A topic which is becoming increasingly popular in school mathematical syllabuses is that of probability and the subject matter which appeared only in the Sixth form previously is now to be found in the Third form. In the classroom, as for all mathematical undertakings, it is always a useful exercise to get children to state what they expect the answer to be before they attempt to solve the problem, and this “playing off of hunches” is extremely interesting in probability.

The following two-player game which we have named Euclid was devised as a source of examples to illustrate the use of a winning strategy algorithm based on the proof of a similar existence theorem[1].

There are many ideas, fundamental to the whole of mathematics, that are difficult to teach explicitly and are rarely examined. They include the concept of mathematical proof, the idea of a converse, necessary and sufficient conditions, and those elements of elegance and rigour without which mathematics can lose its beauty and even its claim to be called mathematics. It is these ideas I shall refer to as mathematical background; it must be emphasised that this background is common to both traditional and “modern” mathematics, and that its cultivation in their pupils would seem a desirable objective for all teachers, both conventional and progressive. Most of these ideas were clearly expressed in the Mathematical Association’s booklet, “Suggestions for Sixth-Form Work in Pure Mathematics”, which seems admirable in everything but its title, since most of the work should be begun much earlier than the sixth-form. (If it is left until then, there is justification for one reviewer’s remark that it is “an attempt to paper over the cracks of sixth-form mathematics”.)

This puzzle is described in Wallis’s Algebra (Cap. CXI), with detailed instructions both for making it and solving it. The apparatus consists of two parts, which are easily made from simple materials, as follows:

Ever since the war, the methods, techniques and results of operational research have increasingly been affecting the organisation and functioning of modern industry and commerce. Diverse in character and embracing general and unsophisticated approaches as well as specialised mathematical ones, operational research is initially concerned with the analysis of situations and the identification of the factors affecting them. Once such analysis has taken place, certain standard optimisation procedures are used, or new ones devised, so as to assign a set of values to the parameters involved that will yield maximum efficiency, minimum cost, greatest speed, etc. One of the simpler and better known examples of the effects of such operational analysis was the institution of the penny post in 1840. Two factors were considered to affect the cost of a postal system: the collection and distribution of mail on the one hand and its actual transportation on the other. The fact that the cost of the latter was only a fraction of the former enabled penny postage to be established. This process of isolating relevant factors and making them interact in certain ways to yield a model of a real situation is one which today’s teachers of mathematics cannot afford to ignore. It is contended here that, if this approach is properly introduced into the classroom and applied to simplified industrial or local problems, it can help

Throughout this article, all numerals are assumed to be in base ten notation, unless otherwise stated.

Under many of the new mathematics programmes currently in use in secondary schools, the topic “Numbers to different bases” is introduced. Indeed, provided such an elementary textbook series has chapters on Sets, Counting to different bases, Symmetry and/or Transformation Geometry, and, possibly, Vectors and Matrices, it is then allowed to write “Modern Mathematics” on its title page almost regardless of the methods employed in the book. However, my topic does not concern the vagaries of modern mathematics textbooks, their content and their methodology, but rather the question of why ten is used as a number base. Undoubtedly many of you already feel that you know, and that the reason is concerned with numbers of fingers and toes. However, one might try to decide more rationally whether ten is a sufficient base to represent the wide range of numbers with which we are normally concerned.

It is the purpose of this note to remark that the mean deviation is quite as respectable and therefore useful a test of dispersion of data as the x2 or other widely used tests—at least in simple statistical situations. That it is not used appears to result from some misunderstanding of its efficiency and perhaps also of its bias as an estimator of parameters of a frequency function.

The best separation of substances by adsorption chromatography requires a de-sorbing solvent whose concentration-volume graph is concave. Various methods have been devised to achieve this, sometimes ignoring the density difference between the dilute and concentrated solvents. A particular method of mathematical interest is that of Baker, Ellis and Hawley [1] where standard components can be used and density differences are reduced by using a floating siphon.

As far as the author is aware, there does not seem to be currently available in the literature a proof of the little-publicised pivotal condensation method of evaluating determinants. The purpose of this note is to supply a simple proof of this.

We propose to construct some triangles somewhat like the Pascal triangle, but more general. The only essential knowledge is that of the binomial theorem, though for some of the results we assume a knowledge of number theory, up to Fermat’s theorem. At the end, we show how matrix algebra could be used to advantage.

Fourteen countries participated. The British team of 8 boys, chosen on the results of the British Mathematical Olympiad held last May—D. J. Aldous, P. M. Bennett, A. J. Mclsaac, S. P. Norton, J. F. Segal, P. D. Smith, A. G. Trangmar, and N. S. Wedd—was placed fifth (after Hungary, E. Germany, U.S.S.R. and Romania). Mr. L. Beeson and I accompanied the team on behalf of the Association (which was asked by the Department of Education and Science to arrange British participation—and has also recently accepted responsibility for the mathematical oversight of the National Mathematical Contest and the British Mathematical Olympiad, presently administered by Guinness Awards on behalf of the organising committee).