Among the greatest advances in knowledge this century one would include relativity, quantum mechanics and the explanation of hereditary replication in living things by the structure of the DNA molecule. But it would not be absurd to include in this category a discovery which is not at all well-known and goes back to 1930, namely, that logic and mathematics as we know them have undecidable propositions.

Do you know precisely how each lesson is to develop?

If the answer to the above question is yes, it is recommended that you look carefully at your technique to see if it is in line with modern ideas. Let me be so immodest as to describe how a lesson developed the other day.

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 :

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

The first section of this paper deals with the Note 3104, A problem in differential geometry, by J. K. Brunton in The Gazette for October 1964. The second section was prompted by it.

The following extends the result given by the author for the Nth power of a square matrix to the case of negative and fractional values of N. See The Mathematical Gazette, 1965, XLIX, p. 421.

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.

The criticism by T. B. M. Neil and M. Singer in The Gazette for October 1965 of C. P. Willans’ formula for the nth prime number is in my view quite unjust. Willans’ formula expresses two properties of the nth prime, namely that the number of numbers x such that 1 ≤ π(x) < n is two less than the nth prime, and that the nth prime is less than 2n. Willans gives a trigonometrical representation of π(x), the number of primes less than or equal to x. The analysis which Neil and Singer give is an exposure of the formula only in the sense that a clear proof is an exposure.

In a recent article [1] in this Gazette, Collings discusses the number of arrangements of n railway trucks on k sidings under a variety of conditions. Let qk (n) be the number of ways of arranging n indistinguishable trucks on k indistinguishable sidings or, what is the same thing, the number of partitions of n into not more than k parts. Let pk (n) be the number of these arrangements which use all k sidings, that is, the number of partitions of n into exactly k parts. Collings describes the determination of pk(n) and qk(n) as an unsolved problem in partition theory; this is not quite correct.

In a recent note (The Mathematical Gazette, 1965, XLIX, 388) the late Prof. D. Markovitch of Belgrade states that the method published by me (The Mathematical Gazette, 1960, XLIV, 241) is not new as he had described the same method in the period 1954 to 1956. By the courtesy of the Editor I have now had an opportunity of studying the full manuscript submitted by Prof. Markovitch together with copies of the published papers.

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.

Since all truth functions in two-valued logic may be represented by polynomials in the arithmetic of residues modulo 2, the question which R. Sibson raises in note 3127 may be expressed in the following form : Find all polynomials Φ(p, q) such that

identically, where p & q is a commutative polynomial which vanishes only when p = q = 0, and p & p = p. Taking p = q = r in (1) we find

Last week I felt that one of my sixth form applied mathematics sets was finding its subject too straightforward. Imaginations had been stuffed with smooth spheres, perfectly elastic and moving as in a vacuum. Methods relied almost exclusively on the less complicated aspects of Newton’s work, with even solid rigid bodies seeming to behave like the ubiquitous point masses. Advanced level text books contain a good store of examples and, needless to say, the class had worked solidly through its own book. “Solidly” was the right word! I wondered how to fan a flagging interest, whether to bring out my own constructional toys for another lesson and to set up a pulley system or project particles from one side of the room to the other, whether to take over the physics laboratory and let the pupils use some proper apparatus. Alas, my pupils are more sophisticated than I and therefore eschew the use of toys, while regrettably the physics staff arrange their physics experiments so much more ably than I could. Besides, it would be entertaining to see the sixth formers investigating a situation which could not be explained with the help of Advanced level mathematics. Let them flounder, as perhaps our forefathers floundered whilst discovering such trivia as Hooke’s Law or Newton’s Experimental Law!

Textbooks dealing with this topic are usually content to solve approximately the equations of motion for a particle moving under constant gravity g, using the fact that the angular velocity w of the Earth is small. (See A. S. Ramsey—Dynamics, Part II (§§4.6-4.74) or L. A. Pars—A Treatise on Analytical Dynamics § 10.10.) These equations of motion may be solved exactly, of course, without placing any restriction on w. The modified equations of motion for the particle moving in a medium offering resistance kv per unit mass, v being the velocity of the particle, may also be solved exactly.

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.