I have been asked to speak about developments in Mathematical Logic since the publication of Principia Mathematica, and I think it would be most interesting if, instead of describing various definite improvements of detail, I were to discuss in outline the work which has been done on entirely different lines, and claims to supersede altogether the position taken up by Whitehead and Russell, as to the nature of mathematics and its logical foundations.

The lack of any obvious pattern amongst the Bernoulli numbers is one of the shocks of analysis which subsequent familiarity with the many beautiful and simple means of deriving them does not altogether assuage. Historically, they first arose in connection with the sums of the rth powers of the first n integers

In connection with a general form of the covering principle and the relative differentiation of additive functions Mr. A. S. Besicovitch has proved that given a unit circle C with centre O, then any set of circles satisfying the conditions

1. Each circle of the set meets (or touches) C;

A. 2. Each circle of the set has radius not less than 1;

3. No circle contains O or the center of any other circle of the set, has less than 22 members.

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.

It is, I think, of happy augury that it should have been found possible to arrange a joint meeting of members of the local branches of the Classical and Mathematical Associations, as well as members of the Yorkshire Natural Science Association, to hear a few words about Greek science. No doubt the classical or humane studies can stand by- themselves and so (in a way) can the study of science based only on good English: but I venture to think that they are both much stronger when they stand side by side and support one another. The idea that there is any necessary antagonism or opposition of interests between them is surely unfounded, and there is room for both in our system of education. It is a mistake to suppose that in our school education there is not time for the one or the other; the question is really one of using the time that there is to the best advantage. No doubt schoolboys in the past have wasted much time in the perfunctory learning of Latin and Greek. It has often been remarked that, while a boy was more or less continuously learning Latin and Greek from the age of nine or ten till he went to the University, a girl who only began Latin and Greek at, say, fifteen or sixteen managed to catch up the boy and take a place alongside him in Part I. of the Classical Tripos. It was therefore probably the inefficient or uninteresting way in which the boy used to be taught Greek and Latin that caused the waste of time. If the subjects were made sufficiently attractive and the bogey of compulsion were removed, there is no reason why the boy of good ability should not during his school years acquire a good working knowledge of both classics and science, as well as of the indispensable amount of modern languages. And, if the boy’s interest can be aroused sufficiently to make him take kindly to Greek and Latin on the one side and mathematics and science on the other, I am clear that each branch will help the other and be better done.

“There is really only one Game, the Game in which each of us is a player acting out his role. The Game is Leela, the universal play of cosmic energy.” Thus begins Harish Johari in Leela: the game of selfknowledge, a serious commentary and religious interpretation of the Hindu board game Leela. Leela is essentially the game Snakes and Ladders, which in the U.S. is the popular children’s game of snakes and ladders, England’s famous indoor sport,” for 50ø in 1943 by the Milton Bradley Company of Springfield, Massachusetts. This game is really a 101 state absorbing Markov chain, which is amenable to mathematical as well as moral analysis. In what follows we investigate only the mathematical side of this diversion, specifically the expected playing time.

I wonder why problems about map-colourings are so fascinating? I know several people who have made more or less serious attempts to prove the Four-Colour Theorem, and I suppose many more have made collections of maps in the hope of hitting upon a counter-example. I like P. G. Tait’s approach myself; he removed the problem from the plane so that it could be discussed in terms of more general figures. He showed that the Four-Colour Theorem is equivalent to the proposition that if N is a connected cubical network, without an isthmus, in the plane, then the edges of N can be coloured in three colours so that the colours of the three meeting at any vertex are all different. (A cubical network is a set of points called vertices joined in pairs by simple arcs called edges, no two of which intersect except at a common vertex, in such a way that just three edges meet at each vertex. An isthmus is an edge whose removal destroys the connection of the network.) It was at first conjectured that every cubical network having no isthmus could be “three-coloured” in this way, but this was disproved by the example of Fig. 1, for which it may readily be verified that no three-colouring exists.

I.The first obstacle may be one of sentiment. It is said that in a certain grassy part of the world a man will walk a mile to catch a horse, whereon to ride a quarter of a mile to pay an afternoon call. Similarly, it is not quite respectable to arrive at a mathematical destination, under the gaze of a learned society, at the mere footpace of arithmetic. Even at the expense of considerable time and effort, one should be mounted on the swift steed of symbolic analysis.

The following problems are merely different forms of one

1. In how many ways can a convex polygon of n + 1 sides be split into triangles by non-intersecting diagonals? We represent such a splitting by a symbol. Take one side of the polygon as base and letter the other n sides in order by a, b, c,…. If a triangle with sides lettered x, y is cut off, denote the diagonal by (xy). If this diagonal and side z form the sides of another triangle of the dissection, denote the third side of that triangle by ((xy)z). Similarly for two diagonals, as the figure illustrates. Finally the symbol for the unlettered base is the symbol for the dissection.

The human body is a complicated machine whose movements involve many different joints, operated by a great many muscles. For that reason it is easy to get bogged down in detail when thinking about walking and running from a mathematical point of view.

Any position of the human body (or of any other jointed mechanism) can be described by giving the angles of joints. The number of angles needed for an unambiguous description is the number of degrees of freedom of the mechanism. For example, the position of a hinge joint is described by just one angle: a hinge allows only one degree of freedom. The human knee is a hinge. The ankle, however, allows rotation about two axes – you can tilt your foot toes up or toes down, and you can also rock it sideways so that the sole faces inwards towards the other foot – so it gives two degrees of freedom. The hip is a ball and socket joint allowing rotation about any axis through the centre of the ball, but any position can be described by just three angles (measured, for example, in three planes at right angles to each other), so it allows three degrees of freedom. In total, there are six degrees of freedom in each leg, making twelve in all, and suggesting that we need twelve equations of motion to describe walking. If we took account of the flexibility of the foot and the movements of the arms, we would need more.

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].

In recent years there has been a great development in the design and use of machines for carrying out calculations of various kinds, and this is a technical advance of some importance, not only because of the saving of time and labour in calculations which would be carried out in any case, but still more because it makes it possible to carry out extensive calculations which would be too laborious and time-consuming to undertake without such mechanical or electrical aids.