Heronian triangles are triangles with integer sides and area. In [1], a classical result about squares in arithmetic progression was obtained by proving that it is not possible for the altitude of a Heronian triangle to divide the base in the ratio of 1 : 2. In this article we shall investigate general ratios of the form 1 : n, where n is a positive integer. These base ratios do not appear to have received previous attention despite the wealth of results about other aspects of Heronian triangles [2].

Twelve months ago, when I was listening to Mike Askew's Presidential Address, my emotions were divided between thoroughly enjoying his presentation and a growing sense of trepidation that it was my turn next! The time seems to have passed very quickly and I'm somewhat nervous, but incredibly honoured, to take the helm as we explore “the case for mathematical fluency”.

In this paper I look at different aspects of mathematical reasoning, and argue that we need to make sure students of all ages engage in a range of mathematical reasoning, particularly given the evidence that teaching for reasoning is a powerful complement to teaching that is more focused on skills and procedures.

The following theorems are famous landmarks in the history of number theory.

Theorem 1 (Fermat-Euler): A number is representable as a sum of two squares if, and only if, it has the form PQ2, where P is free of prime divisors q ≡ 3 (mod 4).

Theorem 2 (Lagrange): Every number is representable as a sum of four squares.

Theorem 3 (Gauss-Legendre): A number is representable as a sum of three squares if, and only if, it is not of the form 4a (8n + 7).

A driver’s estimate of the average velocity of vehicles on a highway is biased by the driver’s sampling of those vehicles that pass the driver or are passed by the driver. Surprisingly, the magnitude of the bias does not head towards infinity as the driver’s speed heads towards infinity. Here, we explore these interesting facts.

John Horton Conway lived to discover the Mathematics behind problems, always working to isolate a pure, essential kernel of truth. He loved to communicate these simple truths to others, often changing the way they thought. Everything John touched turned to Mathematics, and to very beautiful Mathematics. John was generous with his mathematical riches; he gave them to everyone that showed interest — whether at the coffee house, at the sushi restaurant, or in Mathematics departments. He was a magnet for all mathematicians, and he welcomed all who came to him. John would find a way to start with some simple calculation, a game or puzzle and turn it into whatever he wanted to explain. John often chose problems about games and recreational topics, but the insights he derived changed our understanding of several very serious branches of Mathematics.

For n ≥ 1 an integer, we shall denote by σ(n) the sum of the (positive) divisors of n.

Most spirals are continuous spirals, certainly those famous in history such as the traditional Archimedean spiral (r = a + bθ) and the logarithmic spirals (r = aebθ. Here we consider ‘discrete spirals’.

The Lyness equation (1)

\begin{equation}{X_{n + 1}} = \frac{{{X_n} + a}}{{{X_{n - 1}}}},\,(a,{x_1},{x_2} > 0)\end{equation}

In 1942 R. C. Lyness challenged readers of the Gazette to find a recurrence relation of order 2 which would generate a cycle of period 7 for almost all initial values [1].

The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analogue to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley-Hamilton theorem.

Folded paper road maps are found next to sextants in the pile of obsolete navigation tools. GPS navigation apps like Waze and Google Maps, accurate to within a few metres, are available on all smartphones and most new cars. These apps provide drivers with real time traffic conditions and suggest minimum drive time routes, giving drivers the ability to avoid congestion and delays caused by heavy traffic, accidents, road construction and other hindrances.

Calculus students are taught that an indefinite integral is defined only up to an additive constant, and as a consequence generations of students have assiduously added ‘+C’ to their calculus homework. Although ubiquitous, these constants rarely garner much attention, and typically loiter without intent around the ends of equations, feeling neglected. There is, however, useful work they can do, work which is particularly relevant in the contexts of integral tables and computer algebra systems. We begin, therefore, with a discussion of the context, before returning to coax the constants out of the shadows and assign them their tasks.

The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.

The National Archery in the Schools Program (NASP) began in Kentucky, USA in 2002 and has rapidly expanded to thousands of students around the United States. The program teaches archery in physical education classes and organises tournaments for student archers in elementary school and high school. The program goals include improving student motivation, attention, behaviour, attendance and focus, as well as introducing students to an outdoor skill with the hope that this may increase attention to wildlife conservation efforts in the future.

In this paper we state the original identity proved by Euler, and we provide an alternative proof of this result. We then extend the approach to derive a further identity for the sum of reciprocals of squares, along with two other related identities. We conclude by obtaining yet another extension of the identity, and this is accomplished by means of a probability density function, or pdf.

Probability and expectation are two distinct measures, both of which can be used to indicate the likelihood of certain events. However, expectation values, which are often associated with waiting times for success, may, at times, speak more clearly and poignantly about the uncertainty of an event than a theoretical probability. To illustrate the point, suppose the probability of choosing a winning lottery ticket is 2.5 × 10−8. This information may not communicate the unlikely odds of winning as clearly as a statement like, “If five lottery tickets are purchased per day, the expected waiting time for a first win is about 22000 years.”