October 1, 1912 -
Dame Kathleen Timpson Ollerenshaw earned a DPhil in mathematics from Oxford University. Despite being almost completely deaf from the age of 8, much of her adult life was devoted to politics and voluntary public service to education. She also continued working on mathematical problems and at the age of 85 published a book that solved a long-standing and difficult problem about most-perfect pandiagonal magic squares.
Kathleen Timpson was born in Manchester, England, the younger of the two daughters of Charles and Mary Timpson. Her passion for numbers began as a young child. In 1921, the combination of a viral infection and family genetic history led to a sensori-neural deafness. She writes in her autobiography [1, p20], however, that deafness intensified her interest in mathematics since mathematics was "the one subject in which I was at no disadvantage. Nearly all equations are found in books or shown on the blackboard as the teacher speaks...Learning mathematics is rarely as dependent on the spoke word as are most lessons and lectures in other subjects. The mathematics became my lifeline as well as an increasing source of joy." She quickly learned to lip-read, a skill that allowed her to succeed in school and university, and at her first job. At the age of 37 she received her first effective hearing aid. Throughout her life, however, she has never allowed this handicap to restrict her activities.
Ollerenshaw attended the Ladybarn House School until the age of 13. There she had the benefit of being taught by teachers with experience in mathematics. The Headmistress had studied mathematics at Girton College of Cambridge University and took a special interest in Kathleen, emphasizing to her the need for proof and the difference between conjecture and logical mathematical arguments. At this school, also, at the age of 6, she first met Robert Ollerenshaw who would later become her husband. She went on to boarding school at St. Leonards School, St. Andrews, for her secondary education. She continued to be fortunate in having two splendid mathematics teachers who challenged her with problems in pure and applied mathematics.
Reflecting back many years later in her autobiography, Ollerenshaw had the following thoughts about her early life studying mathematics [1, p229]:
Mathematics is a way of thinking. It requires no tools or instruments or laboratories. It may be convenient to have pen and paper, a ruler and a compass, but it is not essential: Archimedes managed very well with a stretch of smooth sand and a stick for his magnificent discoveries in geometry. When Sir Hermann Bondi, as a noted cosmologist, gives a public lecture on relativity, he draws a wild circle with chalk on a blackboard to represent the universe - and this probably conveys the idea of immensity better than any sophisticated computer image. A professional mathematician does not, typically, have a good visual memory for the written word, yet many can give the value of π or the square root of 2 to an incredible number of decimal places - a useless but impressive exercise. I didn't have this gift, but I never forget a mathematical logic proof when once I have understood it - not over decades - and in this way experience accumulates. I never aspired to being a professional mathematician, or to being a professional anything for that matter. If you are deaf, you are glad to 'get by', to keep up with others in an ordinary class-room and not to be condemned as being lazy, inattentive or merely 'slow'. I worked carefully and accurately, getting answers right first time, and so became very fast. I always had time to spare for checking what I had written in mathematics tests or examinations and, for this reason, would do well. Mathematics is the one school subject not dependent on hearing. I was lucky with my teachers, but I was also to a large extent self-taught; reading books about the great mathematicians, solving problems set in magazines. I spent time solving (or devising) mathematical teasers, a habit that has continued through the years.
After finishing school in 1930, Ollerenshaw spent a year studying privately with the mathematician J.M. Child at Manchester University in the areas of higher algebra (a mixture of topics that included number theory, the theory of equations, and functions and series) and geometry, while also preparing for the university entrance examinations at Oxford and Cambridge. She received an Open Scholarship to Oxford, where Robert was studying physiology in preparation for a medical career. At the age of 19 she entered Somerville College at Oxford to read mathematics. She admits in her autobiography that she did not take full academic advantage of her time at Oxford. During her first term, she and Robert became engaged. An excellent athlete, she played hockey, earning her "Blue" against Cambridge, and served as team captain her last two years. Her earlier preparation in mathematics served her well, however, and she passed her examinations at the end of her third year, thereby earning her B.A. degree in mathematics.
Upon graduating from Oxford in 1934, Ollerenshaw did not immediately look for a full-time job. In 1936 she began working in the statistics and Liaison departments of Shirley Institute, a cotton research establishment. She continued to play hockey for various local and regional teams, and competed in figure skating competitions (in February 1939 she was runner-up in the English-style British Pairs Ice-Skating Championship). Kathleen and Robert were married in September 1939, and Robert left almost immediately to serve with the British medical corps in World War II, although he remained in England until sent to North Africa in 1942. Kathleen continued working at Shirley until their son, Charles, was born in 1941.
Through the head of the mathematics department at Manchester University, Ollerenshaw came to know Kurt Mahler, a mathematician who had come to Manchester from Germany in 1938. Mahler mentioned to her a unsolved problem on critical lattices, an area combining aspects of number theory and geometry, which she solved within a few days. Impressed with her abilities, Mahler suggested she consider returning to Oxford for a DPhil degree. So back to Somerville College she went in 1943 as a member of the Senior Common Room. Her supervisor was Theo Chaundy whose lectures she had attended while an undergraduate ten years before. His area was really analysis and differential equations, but he also became interested in critical lattices through working with Ollerenshaw. Over the next two years, while caring for her son while her husband was away at war, she wrote five original research papers which were sufficient for her to earn her DPhil degree without the need of a formal written thesis. She received her degree in 1945 just as the war was ending and shortly before Robert returned to England.
For the next seven years Ollerenshaw raised her two children, a daughter, Florence, having been born in 1946. On occasion she filled in as a part-time lecturer in mathematics at Manchester University, and did manage to keep up with her own mathematical research, publishing additional papers in the area of critical lattices. The year 1953, however, saw the beginning of a life-long involvement in politics and educational issues in England and Wales. Asked to address a meeting of the National Council of Women, her talk on the bad conditions of many of the older schools in Manchester eventually led her to a detailed statistical study of the conditions of school buildings through England. The success of her 1955 report in releasing government funds for school capital building programs strengthened her belief that "if one hopes to influence governments on social issues, it can only be done on the basis of accurately established numerical facts, not on mere opinions and protest. This had been demonstrated by Florence Nightingale after the Crimea War in 1856...I became thought of as a statistician rather than as a mathematician. It was this survey and the publicity it attracted that led to my being described nationally as an 'educationist' involved in a broad range of educational matters." [1, p86]
Meanwhile, in 1954 she was appointed to the Manchester Education Committee. Two years later she won election to the Manchester City Council as a member of the Conservative party, a position she held for the next 25 years. She continued to serve on the education committee, serving as its chair from 1967-1971. She was particularly interested in math education and post-school education. She was also a member of the city council's finance committee, the first and only woman member for many years, and deputy chairman from 1968-1971. From 1958 to 1967, Ollerenshaw was chairman of the Association of Governing Bodies of Girls' Public Schools. She wrote two books and many articles in defense of schools for girls and girls' education in general. From 1967 to 1971, she served as chairman of the education committee of the Association of Municipal Corporations, and thus was often called upon to provide information and advice about educational matters to members of Parliament. In 1975 Ollerenshaw was elected to a one-year term as Lord Mayor of Manchester. In this role she represented the city on formal and ceremonial occasions, but did not run the city. She also presided at city council meetings and civic functions. She has served on the governing bodies of five universities in the northwest region of England, including the Royal Northern College of Music which she helped to establish in the early 1970's.
In 1970 Ollerenshaw was made a Dame Commander of the Order of the British Empire for "services to education" (becoming a dame is the female equivalent of the knighthood). In 1984 Manchester awarded her the Freedom of the City, the highest honor the city could bestow. She has received honorary degrees from the University of Lancaster, Victoria University of Manchester, and Liverpool University, and elected an Honorary Fellow of the Manchester Metropolitan University, Somerville College at Oxford, and the University of Manchester Institute of Science and Technology.
Ollerenshaw's interest and expertise in educational issues led to several trips abroad. In 1963, as a member of a delegation from the British Association for Commercial and Industrial Education, she spent three weeks in Russia visiting technical colleges, school, and universities to learn about Russian post-school vocational education and training. Two years later she received a Winifred Cullis Lecture Fellowship to visit the United States for a three month tour emphasizing mathematics education. In 1970 The British Council sponsored a visit to Japan to have Ollerenshaw talk about the relationship between local and central governments and to lecture about mathematics education in England and Wales. She was also interested in learning why the Japanese were so successful in mathematics education compared to other countries.
Despite the many demands of all her public service, Ollerenshaw did not abandon her own mathematical interests. In 1971 she was appointed to a part-time senior research fellowship in the Department of Educational Research at the University of Lancaster. Her research project was to explore the possibilities of attracting more married women who were trained teachers to return to teaching after being away from school to raise their young children. From 1979 to 1981 she served as the first female president of the Manchester Statistical Society.
In 1964 Ollerenshaw was invited to be a Foundation Fellow for the new Institute of Mathematics and Its Applications, a professional society whose mission was to promote mathematics in industry, business, the public sector, education, and research. She published a number of mathematics papers in the Institute's Bulletin through the years. In 1978 Ollerenshaw succeeded Prince Philip, Duke of Edinburgh, as the president of the Institute. During her term she presented numerous "presidential addresses" on the subject of the mathematical theory of soap film and bubbles. One such talk was given in May, 1979, when Ollerenshaw was invited to give a Friday Evening Discourse at the Royal Institution of London. These lectures had been founded in 1825 by Michael Faraday with the goal of introducing advanced sciences to the general public in an easy-to-understand format. A long list of distinguished scientists have given presentations at the Friday Evening Discourses. Ollerenshaw was only the second woman to do so.
A paper that Ollerenshaw published in 1980 in the Bulletin of the Institute of Mathematics and Its Applications gave one of the first general methods for solving the Rubik cube puzzle (or the Hungarian magic cube as it was often called then) that tried to minimize the total number of moves needed. The algorithm did the bottom face first, then the top corners, then the middle slice edges, and finally the top edges, producing an average of 80 moves. Ollerenshaw worked with the cube so much that she required surgery to fix her "cubist's thumb" tendinitis.
Ollerenshaw's interest in the mathematical theory of magic squares also began in the early 1980s. A magic square of order 4 is one in which in the numbers from 1 to 16 are arranged in a 4x4 array in such a way that the sum of each row, each column, and the two main diagonals add to the same total. Perhaps the most well known 4x4 magic square is the one depicted in the painting Melancholia by Albrecht Dürer. The 17th century amateur mathematician, Bernard Frénicle de Bessey, determined that there are exactly 880 essentially different 4x4 magic squares, i.e. squares that cannot be obtained from one another by rotations or reflections. He did this by an exhaustive search, listing all 880 possibilities. Ollerenshaw and Hermann Bondi, a prominent cosmologist and mathematician, developed an analytical construction of the squares, thereby verifying the number 880. They published their results in a 1982 paper in the Philosophical Transactions of the Royal Society [Abstract], which was reprinted a year later as a book.
After this success, Ollerenshaw began the study of pandiagonal magic squares of order n, where n is a multiple of 4 (no pandiagonal magic squares exist of order n=4k+2.) This research was motivated by a letter from an electrical engineer who was interested in the use of these square in the process of dither printing used for the fast production of pictures in newspapers. Emory McClintock had published an article about pandiagonal magic squares in 1897 (American Journal of Mathematics, Vol. 19, pp.99-120). A pandiagonal magic square, in addition to satisfying the requirement of a magic square, has the additional property that all diagonals, including broken diagonals, i.e. those that wrap around from one edge of the square to the opposite edge, add to the same sum. An example of order 8 from the McClintock paper is given by the following (using the integers from 0 to 63 as favored by Ollerenshaw—McClintock used 1 to 64):
0 | 62 | 2 | 60 | 11 | 53 | 9 | 55 |
15 | 49 | 13 | 51 | 4 | 58 | 6 | 56 |
16 | 46 | 18 | 44 | 27 | 37 | 25 | 39 |
31 | 33 | 29 | 35 | 20 | 42 | 22 | 40 |
52 | 10 | 54 | 8 | 63 | 1 | 61 | 3 |
59 | 5 | 57 | 7 | 48 | 14 | 50 | 12 |
36 | 26 | 38 | 24 | 47 | 17 | 45 | 19 |
43 | 21 | 41 | 23 | 32 | 30 | 34 | 28 |
Here each row and column adds to 252. In addition, all diagonals, such as the one shown in red, also add to 252. A "most-perfect" pandiagonal magic square is one that has the extra properties that any two-by-two block of adjacent entries sum to the same total, and the sum of pairs of integers a distance n/2 apart along a diagonal is constant. For example, the 2x2 adjacent blocks in blue each sum to 126. Notice also that all pairs of numbers four positions apart along a diagonal, such as the 9 and 54 shown in green, have the same sum, 63. For example, along that same diagonal as the 9+54=63, we also have 58+5=63, 27+36=63, and 35+28=63. The same result holds for all such complimentary pairs along any broken diagonal. Unlike the larger class of pandiagonal magic squares, there are no most-perfect squares with odd order. Therefore a most-perfect pandiagonal magic square must have an order that is a multiple of 4.
Ollerenshaw worked on most-perfect pandiagonal magic squares for over eight years. In a 1986 paper she made use of symmetries to determine that there are 368,340 essentially different such squares of order 8. Slowly she figured out how to construct and how to count the total number of squares, first for those whose order is a power of 2, then for squares who order is a multiple of a power of 2, and finally, after another four years of work, for all most-perfect pandiagonal magic squares with order a multiple of 4. Professor David Brée, the head of Artificial Intelligence in the Department of Computer Science at the University of Manchester, assisted Ollerenshaw in organizing, proof-reading, and putting her research notes into an appropriate form to be published as a book. Most-Perfect Pandiagonal Magic Squares: Their Construction and Enumeration was published in 1998 to international acclaim and provided for the first time an algorithm for constructing a whole class of magic squares as well as a formula for counting their number, a remarkable accomplishment for a woman of age 85. In her "personal perspective" at the end of the book, Ollerenshaw writes:
"Only now, with the work complete, is it possible for me to look back and see the process as it developed, savouring in retrospect the challenge presented when proof was first attempted, and recalling each step of the astonishing mathematical adventure that it has been...The manner in which each successive application of the properties of the binomial coefficients that characterize the Pascal triangle led to the solution will always remain one of the most magical revelations that I have been fortunate enough to experience. That this should have been afforded to someone who had, with a few exceptions, been out of active mathematics research for over 40 years will, I hope, encourage others. The delight of discovery is not a privilege reserved solely for the young."
Partial Bibliography of Dame Kathleen Ollerenshaw's mathematical publications
(Access of the online articles from the Journal of the London Math Society requires a subscription.)
Other Publications