Populations of large mammals such as elephants tend to group themselves into herds. As these herds wander about, it may happen that two of them meet and amalgamate, or that a large one may split into two smaller ones, (see John Hillaby, “Elephants as a pest control problem”, New Scientist, 12 (1961), 736-8). Suppose that a given area contains n animals, and let pk (t) denotes the probability that at time t they are grouped into k herds. Conditional on this, let x1(t), …, xk(t) denote the sizes of the herds arranged say, in non-increasing order. The chance that two herds will meet and amalgamate will increase with the number of herds, and it may, for instance be taken to be proportional to the excess of this number over unity. Thus the probability of an amalgamation in the interval (t, t + δt), given that there are k herds at time t, is

Calculus made easy by “F.R.S.” was published seventy years ago. Its anonymous author wrote explicitly for engineering students rather than for future mathematics teachers. He ended by asking those who had enjoyed the book “not to give the author away nor to tell the mathematicians what a fool he really is”.

The purpose of this note is to suggest a system of marking chess-players that is both simple to apply and based on a theoretical background. The non-mathematical reader will not need to understand the formulae, but is recommended to read all the words.

In the British Chess Federation Year Book, 1953–4, pages 39–42, there is an article entitled “ A grading system for British players ”. It describes a system that it admits to be open to some objections. At first sight the system looks reasonable and is fairly simple to apply, though of course any system requires a fair amount of calculation if it is to be applied to hundreds of players. An objection to the system, mentioned in the article, is that it is always profitable to play an opponent whose mark exceeds one's own by more than 500 points, unless the system is modified in an inelegant manner. The publication of the article is stimulating to further thought and the system to be described here is one of the results. It is not open to an objection of the above form.

It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test.

In their recent article [1] B. Grünbaum and G. C. Shephard asked whether, for every pair of natural numbers k, r, there exist sets of k tiles that admit precisely r distinct tilings of the plane. (Two tilings are said to be distinct if they cannot be brought into coincidence by a rigid motion, and it is to be understood that in the tilings each distinct tile must occur at least once.) They gave examples for k = 1, r = 1, 2 and for k = 2, r = 1, and referred in a footnote to a later discovery of a solution for k = 1, r = 3.

Given any skew-symmetric n x n matrix A, we have

det (A - λI)= det (A - λI)′ = det (- A - λI) = (-1)n det (A + λI),

whence we see that the non-zero eigenvalues of A can be arranged in pairs α, - α. Since the set of n eigenvalues of A2 is precisely the set of the squares of the eigenvalues of A, it follows that every non-zero eigenvalue of A2 occurs with even multiplicity, so that the characteristic function ϕ(λ) = det (A2 - λI) of A2, regarded as a polynomial in λ, is a perfect square if n is even, while, if n is odd, then we may write ϕ(λ) =λ{f(λ)}2 for a suitable polynomial f(λ).

The achievements in experimental and theoretical science of the Arab scholar al-Haytham (also known as Alhazen, from his latinized first name al-Hasan) make him as much a figure of the sixteenth and seventeenth centuries as of his own tenth and eleventh centuries.

When his writings become known in the West the importance of his contribution to optics was widely recognized and he was studied by Galileo, Kepler, Fermat, Snell and Descartes. Mathematicians remember al-Haytham chiefly for Alhazen's Problem on the reflection of light from a circular mirror, which he solved by the method of conic sections; Huygens, Gregory, l'Hospital, Barrow (and many others) later took up the problem with the new analytical methods of geometry. Al-Haytham also wrote a commentary on the postulates of Euclid, and his attempted proof of the parallel postulate has similarities to Lambert's quadrilateral and Playfair's axiom in the eighteenth century. His theory of cognition may produce yet further interest in his work.

Since about 1939 my researches have been, in collaboration with Professor J E. Littlewood, mainly in a field of mathematics sometimes described as non-linear mechanics, but as most of the equations in rnechanics are non-linear, it is better described as non-linear vibrations or oscillations. It is rather a curious branch of mathematics developed by different people from different standpoints, straight mechanics, radio oscillations, pure mathematics and servo-mechanisms of automatic control theory The notation varies from equations in v for voltage to θ for the angle of the pendulum, R for the radius of a gas-filled cavity, and x and y of pure mathematicians, with parameters ranging through the Greek and Roman alphabets.

Some time ago I asked what is the average rectilinear distance D between the ends of a string of length L when thrown down at random on a table (The Mathematical Gazette, LII (1968) 165). I have to thank a number of correspondents (J. Clemow, R. 0. Davies, A. R. Everett, M. Fine, J. R. Hunt, G. S. Light and E. Sike) for letters about the problem. Some offered solutions based on simple assumptions regarding randomness. The simplest (Everett) is that all values of D in the range (0, L) are equally probable, and so the answer is \L. I do not find this assumption acceptable, nor the other assumptions put forward.

A tiling of the plane is a family of sets, called tiles, that cover the plane without gaps or overlaps. Usually we are concerned with tilings whose tiles are of a small number of different shapes; familiar examples are the regular and uniform tilings (see, for example, [1]).

Although the mathematical theory of tiling is very old, it still contains a rich supply of interesting and challenging problems. The purpose of this article is to describe some of these. In many cases the mathematical aspect is enhanced by the aesthetic appeal of the resulting tilings.

Let P be a given convex polyhedron and L any plane. Then if L∩P is not empty it is called a plane section of P. Every plane section of P is either a convex polygon, an edge or a vertex of P.

VI. If a convex polyhedron P has v vertices (v ≥4), does there always exist a plane section of P which is a convex n-gon with n ≥ √ v ?

A more general statement of this problem is to ask whether there exists a constant d (0 < d < 1) such that every convex polyhedron P with v vertices possesses a plane section which is an n-gon with n ≥ vd. Problem VI is then equivalent to asking whether d ≤ ½