At a time when the public conscience is being focussed on the disabled, it may be opportune to report on the experience gained at Middlesex Polytechnic between 1979 and 1982 in taking the student Paul Holliman, who has been blind from birth, through its B.Sc. degree in Mathematics for Business. This paper begins with some indications of the system of Brailling mathematics, in the form in which Holliman had learnt it and used in the course. Then the course itself is described, with especial reference to the problems faced in teaching and learning. Attention then switches to the available technology; its scope and limits, and the prospects for future development. A brief final section notes some of the conceptual difficulties which blindness causes.

Last year I ran 1147 miles. Early in that December I had checked my ‘running total’ and found that it had already passed 1000 miles. I pondered on what might be an appropriate figure to aim at and toyed with 1040 miles (20 miles per week), 1095 miles (three miles per day) and 1111 miles (easy to remember). Finally, the combined dementia of runner and mathematician resulted in the choice and achievement of 1147 miles, chosen to give the closest possible daily average to π miles. The quotient 1147/365 exceeds π by about 0-0009.

I am continually surprised by the mathematical structure which surfaces in unexpected situations, and even more so when arranging a tenth birthday party for my elder son. The invasion of the home by some eight rather energetic boys on a cold February afternoon was viewed with some trepidation. The instinct of self preservation was uppermost in the search for activities which would allow the afternoon to pass with the minimum of discomfort and the maximum enjoyment. A family friend is very good at organising team games at parties, but with a space problem (our house does not have a large living room as so many modern houses have these days) it was important to split the children up for their activities, but yet devise a programme which was coherent.

John Gilder’s article on integer-sided triangles in the December 1982 Gazette [1] immediately suggests the question of what other triangles with integer sides have an angle which is a whole number of degrees. This is very simple to answer, for if the sides are integers, the cosine formula implies that the cosines of all three angles must be rational. Only angles of 60°, 90° and 120° fulfil this condition for the appropriate range of values, and it follows that, apart from the familiar right-angled triangle and Gilder’s case of the 60° angled triangle, the only other possible case is a triangle with an angle of 120°.

A popular way of introducing the binomial coefficients is by means of the number of routes in a grid or lattice of lines. In Fig. 1 the grid in the first quadrant of the x−y plane consists of those points for which at least one of x and y is an integer.

Three numbers are selected at random from the range [0,1]. What is the probability that they form the lengths of the sides of a triangle?

The ‘Two surprising limits’ (note 66.31, October 1982) described by William A. Leonard and Harris S. Schultz are in fact special cases of two general limit problems which provide interesting applications of Taylor’s theorem and L’Hôpital’s rule.

In the general case suppose that the radial forces are of magnitude P and that the angle between adjacent forces is 2α (so that α = 20° in the above figure). Consider the portion YY′ of the ring, as in the left-hand figure below. From a consideration of the rotational symmetry of the ring, it is evident that the tensions at Yand Y′are equal, these being marked T in the figure. Similarly, if the shear force at Y is Stowards the centre O, then that at Y′ is S away from the centre. Moreover, the bending moments at Y and Y′ are equal and opposite. These are marked M in the left-hand figure.

In a recent Gazette note (66.31, October 1982) Leonard and Schultz established the limits of two segments of the extended diameter of a unit circle. With reference to Fig. 1, they showed that if an arc of length l and a chord of length l are drawn from a point B on a unit circle, and if a segment, BA, of length l is taken on the tangent at B, then:

The purpose of this note is to show that these results are not only interesting, they are exceptional.

One of the standard results of the theory of Newtonian gravitational attraction is that the potential outside of a spherical shell can be obtained by regarding all the mass as concentrated at the centre of the shell.

1. Presumably all readers of this journal, and certainly all contributors to it do mathematics because they enjoy it. Have you ever stopped to think what a wonderful but strange gift it is, this ability to enjoy mathematics? Outsiders are not, I claim, capable of sharing or appreciating this enjoyment, but the reasons are not easy to find, as we see when we try out a few possibilities. The language is technical—yes, but so is that of art, music, atomic physics and computing, and these subjects suffer from no shortage of enthusiastic followers whose technical knowledge is virtually nil. The subject is hard, and to become a participant requires dedication, even sacrifice; true, but true also of the other examples. Mathematicians are all slightly over the edge of the sanity/insanity divide, a common view but isn’t the same said of mountaineers and marathon runners while we watch their exploits with baited breath?

Frank Budden’s article in the March 1982 Gazette concerned various problems in probability about the formation of triangles from, for example, a glass rod broken into three. That article, and an Oxford Entrance Examination question, set me thinking about the following connected problems. The first is from the 1972 examination.