of the river and the seven bridges. The citizens of Königsberg had tried unsuccessfully to find a route along which they could tour the entire town, traversing each bridge exactly once.

Euler solved the problem by showing why such a route could not be found. Essentially he showed that the region could topologically be considered as having four regions A, B, C and D as shown in the left hand diagram. He then demonstrated that a solution of the problem could then be considered equivalent to finding paths through the network in the right hand diagram. If such a solution was to be possible, each path of the network would be travelled exactly once. The points A, B, C and D could be called nodes of the network. Whereas a node could be visited more than once in a successful tour Euler showed that successful tours depended on nodes being arrived at and departed from different routes each time, requiring even numbers of routes connecting each notes (number of arrivals matching number of departures).

In the case of the Königsberg Bridges problem, as can be seen from the right hand diagram, all of the nodes have an odd number of connecting routes, making the solution impossible.

As an extension of the above, Euler developed a theory for networks, in which lines join nodes and enclose regions. Euler developed the formula

*V*+*R*–*L*=1.

where *V* is the number of verices (nodes) in the network, *R* is the number of regions (enclosed areas) in the network, and *L* is the number of lines in the network, satisfied by a network.

For example, this is obviously satisfied by the right hand diagram describing the bridge network above, in which *R*=4, *V*=4 and *L*=7.

#### The Basel Problem

For several decades there was much speculation about the value of the sum of the infinite series

This problem was known as the Basel problem. It seemed clear that the real sum was a number in the vicinity of 8/5. The problem had received much attention from Pietro Mengoli (1625-1686) and Jakob Bernoulli (1654-1706), brother of Johann and uncle of Daniel. Euler was able to solve this problem in 1735, when he caused a major sensation by showing that the sum had the unexpected value

#### The card problem

In his work on infinite series, Euler also investigated the constant

which bears his name and showed how it can be used for estimating the sum of the finite series

since

This applies to computing the expected number of packets of chewing gum, cereal, etc one needs to buy when the manufactures place a hidden card inside the packet, numbered as to form a collector’s set. If there are *n* cards in the set it can be shown that the expected number of packets *N* one needs to buy before completing the set is

For *n*=25, it turns out after much calculation that the exact value is *N*=95.4. However Euler’s formula gives with much less effort the very accurate approximation 94.9.

#### The Gamma function (Extension of the factorial function)

Euler extended the concept of factorial, so useful in combinatorics, infinite series and elsewhere, defined as

where *n* is integer. Euler was able to show that the function

satisfied

for *n* integer and had wide power as a generalised factorial function, that is for arguments which are not necessarily integer. The function also has the unexpected value

#### Fermat’s Last Theorem

Pierre de Fermat (1601-1665) posed one of the most famous Theorems in Mathematics, stating that the equation

has no integer solutions for *x*, *y* and *z* when *n* is a positive integer for *n* greater than 2.

Fermat himself was able to construct an argument to show that there was no solution for *n*=4. The next advance was not until 1765, when Euler was able to announce a proof for the case *n*=3 to his friend Christian Goldbach.

In later years Dirichlet, others and computers were able to extend the cases, but it was not until virtually the end of the twentieth century that this theorem was to be finally proved, by the English mathematician Andrew Wiles.

Written by Peter Taylor, June 1997, revised March 2001.

**References**

- Boyer, Carl B, revised by Merzbach, Uta C,
*A History of Mathematics*, 2nd ed, Wiley, New York, 1991. - Dunham, William,
*Euler: The Master of us all*, Dolciani Mathematical Exposition No 22, Mathematical Association of America, Washington DC, 1999. - Green DR, Euler,
*Mathematical Spectrum*,**15**, 3, Sheffield, 1982/3. - Gullberg, Jan,
*Mathematics: From the Birth of Numbers*, Norton, New York, 1997. - Singh, Simon,
*Fermat’s Last Theorem*, Fourth Estate, London, 1997.

This T Shirt, which celebrates Euler’s solution to the Königsberg Bridges problem, is available from the AMT Publishing.

#### GAUSS Carl Friedrich (1777-1855)

Carl Friedrich Gauss has born in Braunschweig, Germany. His parents were poor. His father Gerhard, a labourer, canal tender and bricklayer, encouraged him only to what he saw as useful, labouring tasks.

His mother, Dorothea, who herself had a talented younger brother Friedrich, recognised his talents and encouraged him to pusrue them. In return Gauss looked after her, especially in her later years after she had become blind and until her death in 1839.

Gauss’ talents came to the notice of a school master when he quickly solved the task of adding the numbers from 1 to 100 and later he received the sponsorship of the Duke of Braunschweig to study at College, where he maintained an interest in Philology as well as Mathematics.

He later studied at the University of Göttingen, where he now focussed on Mathematics.

As a student he made major discoveries, including the Method of Least Squares and the discovery of how to construct the regular 17-gon. The latter result was highly significant. Since the time of Euclid mathematicians had known only how to construct with compass and straight-edge regular n-gons in which *n* was a multiple of 3, 5 powers of 2 or combinations thereof. Gauss’ discovery added to these numbers prime numbers of the form 2^(2^*n*)+1. For *n*=0 and 1 this included 3 and 5 but for *n*=2, 3 and 4 this added 17, 257 and 65,537 to the list.

In later life, after having a profound influence on mathematics, Gauss still regarded this as one of his greatest achievements and asked that a regular 17-gon be placed on his tombstone (unsuccessfully, as it happened).

Gauss went on to be awarded a Doctorate in Philosophy at the University of Helmstedt in 1799, with a thesis which proved that every rational integer function of one variable can be resolved into real factors of the first or second degree. This was a major unsolved problem, commonly known as the Fundamental Theorem of Algebra, and had been believed to be true by Euler.

Gauss in 1801 published a major work in Number Theory, “Disquisitiones arithmeticae” which recast much of 18th century Number Theory, but many of his own discoveries, including the Law of Quadratic Reciprocity, for which he found independent proofs.

For some years Gauss lived in Braunschweig and had six children from two marriages. He is believed to have many descendants in Germany and the United States. He was supported by the Duke of Braunschweig until his death and in 1807 received a flattering offer from St Petersbourg, which had never satisfactorily replaced Euler.

The Baron Alexander von Humboldt, an amateur patron of the sciences managed to get him appointed as Professor of Mathematics at Göttingen, and Director of the Göttingen Observatory, posts which he held to his death.

It is impossible in a short resume to refer to all of his discoveries, not only to mathematics, but also physics, statistics (he introduced the normal distribution), astronomy and geodesy.

As an analyst, he was the first to develop adequate standards of proof of results involving infinitely many numbers. He anticipated the development of non-Euclidean geometries.

For a great mathematician, Gauss published very little, sometimes having his results independently discovered by others. He kept a methodical diary which recorded his results, but applied to himself the strictest standards about the way in which his work would be published.

Gauss asserted that “Mathematics is the Queen of Sciences, and the Theory of Numbers is the Queen of Mathematics”.

Written by Peter Taylor, June 1998.

This T Shirt, which celebrates Gauss’ discovery of the 17-gon construction, is available from the AMT Publishing.

#### NEWTON Sir Isaac (1642-1727)

Sir Isaac Newton was born to a farming family in Woolsthotpe, Lincolnshire on 25 December 1642 (4 January 1643 in the Gregorian calendar). He received a bachelor’s degree at Trinity College Cambridge in 1665, became a Fellow of Trinity College in 1667 and Lucasian Professor of Mathematics at Cambridge in 1669.

During his period at Cambridge he became responsible for some of the most enduring results in mathematics and physics. He co-discovered the Calculus, with Gottfried Wilhelm Leibniz.

As a physicist he discovered the composition of white light. He explored the mechanics of planetary motion and proposed the three fundamental laws of mechanics, the law of gravitation and the inverse square law.

In 1687 he published the “Principia”, generally regarded the most significant scientific book ever written, and embodying his discoveries.

In 1672 Newton was elected as a Fellow of the Royal Society.

Newton became Warden of the Royal Mint in 1696 and Masterof the Mint in 1699.

In 1703 Newton was elected President of the Royal Society and re-elected as such every year until his death.

He published “Opticks”, which discussed Newton’s rings and diffractions as well as his theories on the composition of light. in 1704. He was knighted by Queen Anne in 1705. In his later life he did not engage in scientific research. He took a direct interest in the operations of the Mint and became wealthy.

Newton’s life was clouded in controversy with a number of people, such as Hooke, over the origin of the inverse square law and who had claimed he stole ideas in optics from him, and with Leibniz over the origins of the calculus.

Newton died in London in 1727.

Written by Peter Taylor, January 2001.

A number of historical references were used in composing this short biography. They included

*Men of Mathematics*, ET Bell, Simon and Schuster, New York, 1937, 1965.*An Introduction to the History of Mathematics*, Howard Eves, 4th ed,, Holt, Rinehart and Winston, New York, 1976.

Newton is the subject of the Australian Mathematics Trust T Shirt in 2003.

This is available for sale through AMT Publishing.

#### NOETHER Emmy (1882-1935)

Emmy Noether is one of the most significant female mathematicians in history. She was born in the Bavarian town of Erlangen. Erlangen at the time had one of Germany’s three “free” Universities (i.e. independent of the churches), the other two being at Halle and Göttingen. The Erlangen University had been cast into the mathematical spotlight by one of its mathematicians named Felix Klein, who had given significant insights into the concept of a group in geometry, insights which became known as the “Erlangen Program”. Emmy Noether’s father, Max Noether, was a mathematician at Erlangen. He was a significant mathematician in his own right and became a Full Professor at that University.

Women were not officially allowed to study at German Universities, or to hold normal teaching positions. Nevertheless Emmy became known while enrolled as an audit student and was able eventually (in 1907) to graduate with a PhD summa cum laude at Erlangen under the supervision of Paul Gordan (whom David Hilbert had described as “King of the Invariants”).

In 1915 she moved to Göttingen where she was given a licence to teach without being paid. Hilbert was in fact one of her colleagues there. Her most productive years were during the 1920s, when she produced a number of significant results. She is best known for her work in abstract algebra, particularly working with structures such as rings. She also did important work on the theory of invariants, which had an influence on the formulation of Einstein’s general theory of relativity.

Also during the 1920s she spent short periods as Visiting Professor at Frankfurt and Moscow. In 1933 the Nazis withdrew her licence to teach. She left Germany and emigrated to the US, where she took up a Faculty position at Bryn Mawr, a Women’s College in Pennsylvania. Bryn Mawr was not far from Princeton, where Einstein had recently arrived. Emmy Noether also gave weekly lectures there. She died suddenly on 14 April 1935 at Bryn Mawr. It is significant that Albert Einstein wrote a deeply respectful commentary which was published in the *New York Times* on 1 May 1935. In this commentary Einstein said

*Within the past few days a distinguished mathematician, Professor Emmy Noether, formerly connected with the University of Göttingen and for the past two years at Bryn Mawr College, died in her fifty-third year. In the judgement of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of importance in the development of the present-day younger generations of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relations. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper presentation into the laws of nature.*

#### A Brief Insight into Emmy Noether’s Work

Emmy Noether’s results were mainly in the area of algebraic structure. Einstein (above) gives some broad ideas on the potential outcome of a knowledge of these structures. More specifically, today, a knowledge of these structures gives insight into the optimal way in which computers may be designed, computation can be performed and how data can be optimally stored.

Emmy Noether worked on such structures as ideals, rings and chains.

A *ring* is an abstract structure in which the objects are subject to two operations (such as addition and multiplication) and satisfy a number of axioms (rules). These axioms require the existence of certain laws, such as the associative law which must be satisfied by these operations, and the ring must include a zero element. The simplest example is the ring of integers **Z**, which consists of the well known numbers …,-3,-2,-1,0,1,2,3,… (i.e. the positive and negative integers, including zero). Any two numbers can be added or multiplied (the two operations) to give a result which is also in the ring. For example 4 and -3 are two members of **Z** and 4x-3=-12, another member of **Z**.

An *ideal* of a ring is a subset of the ring (i.e. a structure whose elements are all in the ring), which is a ring itself, and furthermore satisfies the rule that if any element of the ideal is multiplied by any member of the ring, the result is a member of the ideal. In the case of **Z** the only ideals are the sets of integers divisible by a given integer. For example the ideal 2**Z** is the set of integers divisible by the number 2. In this case it is the set …,-4,-2,0,2,4,… . Multiplication of any of these elements even by any number (even or odd) still leads to an even result. Also there is are ideals 3**Z**, 4**Z**, etc.

A *chain* is a relationship in which ideals are linked by the subset relation. For example all numbers divisible by 6 are also divisible by 3. So we can say 6**Z**<3**Z** (where we use here the symbol “<” to mean “is a subset of”, rather than the normal symbol, because of font constraints) is a chain in which the first term is 6**Z**. In fact we would call this an ascending chain in which the first term is 6**Z** because each term (ideal) in the chain is a subset of the next. Since numbers divisible by 6 are also divisible by 2, this ideal also leads to the chain 6**Z**<2**Z**. Chains can be extended in length. For example we can also write 6**Z**<3**Z**<**Z** or 6**Z**<2**Z**<**Z** are both chains commencing with 6**Z**. Note that both of these chains cannot be further extended upwards. We say that they are *finite*.

Emmy Noether’s name is perpetuated as the name for a ring in which every (ascending) chain of ideals is finite, as it is demonstrably in the case of **Z**.

The 1999 T Shirt of the Australian Mathematics Trust indeed commemorates Emmy Noether, after whom one of our Mathematics Enrichment courses is named, but also lists each of the eight chains which commence with 18**Z**.

18**Z**<9**Z**<3**Z**<**Z**

18**Z**<6**Z**<3**Z**<**Z**

18**Z**<6**Z**<2**Z**<**Z**

18**Z**<9**Z**<**Z**

18**Z**<6**Z**<**Z**

18**Z**<3**Z**<**Z**

18**Z**<2**Z**<**Z**

18**Z**<**Z**

**Reference**

*Emmy Noether*, Auguste Dick, Translated by HI Blocher, Birkhäuser, Basel, 1981.

Note that the photo above is reproduced from this reference with permission from the publisher. It shows Emmy Noether en route from Swinemünde to Königsberg (aboard the Steamship *Danzig*) to attend the annual meeting of the German Mathematical Society, September 1930. The photo was taken by her collaborator Helmut Hasse.

Written by Peter Taylor, March 1999.

This T Shirt, the finite ideal chains of 18**Z**, is available from the AMT Publishing.

#### PÓLYA George (1887-1985)

George Pólya was born in Budapest on 13 December 1887. His father Jakab (who died in 1897) had been born Jakab Pollák, of Jewish parents, and with a surname which suggested Polish origin. It is likely that ancestors had emigrated from Poland to Hungary, where a lesser degree of anti-Semitism existed. However Jakab converted to Catholicism believing that this would help him advancing in a career and changed his name to the more Hungarian Pólya. George’s mother had also been of Jewish background with similar history. Her paternal grandfather, Max Deutsch, had in fact converted to Presbyterianism and worshipped with Greek Orthodox Romanians.

George’s father Jakab had been a solicitor with a great mind, but one who was prepared to pursue a case in which he believed with no fees. He was not financially successful despite the time he lived in being considered a golden age for Hungary.

As a student George attended a state run high school with a good academic reputation. He was physically strong and participated in various sports. His school had a strong emphasis on learning from memory, a technique which he found tedious at the time but later found useful. He was not particularly interested in mathematics in the younger years. Whereas he knew about the Eötvös Competition and apparently wrote it he also apparently failed to hand in his paper.

He graduated from Marko Street Gymnasium in 1905, ranking among the top four students and earning a scholarship to the University of Budapest, which he entered in 1905. He commenced studying law, emulating his father, but found this study boring and changed to language and literature. He had become particularly interested in Latin and Hungarian, where he had had good teachers. He also began studying physics, mathematics and philosophy. His development was greatly influenced by the legendary mathematician Lipót Fejér, a man also of wit and humour, who also taught Riesz, Szegö and Erdös. Fejér had discovered his theorem on the arithmetic mean of Fourier Series at the age of 20.

Pólya soon concentrated his studies on mathematics and in 1910 finished his doctorate studies, except for his dissertation. He took a year in Vienna and returned to Budapest in 1911-12 to give his doctoral dissertation and met Gábor Szegö, seven years younger, who was to become one of his major collaborators.

In the fall of 1912 he went to Göttingen for postdoctoral study and met David Hilbert, Richard Courant, Felix Klein and Hermann Weyl. In 1913 he was offered a position in Frankfurt, but was discouraged from staying in Germany and turned the job down after being told he was a “bloody Jew” by a ruffian on a train and went on to the University of Paris for further postdoctoral work.

In 1914 he took up a position at the Eidgenössische Technische Hochschule (ETH) in Zurich, an institution which boasted the names of the physicists Röentgen and Einstein (1900) among its graduates. This position was arranged by the mathematician Adolf Hurwitz (1859 to 1919), who had studied at various times under Kronecker, Klein and Weierstrass and was the other great influence on Pólya. The ETH was next door, and closely associated with, the University of Zurich and Polya had joint teaching rights with the University.

In 1914 Pólya was called up by Hungary to fight in the war, but by this time he had adopted Russell’s pacifism and refused to go. The fear that he might be arrested for being unpatriotic meant that he did not return to his native country until after World War II. In Zurich he met his future wife, Stella Weber. They married in 1918 and were still together 67 years later when Pólya died. They had no children.

Inspired by walks in the woods near Zurich, Pólya in 1912 published one of his major results, the solution of the random walk problem. In this problem one walks in an infinite rectangular grid system, at each node having an equal probability of walking to each of the adjoining nodes on his next leg. Pólya was able to show that in the two dimensional case it was almost certain (but with probability 1) that one would eventually return to the original position, but one would almost never (with probabilty 0) return to the origin in the case of three or more dimensions.

Pólya was interested in chemical structure, which led him in 1924 to publishing the classification of seventeen plane-symmetry groups, a result which was later to inspire the Dutch artist M.C. Escher.

In 1924 he spent a year in England, working with G.H. Hardy and J.E. Littlewood at Oxford and Cambridge. This collaboration led to publication in 1934 of the book *Inequalities*, which included a new proof by Pólya of the AM-GM inequality based on the Maclaurin expansion of the exponential function.

In 1925 Pólya, with Szegö, published arguably one of his most influential books, *Aufgaben und Lehrsätze aus der Analysis*, volumes 19 and 20 of the series *Die Grundlehren der Mathematischen Wissenschaften* published by J. Springer, Berlin. A whole generation – the generation of Erdös, Szekeres and their circle, and later, learned their mathematics not so much from the lectures they attended but by trying to solve the problems of this book one after another and debating their solutions with each other. Problem solving as a method of teaching and learning may never have been practiced on such a scale, and with such success, before (or since).

One of Pólya’s most famous results, the Pólya Enumeration Theorem, was published in 1937. This also arose from his interest in chemical structure and looking at possible configurations of the benzene ring and other figures with 6 vertices. Generalising a theorem by Burnside in Group Theory, Pólya showed how one can determine the number of different assignments of atoms, or colours, to vertices as sides of geometrical figures.

A special case of this is the Necklace Theorem, which shows how many necklaces of *n* beads can be constructed with *k* colours available, assuming there is an infinite supply of each colour.

In the case where *n* is prime and necklaces are regarded as unchanged by rotation, the number of configurations is *k*+(*k*^*n*–*k*)/*n*. Thus if there are 5 beads and 3 colours the number of necklaces is 3+(3^5-3)/5=3+240/5=3+48=51.

In the case where *n* is composite there is also a formula but it is a little more complicated. However more details of this and another version of the theorem can be found here.

In 1940 the Pólyas became increasingly concerned, with George’s Jewish background, of the possibility of a German invasion of Switzerland, and decided to leave for the United States.

He was offered a research position by his old collaborator, Gábor Szegö, now at Stanford, but he did not initially accept it, going instead to Brown University. In 1942 he did move to Stanford, however, where he stayed until his retirement from teaching in 1953. After 1953 he stayed at Stanford, living at Palo Alto until his death, as a Professor Emeritus.

In 1945, Pólya published one of his most famous books, *How to Solve It*. Then in 1951 he published, with Gábor Szegö, *Isoperimetric Inequalities in Mathematical Physics*.

Gábor Szegö had been a winner of the Eötvös Competition in 1912 and Pólya saw the value in competitions. In 1946 Pólya and Szegö founded the Stanford University Competitive Examination in Mathematics. In its first year 322 students from 60 schools in California entered. The competition grew to having typically 1200 students from 150 schools in 3 western states. However the competition was terminated in 1965 when Stanford shifted its emphasis to postgraduate study. Pólya however continued his activity in this area by publishing problem material in books and journals.

Pólya was particularly interested in the high school curriculum and was concerned about the new maths curriculum. He eventually saw the curriculum change back to basics and was not happy with the way this happened either.

In 1954 he published the two volume book *Mathematics and Plausible Reasoning* and in 1962 and 1965 a further two volume set entitled *Mathematical Discovery*.

From his retirement in 1953 Pólya took an active interest in improving the standard of teaching and took steps to establish, with NSF funding, eight-week Summer Institutes for mathematics teachers, first at the college level (1953-1960), then later for teachers of high school and eventually moving the Institutes to Switzerland.

George Pólya died on 7 September 1985.

#### References

*George Pólya, Master of Discovery, 1887-1985*, Harold and Loretta Taylor, Dale Seymour Publications, 1993.*The Random Walks of George Pólya*, Gerald L Alexanderson, MAA, 2000

With thanks to a useful suggestion from Laci Kovács.

Written by Peter Taylor, June 2000. Adapted slightly January 2008 with help from Kevin McAvaney.

This T Shirt, which celebrates the Necklace Theorem, is available from the AMT Publishing.

#### TURING Alan (1912-1954)

Alan Mathison Turing was born on 23 June 1912 in a nursing home in Paddington, London. He was the second of two boys. His grandfather, John Robert Turing had taken a mathematics degree in 1848 at Trinity College, Cambridge, being placed 11th. However he gave up mathematics to take up the ministry. He fathered ten children. The second son, Julius Mathison Turing, born in 1873, was reputedly not as able as his father in mathematics, and studied literature and history at Corpus Christi College, Oxford, graduating in 1894. He became a member of the Indian Civil Service.

Julius Turing was to meet his future wife and Alan Turing’s mother, Ethel Stoney, on a voyage home in 1907. Ethel Stoney was descended from a family originally from Yorkshire, but which had become Protestant landowners in Ireland, and she had been born in Madras, where her father had gone as an engineer in 1881. She had a distant relative who was the Irish scientist responsible for coining the word “electron”. Both Alan Turing’s parents had come from families which were resourceful, although having funds.

Alan Turing was conceived in India but born in London and never actually went to India. His parents decided he should not be exposed to the heat and other risks of Madras and left him and his older brother in the care of another family (later, when Turing was at school, his father did prematurely retire and his parents returned to Britain).

In 1926 Turing sat the Entrance Exam for Sherbourne College and was accepted. This was one of the original English Public Schools, based in Dorset. At this school Turing showed a talent in mathematics and interest in some of the sciences but little interest in the humanities.

Whereas his first choice had been to study at Trinity College, Cambridge, Turing was unsuccessful in gaining a scholarship there. He was however successful in his second choice, King’s College, which he entered in 1931, reading for the Tripos degree in mathematics. He enjoyed his new-found freedom here, with a room to himself and the academic climate enriched at Cambridge with the presence of GH Hardy (who returned from Oxford in the same year to take up the Sadleirian Chair), Eddington, Dirac and the eminent topologist MHA Newman. During this period Turing remained a Pure Mathematician, although he dabbled with interests in quantum physics. He was particularly interested in theoretical studies which had interface with the physical world.

An example of this was his proof of the Central Limit Theorem in 1933. Turing was inspired to investigate this by a course given by Eddington, where he noticed that irrespective of the distribution of a set of *n* sample points, the mean of these points always had the same mean as that of the original sample, had a variance equal to the variance of the original distribution but divided by *n* (making it smaller for large numbers of data points, as one would expect) but furthermore was **normally** distributed. After proving it, Turing was told that this very difficult and famous theorem had been proved in 1922 by Lindeberg. However it seems that his original work was acknowledged as such.

While at Cambridge Turing flirted with the anti-war movement. He studied the Soviet system but concluded there was a totalitarian aspect to it and became identified with the middle-class progressive opinion espoused by such journals as the *New Statesman*. Turing also thought deeply about religion and agnosticism.

In 1934 Turing wrote the Tripos examination and passed with Distinction. He and eight others were made B* Wranglers. Turing was not particularly interested in the exact status, regarding this as resulting only from an exam. He was interested in further things and had been inspired by a course given by Newman on the Foundation of mathematics. This course had largely been given in the spirit of Hilbert’s attempts to define mathematics. In 1900 at the International Congress of Mathematicians Hilbert had posed a number of unsolved problems. Hilbert had continued to consolidate his position as one of the world’s leading mathematicians as Professor at Goettingen. Germany had been forbiddden representation at the 1924 Congress but Hilbert did attend in 1928 and posed three fundamental questions:

- Was mathematics
*complete*, in the sense that every statement could be proved or disproved, - Was mathematics
*consistent*in the sense that no apparently incorrect statement such as “1+1=3”, could be arrived at by a sequence of valid steps of proof, and - Was mathematics
*decidable*, in that did there exist a definite algorithm which could, in principle, be applied to any assertion and which was guaranteed to produce a correct decision as to whether that assertion was true?

It is believed that Hilbert expected the answer to each question to be “yes”. As it happened the Czech logician Kurt Gödel answered the first two questions, answering the first in the negative by producing self-referential statements such as “This statement is unprovable”. He did answer the second question in the affirmative. To answer the third question in the negative one would need to find an assertion for which no algorithm existed. Indeed, the definition of algorithm itself needed to be carefully addressed.

Turing was appointed to a Fellowship at Cambridge after his graduation, granting him considerable scope to continue research and in addition to research in group theory, pondered Hilbert’s third question. He found the answer to be in the negative by devising a concept called a computable number and mecanical processes (later to be called Turing Machines). A Turing machine (as it later became known) was an abstract mechanical process which changed from state to state by a finite set of rules depending on single symbols read from a tape (rather like the program on a modern computer). A Turing machine could write a symbol on to the tape or delete a symbol from it. He defined a computable number as a real number whose decimal expansion could be produced by a Turing machine starting with a blank tape. He was able to demonstrate that the irational number pi was computable. However he was able to produce a diagonalisation argument not unlike that of Cantor when proving the uncountability of the real numbers to show that there was an uncountable number of numbers which are not computable. He was also able to describe such a number.

This result was also independently produced by the Princeton mathematician Alonzo Church, but Turing’s method, involving the idea of the Turing Machine (a term later coined by Church) provided an avenue for more significant later development.

After some lectures given at Cambridge by Princeton’s John von Neumann and discussion with Newman, Turing moved himself to Princeton in 1936 on a Fellowship which enabled him to collaborate with Church and others. During his time in Princeton Turing helped develop some machines to multiply numbers, and to compute zeros of the Riemann zeta-function (which he thought he might be able to prove false by finding a zero with the wrong value). The Riemann zeta-function problem, which can establish the way in which primes “thin out” for large numbers, was posed as one of the important unsolved problems in mathematics by Hilbert in 1900. It remains, in 2002, as the Everest of unsolved mathematics problems. Turing also started to think of how to develop high quality cipher which was easy to encrypt.

During 1937 Turing had his Fellowship renewed at Princeton for a second year. His paper *Computable Numbers* was published while he was at Princeton and he also completed a PhD further exploring Gödel’s theories.

In 1938 he was offered an attractive position at Princeton by von Neumann and also had his Fellowship at King’s renewed. He decided he wished to return to King’s, and commenced lecturing there and working on further developments with calculating zeros of the Riemann zeta-function.

After the outbreak of war Turing’s life changed from that of a University Pure Mathematician. He was recruited by HM Government’s Code and Cypher School, which had become concerned at its inability to crack the German Enigma Code. Turing commenced a new phase of his career, unable to publish or talk about new results.

Turing arrived at Bletchley Park as one of a team of mathematicians charged with decoding the German Enigma messages. This was an immense mathematical challenge, not only mastering the principles used by the Germans, but also dealing with variations between the services and changes from time to time. It is history that Turing not only met these challenges, but was in effect the leader of the powerful group. As part of the process Turing and associates produced a machine known as a Bombe, based on Polish ideas, which was able to decode Enigma. From late 1940 these Bombes were able to decode German Luftwaffe messages. The German Naval Enigma codes produced more of a challenge, but they were able to decode them by mid 1941.

Of young mathematicians who later became more famous the mathematicians at Bletchley Park included William Tutte (who later moved to Canada) and Peter Hilton, who later worked in the US. Another noted resident was Chess Master Harry Golombek, who was able to defeat Turing at chess, turn the board around, and recover from the resigned position. Later in the Bletchley Park operations MHA Newman arrived himself to add his own expertise.

Turing rose to the point where he travelled to America to advise on British progress and to exchange ideas. He had personal clearance from Churchill and the White House and worked also for a few months at the Bell Labs in New York. He also advised the Allies on safer encryption of their own messages.

Later in the war the Germans improved their code and whereas Turing was not involved in the later deciphering, his ideas were still used. By late 1943 Bletchley had got on top of their work and Turing became to some extent redundant, allowing him to transfer to nearby Hanslope Park, where, under the secret service, he was able to commence a speech encipherment project. This project, code named Delilah, after the “betrayer of men”, involved Turing and Don Bayley, the construction of a machine and the use of a surprising amount of undergraduate mathematics, particularly in the fields of Fourier and Numerical Analysis. They succeeded in getting the machine to work by the end of the war, but it had made no impact on the war effort and in fact had a little too much crackle to have a future. It seemed other organisations had also made other plans to deal with tasks which Delilah could potentially solve.

At the end of the war Turing’s King’s fellowship had been extended and he had the prospect of an eventual Lectureship there. However the War had more than interrupted his career. He now wanted to construct a “brain” and set about thinking of how this could be done. The brain was to draw on the abstract Turing machines and his experience built up with decryption and machines as acquired during the war, and was to become the modern computer. In his remaining time at Hanslope Park his ideas were developed.

In 1945 he was recruited to the National Physical Laboatory to provide the mathematical direction to development of a computing machine known as ACE (Automatic Computing Engine). Whereas there were other computing projects in the US (involving von Neumann) and there had been others, including those with which Turing had been involved for special purposes, this project planned the nearest thing to a true computer able to perform programmed tasks. Other mathematicians working at the NPL at the time included Leslie Fox and JH Wilkinson, both Numerical Analysts. The ACE program progressed rapidly in its early stages, and during Turing’s stay had rather complete plans, but the production of such a machine seemed a long way off in 1947, when Turing, in some frustration, returned to King’s College to resume his Fellowship.

In the meantime, Newman had been appointed after the war to a Chair in Pure Mathematics at Manchester, a promotion from his Lectureship at Cambridge. He had developed an interest in computing machines through his time at Bletchley. Newman also had funding from the Royal Society to develop computing equipment. Also, a key electronics expert, FC Williams, had been appointed to a Chair in Engineering at Manchester.

Newman offered Turing a Readership in Pure Mathematics at Manchester to enable him to supervise the mathematical work on a computer being developed by Williams and he took up this appointment, resigning formally from the NPL in 1948. Williams actually got the first electronic computer working before Turing arrived, but Turing kept busy writing all the mathematical subroutines for this machine, which was eventually devloped for manufacture by Ferranti.

Whereas the Manchester computer was the first, it should be acknowledged that Wilkes had also developed a computer at Cambridge. This became the first one really available for serious mathematical work. It should also be noted that the NPL’s ACE eventually was completed, consistent with one of Turing’s plans, in 1950, with Wilkinson still involved. And of course computers were being developed in the US also.

In 1950 Turing published in *Mind* the highly significant *Computing Machinery and Intelligence*. In this paper he posed a test for whether a machine had acquired intelligence, a test still used 52 years later.

In 1951 Turing was elected a Fellow of the Royal Society.

In 1952 Turing was arrested and convicted for homosexual activity. This had a major effect on him. He had still, unknown to his colleagues at Manchester, being doing secret work for the government. As a result of his conviction his security cover was stopped and police started investigating his activities and his associates around the world.

Turing’s last two or three years at Manchester were spent on developing his ideas to biological forms, in particular a field called morphogenesis, the development of pattern and form in living organisms. He also developed ideas on quantum mechanics, particle theory and relativity.

Turing was found dead on 7 June 1954 during a period in which he was conducting electrolysis experiments. Beside him was a half-eaten apple containing potassium cyanide. Whereas his mother maintained death was accidental the court decided that the poison was self-administered.

A note should be made about Turing’s athletic fitness. He was a dedicated cyclist, always commuting around cities by bicycle rather than cars. In his earlier years he was also a competitive rower. In his later career he took up athletics seriously, showing proficiency as a distance athlete. He almost made the British Olympic team in the marathon, having finished fifth in the 1947 AAA Marathon. He was also successful at distances from 800 metres and up and was a stalwart of the Walton Athletic Club.

Certainly, despite his short life, Turing achieved much, and as much as anyone in the twentieth century, inspired the development of the modern computer.

**References**

- Hodges, Andrew,
*Alan Turing, the Enigma*, Vintage Edition, Random House, London, 1992 (first published Burnett 1983).

Turing is the subject of the Australian Mathematics Trust T Shirt in 2002.

This is available for sale through AMT Publishing.

Peter Taylor

Australian Mathematics Trust

Canberra

14 May 2002