David Hilbert, one of the giants of mathematics, delivered a lecture at the International Congress of Mathematics at Paris in 1900. The first part of the lecture, a preamble to his announcement of the now-famous 23 problems, began with the words:

It is well known that Lewis Carroll enjoyed the company of children and entertained them with fantastic stories, whilst on boating expeditions or on beaches, but his diaries also reveal that he enjoyed teaching children mathematics and other school subjects. On April 16th 1855, he recorded his concern with the education of his younger sister Louisa, who had an aptitude for mathematics: ‘Went into Darlington – bought Swale’s Chamber’s Euclid for Louisa. I had to scratch out a good deal he had interpolated, (e.g. definitions of words of his own) and put in some he had left out. An author has no right to mangle the original writer whom he employs: all additional matter should be carefully distinguished from the genuine text. N. B. Pott’s Euclid is the only edition worth getting – both Capell and Chamber’s are mangled editions.’ Three days later he recorded: ‘Advanced Louisa’s mathematics to simple Equations (third day of Algebra), and the first 12 propositions of Euclid.’ On the next day, he left his home at the rectory of Croft, and returned to Oxford for the Easter Term.

In the recent Gazette [1], John Sharp has given us a most entertaining account of the number η (eta) which is the only real root of the equation x3 - x2 - x - 1 = 0 , and its connection with the snub cube. I have been investigating this a bit further, with some results that may be of interest.

For convenience, if P, P′ are points in the Euclidean plane R2, [PP′] will denote the closed segment . Excluding the endpoints, we get the open segment ] PP′ [. Further, if Px is a half line, ]Px will stand for Px - {P}.

Consider the vulgar fraction 257/47. We can convert it to an integer plus proper fraction. If we now invert the fractional part and repeat the process on it, we get

In this paper a magic cube of order n is a 3-dimensional matrix Qn = {qn(i,j,k); 1 ≤ i,j,k ≤ n} containing natural numbers 1,2, ... n3 such that the sums of the numbers along each row (n-tuple of elements having the same coordinates on two places) and also along each of its four great diagonals are the same. This contrasts with a previous article [1] which considered magic cubes with no requirement for the diagonal sums.

How big is n!? For example, to pluck a number out of thin air, what is the order of magnitude of 272! ?

This is the third note of a series on the factorial. The two previous notes [1] and [2] dealt with the factorial function x! while here we only consider n! when n is an integer, but its main results also apply to non-integer arguments. This note is longer and more elaborate than the previous ones, but the principal result is in the opening section. Only aficionados need go any further.

From the date of my own A level studies (many moons ago!) I have been dissatisfied with the presentation of the law of elastic restitution. The title Newton’s ‘Experimental’ Law implies that it is a purely empirical phenomenon, yet if there is a wide range of conditions in which it is approximately valid (and it isn’t worth using at all if this isn’t true), then there must be some reason for it. Equally unsatisfactorily, the law implies loss of energy in most circumstances. Even in my school days I could see that the usual argument about conversion into heat, sound etc didn’t really hold water, as this process couldn’t conceivably be instantaneous. A recent article by O’Connor [1] stimulated me to crystallise my thoughts on the subject [2], and I was later able to develop a simple and plausible mechanical model exhibiting restitution, which also conserved energy [3]. My presentation in [3] was mainly focussed on the physical consequences of the model, and solution of the differential equations involved was numerical. Yet, in the simplest case, the model is amenable to fairly elegant exact analytical techniques, which it is the aim of this paper to present. Apart from the interest of the model, the methods used are a nice illustration of the role of normal modes in the solution of simultaneous linear differential equations—a topic approachable by A level students of Further Maths, or by first-year undergraduates.

On 15 March 1997, when boarding an aeroplane bound for Spain, I was handed a copy of the Independent newspaper. One page of the paper was devoted to the Hale-Bopp comet; from two pieces of information about the comet’s orbit I set out to find out as much as I could about the orbit using A level mathematics, and in some cases further mathematics. I was interested in finding the fastest and slowest speeds of the comet, and the size of its orbit in relation to the planetary orbits. I also found speeds and times at other points on the orbit, where distances from the sun could be calculated. These findings, along with an approximate calendar for the next orbit of the comet, are presented in a diagram at the end of this article. During the course of my working, I made two discoveries of a more general nature. The first of these is that the radius of curvature of a conic at the point nearest to a focus is equal to the semi-latus rectum. The second was one of Kepler’s laws, that for an elliptical orbit T2 ∝ a3, where T is the time of a complete orbit, and a is the semi-major axis.