“I know my ones. I know my twos. I know my threes… Ah, the finger man.” An unforgettable exchange between a defiant, remedial student, and a confident, driven math teacher in the movie Stand and Deliver [1]. This back-and-forth exchange presented a teachable moment for the determined teacher giving way to a class presentation of the finger manipulation for multiplying by nine, a technique familiar to most every elementary school student in the U.S. Little did I know this simple idea would lead to the heuristic thinking outlined in the sequence of topics below connecting Fermat and finger multiplication.

The well-known Fibonacci and Lucas numbers continue to faxcinate the mathematical community with their beauty, elegance, ubiquity, and applicability. After several centuries of exploration, they are still a fertile ground for additional activities, for Fibonacci enthusiasts and amateurs alike.

Fibonacci numbers Fn and Lucas numbers Ln belong to a large integer family {xn}, often defined by the recurrence xn = xn−1 + xn−2, where x1 = a, x2 = b, and n ≥ 3. When a = b = 1, xn = Fn; and when a = 1 and b = 3, xn = Ln. Clearly, F0 = 0 and L0 = 2. They satisfy a myriad of elegant properties [1,2,3]. Some of them are:

In this article, we will give a brief introduction to the Q-matrix, employ it in the construction of graph-theoretic models [4, 5], and then explore some of these identities using them.

In 1960 C.H. King studied the Q-matrix

In [1], W. D. Munn proved the following result.

Theorem 1: Infinitely many ellipses pass through the four vertices of a given convex quadrilateral.

Much of the geometry that I studied as an undergraduate in the 1950s concerned complex projective space, in which convexity plays no part. So I found Theorem 1 especially piquant and sought to understand it better. This article is the result. After examining the convexity of quadrilaterals in general, especially those inscribed in conics, I consider the following problem. Let P be a variable point in the plane, distinct from the vertices of a given convex quadrilateral ABCD. It is well known that there is a unique conic, S (P), through the five points A, B, C, D and P. How does the nature of this conic depend on the position of P? As a spin-off, we get a very short proof of Theorem 1. Finally I look at what happens when the quadrilateral ABCD is not convex. In this case, S (P) is always a hyperbola, but the distribution of A, B, C and D on its branches is still of interest.

In the course of a coffee-table conversation with my friends regarding the nature of static equilibrium of different solid objects the situation involving a uniform hemisphere came up. Intuition (and perhaps experience) tells that a uniform hemisphere as shown in Figure 1 resting on a flat surface will be at stable equilibrium, and so will an oblate hemispheroid as shown in Figure 2. Things get complicated when we move to a prolate hemispheroid like the one shown in Figure 3, for the nature of its equilibrium is less obvious. The intuition does come to mind though that if the prolate hemispheroid is made indefinitely taller, keeping its equatorial radius fixed, then the equilibrium should eventually become unstable. Intrigued, we decided to probe into the matter quantitatively.

The problem of finding a closed-form evaluation of baffled the pioneers of calculus such as Leibniz and James Bernoulli and, following the latter’s promulgation of the problem, it became known as the Basel problem after his home town (which was also Euler’s birthplace). Euler’s early sensational success in solving the Basel problem by identifying is extremely well-documented. In this paper, we give the full details of his subsequent derivation of the general formula

(1)

where (Bn) is a sequence of ‘strange constants’. Euler’s polished account of his discovery, in which he popularised the designation of the strange constants as ‘Bernoulli numbers’, appears in Chapter 5 of Volume 2 of his great textbook Institutiones calculi differentialis [1; E212]: see [2] for an online English translation. Here, we will focus on his initial step-by-step account which appeared in his paper with Eneström number E130, written c1739, carrying the rather nondescript title De seriebus quibusdam considerationes, ‘Considerations about certain series’. (For convenience, we will just use ‘Eneström numbers’ when referencing Euler’s work: all are readily available on-line at [1].) Euler’s proof is notable for its early, sophisticated and incisive use of generating functions and for his brilliant insight that the sequence (Bn) occurring in the coefficients of the general ζ(2n) formula (1) also occurs in the Euler-Maclaurin summation formula and in the Maclaurin expansion of . By retracing Euler’s original path, we shall not only be able to admire the master in full creative flow, but also appreciate the role played by recurrence relations such as

(2)

which, as our ample list of references (which will be reviewed later) suggests, have been rediscovered over and over again in the literature. Moreover, our historical approach makes it clear that, while deriving (2) is relatively straightforward (and may be used to calculate ζ(2n) recursively as a rational multiple of π2n), it is establishing the connection between ζ(2n) and the Bernoulli numbers that was for Euler the more difficult step. Even today, this step presents pedagogical challenges depending on one’s starting definition for the Bernoulli numbers and what identities satisfied by them one is prepared to assume or derive.

Given two triangles ΔABC and ΔDEF, if each side of ΔDEF contains a vertex of ΔABC, then we call ΔDEF an outscribed triangle of ΔABC. Given ΔABC, let ΦΔABC be the set of all outscribed equilateral triangles of ΔABC. Clearly ΦΔABC is non-empty. In the following we will determine the area of the largest member of ΦΔABC when each angle of ΔABC is smaller than 120° and show that this largest member can be constructed by ruler and compass from ΔABC. The corresponding problem on quadrilaterals has been considered in [1].

The Royal Statistical Society and the Institute and Faculty of Actuaries have published a report, written by Roger Porkess [1], which draws urgent attention to the issue of statistical education: is it fit for purpose? Statistics is an ever-present part of real life, and there can be no doubt that for most pupils it is statistics that will be the most directly used part of their mathematical education.

The report follows an investigation into the perceived statistical needs of various subject ‘communities’, and is a follow-up to The future of statistics in our schools and colleges [2] by the same author and publishers. It is aimed at a wide readership, from policy-makers through curriculum providers and teachers (fairly low in the pecking order!) to authors and other providers of resources. There is no doubt that all such interested parties should be considering the issues covered by this study, and it should be read widely. The title of the report implies suggestions for cross-curricular work, but it is more essentially a plea to rethink the role of statistical education – a plea articulated more fully in [2]. It is the purpose of this article to further the debate.

The article ‘Reflections on the Woolwich Ferry’ [1] brought back some childhood memories. I remember being whisked away with my siblings from the verdant pastures of north-west Kent by bus and tram to Woolwich for a ride on one of the ‘Free Ferry’ paddle steamers; there were four of them and their passing in the mid 1960s was not universally welcomed. It was indeed a real treat that consumed just a little more than a full working day. The depths of a winter school holiday could find us huddled, privileged below the ferry decks on a gantry near the fire hole to receive the welcoming hot blast as its doors opened to accommodate the hand-stoked coke. It was hand-stoked to reduce smoke output, and each steamer could make about eight knots. Presumably health and safety considerations now would not allow for such a luxury of positioning.

However, it was not always wine and roses. To borrow a phrase from one of Kipling's poems, it was sometimes a case of ‘not with this wind blowing, and this tide’, for there were rare occasions when journeys were temporarily delayed by exceptionally windy weather. The elements below (river velocity effect) conspired with the elements above (wind velocity effect) to delay ferry movements. The earlier article proposed a velocity model to deal with effects due only to the river flow and it is the purpose of this note to extend that argument by considering also a velocity effect due to the wind, because the superstructure on a ferry can be as susceptible to the wind as is the keel to the tide.

Given positive numbers a > b, consider the ‘agm iteration’ given by a0 = a, b0 = b and

At each stage, the two new numbers are the arithmetic and geometric means of the previous two. It is easily seen that bn < an, (an) is decreasing, (bn) is increasing and an + 1 − bn + 1 < ½(an − bn), and hence that (an) and (bn) converge to a common limit, which is called the arithmetic-geometric mean of a and b. We will denote it by M (a, b).

A well-known exercise in classical differential geometry [1, 2, 3] is to show that the set of all points (x, y, z) ∈ ℝ3 which satisfy the cubic equation

is a surface of revolution.

The standard proof ([2], [3, p. 11]), which, in principle, goes back to Lagrange [4] and Monge [5], is to verify that (1) satisfies the partial differential equation (here written as a determinant)

which characterises any surface of revolution F (x, y, z) = 0 whose axis of revolution has direction numbers (l, m, n) and goes through the point (a, b, c). This PDE, for its part, expresses the geometric property that the normal line through any point of must intersect the axis of revolution (this is rather subtle; see [6]). All of this, though perfectly correct, seems complicated and rather sophisticated just to show that one can obtain by rotating a suitable curve around a certain fixed line. Moreover, to carry out this proof one needs to know a priori just what this axis is, something not immediately clear from the statement of the problem. Nor does the solution give much of a clue as to which curve one rotates.

A search of the literature failed to turn up a treatment of the problem which differs significantly from that sketched above (although see [1]).

The polynomial (1) is quite famous and has been the object of numerous algebraical and number theoretical investigations. See the delightful and informative paper [7].

What are (themed) cryptic crossnumber puzzles and where do they appear?

I'll start by answering the second question first, as that is the easier. This rather unusual genre of puzzle appears quarterly as part of the series of Listener crosswords, which found a new home in The Times newspaper following the demise of The Listener magazine in 1991. There is also a monthly on-line crossword magazine called The Magpie, which has a numerical puzzle in each issue. As a result, fans of crossnumber puzzles have sixteen a year to tackle. Themed cryptic crosswords appear much more frequently – perhaps ten times as often.

All Listener puzzles have a title and the setter is given a pseudonym. They are also numbered, and in what follows this reference is given in square brackets following the letter L. Puzzles from the The Magpie are referenced according to the issue of the magazine, using M.

We start from two simple identities:

For any p > 0 and 0 ≥ b ≥ a, now let

Can we formulate statements about Gp(a, b) that generalise (1) and (2)? We cannot hope for equalities, but perhaps we can establish inequalities which somehow reproduce (1) when p = 1 and (2) when p = 2. For (1), this might mean an inequality of the form Apap ≤ Gp (a, b) ≤ Bpap for certain constants Ap and Bp, and for (2) a similar statement with ap replaced by ap + bp. However, these are not the only possibilities, as we shall see.

Many mathematical puzzles involve using a scale to compare the weights of two objects, or two groups of objects. The present work studies a similar but different type of question involving a scale.

We are given a set of marbles, whose weights are all different. The only way to distinguish them is to use a special kind of scale. The scale has three trays and each can accept exactly one marble. The scale then indicates which is the heaviest, the lightest and consequently the middle one of the three marbles.

The paper studies the question of ordering 15 different marbles with as few weighings as possible. The problem is taken from the website Enigmes, casse-têtes, curiosités et autres bizarreries [1] and Toppuzzle [2] where the best reported strategy requires 23 weighings. The problem can also be found at the website Trick of Mind [3] where a strategy requiring 22 weighings is proposed. These websites contain many challenging mathematical puzzles, some are classical problems, others are original creations. The author is not aware of any other work on this problem.

The present paper describes a strategy that improves on the value of 22 weighings, and is structured as follows. Section 2 introduces some useful definitions and provides strategies to sort up to 9 marbles. Section 3 proves our main result, that 20 weighings is always sufficient to sort 15 marbles.