... exactitude, formal proofs, and total accuracy at all times -- with checks. She emphasised the need for 'proof' and the difference between conjecture and logical mathematical proof.

He felt he wouldn't be needing mathematics and I would make better use of it. This slide-rule was of a superior design: large, with a powerful cursor. He had made a leather case for it and I counted this as the mark of true love.

Before falling asleep, I 'drew' with my finger any relevant geometrical figure or algebraic equation on the partitioning of the dormitory cubicle that formed a bedside wall. The result would be miraculous. Without fail, on waking in the morning, the details, the logical argument required or the facts that I needed to recall were clearly imprinted in my mind and, because of clarity, any required solution would often be clearly 'written' on the partition.

With the recent experiences with J M Child, I found one question not only easy, but ridiculously so.

Looking back, I threw away the academic opportunities that were available, unaware of the guidance I needed.

There is a strong parallel between mountain climbing and mathematics research. When first attempts on a summit are made, the struggle is to find any route. Once on the top, other possible routes up may be discerned and sometimes a safer or shorter route can be chosen for the descent or for subsequent ascents. In mathematics the challenge is finding a proof in the first place. Once found, almost any competent mathematician can usually find an alternative often much better and shorter proof. At least in mountaineering we know that the mountain is there and that, if we can find a way up and reach the summit, we shall triumph. In mathematics we do not always know that there is a result, or if the proposition is only a figment of the imagination, let alone whether a proof can be found.

It was moreover a matter of geometry -- pure mathematics -- a nice problem that had a neat and successful solution. The requirement was that rain falling on a tent or coat should run directly downward and not soak through the woven fabric. It fell to me to devise a weaving pattern so that this could be achieved with cotton.

Critical lattices relate to whole numbers in two or more dimensions and lead, by geometrical methods, to solutions concerned with 'close packing', for example, how best to stack tins in a cupboard or oranges in a box.

The links in the USA were with the Association of American University Women (AAUW) and with Rotary. I had to prepare four standard speaking scripts to be used during the tour as material for my hosts to select from and for press, radio, television and other publicity.

It confirmed my experience that the attitude toward mathematics, of parents, teachers and the general populace, is of critical importance, and that (given good class discipline) the mathematical competence and enthusiasm of the teachers matters more than the size of classes.

A cluster of bubbles blown with a bubble-pipe gives a spectacular illustration of the 'closest packing' of pliable solid objects. The bubbles in the centre of a cluster (or 'froth') take up shapes that fill the space available without extraneous air gaps and give minimum surface area for the volume of air which they contain.

Every true mathematician sees mathematics everywhere -- in a child's swing or a pendulum, in the outline shape of a tree and that of its leaves, in the clouds, in the way a circular tube is made from straight strips of paper.

This was the first time in all the thousands of years during which magic squares have fascinated mathematicians and laymen alike, that a method of construction had been found for a whole class.

Dame Kathleen represented Rusholme ward as a councillor and alderman, then on reorganisation as an elected councillor again. She was Lord Mayor from 1975 until 1976. In 1984 she was granted freedom of the city, the highest honour a city can bestow.