No field of endeavour, trivial or non-trivial, escapes man’s quest for the superlatives—always the smallest or the largest, the fewest or the most is his goal. Whether this drive stems from a thirst for knowledge or a desire for leisure in which to pursue other work is a good question. In this massive and perpetual search for the ultimate a trivial game is no exception. No doubt a fantastic amount of time has been spent seeking to determine how many little counters could be arranged in a certain pattern, then this pattern re-arranged by jumping one unit over the other, removing the counter jumped, until the greatest number is lined up to be taken in a single move, or the pattern is reduced to a single unit with the fewest moves. Erom available published accounts, the single move limit has been 8 and the fewest moves have been 16 since this particular pattern called a 33-Solitaire board was conceived, the 33 referring to the number of spaces in this pattern.

A sequence of athletic records forms by definition a monotonic sequence. Since it is reasonable to assume this to be bounded, it follows that a limit must exist to future performance. It has been proposed [1, 2] that this can be estimated by use of a curve-fitting procedure on existing data, such as may be found in many compendia of athletic records (e.g. [3]).

In an article [1] in The Mathematical Gazette, S. N. Collings discusses and proves certain properties of projective polygons. Some of these results are contained in a previous article [3], published in the American Mathematical Monthly, or can be proved from the results obtained in that article. However, the language of [3] is different from that of [1], and the purpose of this note is to show how some of the results of [1] could be translated into the language of [3].

Given points P1 P2, …, Pn on a circle centre 0, they have a centroid G. We shall show that they have also a generalised orthocentre H, a generalised nine point circle with centre N, and that OGNH lie on a line forming a harmonic range.

An excellent S-level question asks how far a tender capable of 12 knots has to travel to intercept a ship travelling at 20 knots along a course passing within 1 nautical mile of port, the tender leaving port at the latest possible moment. The problem leads to interesting ramifications.

One of the best known of all classical elementary combinatorial problems is the enumeration of the derangements of n letters A1 A2,…, An, i.e., the permutations of these letters in a row so that, for every i, At is not the ¿th letter. The question is frequently presented in the form of an office boy who puts n differently addressed letters into n envelopes; we then seek the number of ways (w„) in which he can send every letter to a wrong address. Textbooks almost invariably solve the problem by deducing from general reasoning the recurrence relation :

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.

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.

An Editor who asks for comments deserves what he gets, though he may not always be able to take it. The problem of the “slipped disc”, raised originally by Miss W. M. Griffin (see The Mathematical Gazette, XLIV (1965), p. 389), has aroused a great deal of interest and several notes have been received, including one from Miss Griffin herself. In the present circumstances, however, when a good deal of material is being returned to authors through lack of space, it seems, regrettably, to be necessary to confine publication to one article, given below, which introduces an entirely new point of view. Thanks are due to all those who offered comments on an interesting but awkward problem.

The problem of a disc supported on two pegs and disturbed by a horizontal force has been recently discussed by Griffin, Skelsey and Scorer (1965) mainly in terms of limiting equilibrium. However the way in which limiting equilibrium is approached, and breaks down through a transitional slipping at one peg, can be discussed within the simple laws of friction, provided only that we assume the peg supports are not perfectly rigid and obey some law of elasticity in providing a reaction opposite to their displacement. In the case when the pegs are far enough apart for the disc to be jammed between them, the reactions at the pegs may be indefinitely large, limited only by the breaking strength of the pegs or the possibility that the pegs may be forced so far apart as to allow the disc to pass between them. If the horizontal force on the disc is increased, equilibrium is then broken by slipping at one of the pegs, which is unrestrained and explosive. The cases of small and large friction coefficient are indistinguishable as limiting equilibrium states, because the disc is performing accelerated rotation about the other peg when either the pressure on the first peg becomes zero or slipping starts at the second peg. Thus in the case θ>λ>90θ – 2θ considered by Miss Griffin the limiting equilibrium state considered is not always attainable. Also the case quoted by Miss Skelsey under diagram 15 wherein equilibrium is never broken is physically impossible unless one of the pegs has a negative elastic constant or some additional mechanism is provided to increase the pressure between the pegs as the disturbing force is increased.

Space does not allow much room for “reporting” in The Mathematical Gazette, though that deficiency is greatly offset by the Newsletter. As it happens, however, two very contrasting items arrived while this Number was in the proof stage and so they are included, as typical of much else, to give some information about mathematical activities in general.

This is the most ambitious attempt yet made to help mathematics teachers and pupils through the medium of television. It is designed for all but the least able pupils in the first year of the secondary school (i.e. of about 11 years of age), and will extend for two years. It aims to act as a guide-line for teachers who feel they would like to modernise their mathematics teaching both in content and method.