In an interesting Note [1] a method for constructing the directrices of a conic section using simple tools, viz. a pencil, a pair of compasses and a straight edge, was presented and it was wondered if the method might have been known before. From earlier days at school this author was aware that for a specific conic section (parabola, ellipse or hyperbola) displayed without reference on a piece of paper it was possible to determine not just the directrices but also the focal points and the usual axis set, again using just the simple tools. Others might also be aware of these facts, but if they are not so well known it might be of interest to present an argument in recipe format that will enable readers to appreciate a route to fathom the business, although there might well be other paths leading also to the same end-result, depending on the starting point.

A Proof Without Words (PWW) contains data, the claim that is to be proved, and one or more diagrams, sometimes without anything else and in other cases with a few mathematical expressions, without any verbal justifications [1]. It is assumed that students and researchers who possess the related appropriate mathematical knowledge will view the drawings and the expressions, will be able to follow and justify each step in the proof and develop their own visual proof abilities. PWW is very much like a cartoon which contains a drawing with sometimes only a few words or sometimes no words at all and the observers are expected to understand the context or the projected message.

A surveyor often encounters problems like Problem 1:

Determine the length of side AC of the right triangle ABC with the right angle at B. A lake intersecting the sides AC and CE prevents the direct measurement of their sides.

In the present paper, we show that the point of intersection of the bimedians of a cyclic quadrilateral belongs to the nine-point circle (Euler’s circle) of the triangle with one vertex at the point of intersection of the quadrilateral’s diagonals and the other vertices at the points of intersection of the extensions of its pairs of opposite sides.

This paper presents some new theorems about the Pascal points of a quadrilateral. We shall begin by explaining what these are.

Let ABCD be a convex quadrilateral, with AC and BD intersecting at E and DA and CB intersecting at F. Let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. By using Pascal’s theorem for the crossed hexagons EKNFML and EKMFNL and which are circumscribed by ω, the following results can be proved [1]:–

1. (a) NK, ML and AB are concurrent (at a point P internal to AB)

2. (b) NL, KM and CD are concurrent (at a point Q internal to CD)

One of the nicer things about growing older is that you tend to find a great deal of pleasure in the little rituals of life. Things like morning coffee or tea, a warm bath, or a glass of good wine at the end of the day. There are some rituals, however, that are not at all enjoyable. Take, for example, the monthly ritual of paying bills. I still do this by cheque, so I need to make sure my bank account balance is sufficient to pay the cheque, write the cheque, and subtract the cheque amount from the account balance. Now there is a little pleasure when one actually adds money to the account balance, but the one time that gives me more than a modicum of joy is when the pence portion of the account balance comes out to be exactly zero. This happens very rarely and unpredictably. As a statistician I became intrigued with trying to find the probability of this event occurring. I did an extensive literature search both in traditional hard copy journals and also online, but I could find no reference to this problem.

When defining a group, do we need to include closure? This is a detail that is often touched upon when the notion of a group is introduced to undergraduates. Should closure be listed as an axiom in its own right, or should it be regarded as an inherent property of the binary operation? There is no clear answer to this question, although there are firm opinions on both sides. Indeed, a very brief survey of group theory textbooks found in [1, pp. 458-459] suggests that there is a rough 50 : 50 split between authors who include closure explicitly and those who do not. In this Article, we go back to the beginning of the twentieth century to provide some historical perspective on this problem.

We consider the second order homogeneous linear differential equation (H)

$${ ay'' + by' + cy = 0 }$$

Magic squares have long been popular in recreational mathematics. Their potential for introducing students to ideas in linear algebra was recognised over forty years ago in [1] and later in [2]. More recently they have proved to be a fascinating topic for undergraduate exploration, especially when students have access to a computer algebra package [3]. Some results on powers of magic square matrices can be found in [4], [5] and [6]. (Readers who google the title ‘Odd magic powers’ of Thompson’s paper [5] will be treated to a wide variety of non-mathematical exotica!)

The launch and subsequent motion of a projectile provide a context for several quantities that yearn to be optimised. Most notable is the horizontal range of the projectile, a problem dating back to Galileo and still studied in modern times; see, for example [1], [2], [3], [4]. In a different direction, the articles [5] and [6] provide a solution to the problem of finding the angle of launch that results in the trajectory of longest arc length.

Two common methods of approximating difference problems are by rounding or truncating the minuend and the subtrahend before calculating the difference. Since rounding generally gives a better approximation of a number than truncation, one might expect that rounding also generally gives a better approximation in difference problems. We show that intuition to be false.

Steiner’s proof of what is now called the Steiner-Lehmus theorem was published in 1844, the same year as the book The three musketeers, written by the French author Alexandre Dumas. The motto One for all, all for one (Einer für alle, alle für einen; Un pour tous, tous pour un; Uno per tutti, tutti per uno) of the three musketeers came into widespread use in Europe in the 19th century, and its essence is that the three musketeers are inseparable; each member pledges to support the group, and the group supports each member. Now there are three classical geometries of constant curvature, namely Euclidean, spherical and hyperbolic geometries, and one can argue that, like the three musketeers, these geometries should be considered as being inseparable; that is, an idea, theorem or proof in any one of them should automatically be considered in the other two. The issue here should be not only to decide whether a particular result is true, or false, in a given geometry, but to understand which particular properties of the geometries make it so.