# Discovery has no upper limits

November 20, 1998

Most mathematicians are young and male, but John Ashworth finds much to celebrate in Kathleen Ollerenshaw's first book on the subject

If the first nine numbers, 1-9, are written in the shape of a 3 x 3 square (left), they form a unique "magic square". Do you find that beautiful? Fascinating? No? Before you stop reading, add up all the columns, rows and diagonals. Note the positions of the even numbers. Changed your mind? Would you like to try to make a 4 x 4 magic square with the numbers 1-16? There are 880 of them, so it is not too difficult.

Mathematicians have known that there are only 880 magic squares of order four (or n=4) since 1693, but the proof that this is the complete list was not given until 1982, when Sir Hermann Bondi published a paper with one of Britain's mathematical stars. Mathematicians are usually male, young and work for universities or research establishments so it is something of a surprise to find that Bondi's collaborator, Dame Kathleen Ollerenshaw, is 85, female and has not been employed as a mathematician for 40 years.

Even more surprising is that this autumn she published her first book on mathematics (with David Bree), Most-perfect Magic Pandiagonal Squares: Their Construction and Enumeration, which has been described as "a work of great scholarship" by Dr Hiorns of Oxford University. It is a remarkable book, containing innovatory methods leading to results of remarkable beauty and solving fascinating mathematical problems.

What Ollerenshaw and Bree have done is to provide quite general methods of constructing a subclass of magic squares of any size. What is surprising is how many of such things there are. With n=8 there are 36,864; n=16, 2.66355 x 1013 and with n=36, 7.04210 x 1041. This is an unimaginably large number, and this, together with the symmetry properties of this subclass of magic squares, suggests their use in cryptography. Other uses might be in image processing and parallel computing. I suspect that their really useful applications are yet to be found.

The driving force behind the work reported in this book was the sheer love of mathematics. Ollerenshaw had several dedicated female teachers who stimulated her innate love of problem solving, channelled it into maths and then developed that into the talent that enabled her to get a scholarship to Oxford in 1931. During the war she worked as an applied mathematician at Manchester's Shirley Institute and then, after marriage and a baby, she got the chance to go to Oxford as a mathematics don, where she also seized the opportunity to get a DPhil.

After the war she continued to do research on the geometry of numbers and teach part-time at the University of Manchester. In the 1950s, she began to develop a new career as a member of the Manchester Education Committee. She became involved with Lancaster and Salford universities, and ended up Lord Mayor of Manchester in 1975.

All this time she kept up with mathematics and remained fascinated by combinatorics and mathematical puzzles. Many mathematicians have this ability to retreat into a mathematical world, but for Ollerenshaw it was also part of the way she coped with the deafness that resulted from a childhood illness.

In a world that is increasingly specialised and where careers, especially academic and research careers, seem to become ever narrower, it is timely to be reminded that other possibilities exist. Part of the fascination of Ollerenshaw's work is, as she says in a note in her book, that: "The delight of discovery is not a privilege reserved solely for the young." The older I get the more comforting and challenging I find that becomes.

John Ashworth is chairman of the British Library Board. Most-perfect Magic Pandiagonal Squares: Their Construction and Enumeration is published by the Institute of Mathematics and its Applications, Pounds 19.50.

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