A man has three vessels, whose capacities are 3, 5 and 8 pints respectively. The largest is full of water. He desires to divide this water into two equal parts by using these vessels only What are the simplest ways of doing this?

Graphical solution.

Call the vessels X, Y and Z respectively, and the amounts of water they contain x, y and z respectively.

Some seven years ago an attempt was made to tell something of the story of Francis Walkingame, the author of The Tutor’s Assistant, which was first issued in 1751 and became the most popular arithmetic text book for over a century afterwards. Recent contributions to the Gazette shows that this author and his famous book are still unknown to many readers. Interest was aroused by Dr. T. M. Flett’s description of Edward Winder’s Exercise Book, which was clearly based on an edition of The Tutor’s Assistant of about 1830. In his description of John Stewart’s Exercise Book, written at Dundee some thirty years earlier, Mr. W More remarked that the rules were the same as Winder used and quoted additionally the Rule of False in very similar words to those used by Walkingame. This example is interesting as showing that Walkingame’s book was used in Scotland at the turn of the century. Richard Hartley’s 1846 letter gives a list of the later sections in Part I (Integers) of Walkingame and shows that he had just started Part II on Vulgar Fractions.

The considerations which I propose to put before you to-day arose from an attempt, when teaching students, to build up the classical laws of Newtonian Mechanics inductively from laboratory experiments.

In this task we are immediately confronted with two fundamental notions, those of mass and force.

It will probably be admitted that no definition of a physical quantity can be accepted as adequate unless it implies a criterion leading to an unambiguous measure of the thing defined.

The simplest overhand knot, if tied in a strip of paper and carefully pressed flat, forms a regular pentagon (Fig. 1). All higher order polygons can be produced in this way with the single exception of the regular hexagon which requires two strips. The reason for this exception is interesting.

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”.)

In many branches of mathematical physics it is of some importance to know how a given potential field is disturbed when a finite sphere is introduced into it. The new potential is usually obtained as the superposition of the former one and a suitable “perturbation potential” which becomes vanishingly small at large distances from the sphere. Such problems arise in electrostatics, hydrodynamics and steady heat flow. The nature of the perturbation potential depends on the boundary conditions on the surface of the sphere. In this note a method is given for finding this perturbation potential for several different types of boundary condition. The field need not be axially symmetric. The method is useful in several cases for determining image systems directly and without using special geometric properties of inverse points in a sphere.

In a great many problems in probability, one is given one set of quantities Xj expressed in terms of a second set of quantities Yk, and one needs an expression for the Yk in terms of the Xj. Basically, of course, this is just a problem in solving equations, but the results can be expressed in a particularly elegant form when the coefficients involve the binomial quantities . This situation is one that occurs very frequently, and the resulting formulae simplify the derivations of many identities. The present expository account is intended to collect together various results in a convenient form. We shall need to recall briefly a few of the main properties of the binomial coefficients.

Everybody is familiar with the game of “Snakes and Ladders”, but not so many realise that the game constitutes an interesting example of a Markov chain. We feel that a good way of introducing Markov chain theory is by solving the problem, How long does a game of Snakes and Ladders last? Doubtless the reader has found that the last game before sending a child to bed seems to drag on interminably, and we used a computer to determine the actual average number of moves when 2 people play on a typical board.

I consider here another 3 problems, whose solutions are “not easy to see, but easy when seen”. Each of them involves in some way the notion of the convex hull of n points in a plane.