"What should we use?" seems to be the question when one approaches a plane geometry problem. In many ways, Euclidean geometry is a laboratory in the realm of logic, an ideal place where one can see how alternative methods can be employed to solve problems. What detail might represent a hint? And from among many choices, what method could one consider? Does the geometric structure suggest a certain type of approach?

Fate laughs at probabilities.

—Edward Bulwer-Lytton, 1st Baron Lytton

Consider the problem of finding triples of numbers (x1, x2, x3) and (y1, y2, y3) satisfying (1)

$${x_1} + {x_2} + {x_3} = {y_1} + {y_2} + {y_3}$$

(2)

$$x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2.$$

Many papers and indeed books have been written about the problem of dissecting a number of squares of different integer side length and reassembling the pieces to form a single square (see [1], [2] and [3]). For example, in the case of 32 + 42 = 52, we can achieve a four-piece dissection as follows:

In [1] de Villiers points out the duality between sides and angles of quadrilaterals. The first objective of this article is make explicit two new dualities in the quadrilaterals. For this we will consider the diagonal segments: if diagonals AC and BD of a quadrilateral ABCD intersect at O, we call the diagonal segments OA, OB, OC, OD. According to [2, pp. 179-188], hierarchical classifications of the convex quadrilaterals are made taking as classification criteria the quantity and position of sides, angles and diagonal segments. In hierarchical classifications the more particular concepts form subsets of the more general concepts. Through classification according to the number and position of equal sides it is possible to define six families: four sides equal, at least three equal, two opposite pairs equal, two consecutive pairs equal, at least one opposite pair equal, at least one consecutive pair equal. In the families two opposite pairs equal and two consecutive pairs equal, the pairs may be equal to each other and then the four sides are equal. So in these two families the possibility DA = AB and AB = BC is excluded. Families analogous to the previous ones can be defined taking as a criterion of hierarchical classification the quantity and position of equal angles or the quantity and position of equal diagonal segments.

This is a follow-up to a recent Gazette Article by Abel [1]. We investigate probabilistic equalisation problems proposed in [1] using the Pólya urn, and characterise them through simple random walks.

Time spent in national pandemic ‘lockdowns’ and ‘tiers’ for most of 2020 has created an opportunity to revisit some ideas previously committed to paper, but unfinished. I now return to one of them, to continue and share the line of thought, if not to complete it.

Figure 1 shows the photograph of a tractrix made of cards. This is a simple yet captivating way to make a tractrix — an arrangement that has recently appeared in some engaging pedagogical resources and literature [1, 2]. When closely spaced cards lean like this, one after another, over a horizontal plane, the contour created is a tractrix — a curve of significance in mathematics, since its revolution around its asymptote produces a pseudosphere: a curved surface with constant negative Gaussian curvature [3, 4].

Consider the general cubic equation:

$${x^3} + a{x^2} + bx + c = 0.$$

A change of variable: x = z + t leads to: (1)

$${z^3} + (3t + a){z^2} + (3{t^2} + 2at + b)z + ({t^3} + {at^2} + bt + c) = 0.$$

In 1631, Johannes Faulhaber published the result that sums of the form

$$\sum\limits_{i = 1}^n {{i^k}} = {1^k} + {2^k} + \,...\, + {n^k}$$

The first solution to the quartic equation is attributed to Lodovico Ferrari, a student of Geralamo Cardano. The solution was published alongside the solution of the cubic in Cardano’s book Ars Magna [1]. In this Article, we introduce a new canonical form to simplify the derivation of the roots of the equation (1)

$${z^4} + {z^3} + f{z^2} + g = 0\quad \textrm{with}\quad f,g \in \mathbb{R}.$$

Any mathematics student who has ever used the cosine rule to investigate simple properties of an integer triangle will immediately have realised that the cosine of each angle of the triangle must be a rational number. It is clear, however, that the same is not in general true for the sines. In [1], it is shown how to use a property of the sines of the angles of an integer triangle to categorise the triangle as being of a particular class. In this article, we develop some of the concepts and results of [1] to derive a method for generating integer triangles of a given class. Finally, we apply our results to find all primitive integer triangles in the particular case of Heronian triangles.

Figure 1 shows a triangle ABC with the midpoints A′, B′ and C′ of its sides. The line segments AA′, BB′ and CC′ are called the medians, and the point G of their intersection the centroid. The line segments AG, BG and CG will be called, for lack of a better name, the semi-medians. It is interesting that the medians of any triangle can serve as the side lengths of some triangle. This property of the medians is referred to as the median triangle theorem in [1, §473, page 282], and is discussed, together with generalisations to tetrahedra and higher dimensional simplices, in [2].

We describe various methods to derive monotonic infinite series for fractions near π and obtain a variety of series for the special case of its convergents. These series immediately show that π is clearly different from these fractions, replicating with series the results in Dalzell [1, 2] and Lucas that used integrals with non-negative integrands to represent the gaps between π and fractions.

Ask anyone who has studied mathematics to a moderate level how many trigonometric functions there are and one is likely to be presented with a range of answers depending on what the person being asked is most likely to remember. Perhaps the ‘calculator button’ three of sine, cosine, and tangent will come to mind as these are the three trigonometric functions found on any standard scientific calculator. At a stretch, perhaps the names for their respective reciprocals, cosecant, secant and cotangent, will be recalled. Beyond the modern standard six, looking at calculus or trigonometric texts published prior to 1900 one soon discovers others going by strange names such as versine, haversine, or coversine (see, for example, [1, pp. 53, 63]). There are at least six others with as many as perhaps ten to twelve having received a name at one time or another. Today all these additional trigonometric functions considered important enough to grace the pages of texts in centuries past have fallen by the wayside, to be largely forgotten in favour of the modern standard six. Of course the pedant amongst us would say there is only one trigonometric function, the sine function, which currently stands as the preferred fundamental trigonometric entity, with all others being simple variations of this function, and they would not be incorrect in asserting this. But having the current standard six seems about the right balance between the minimalistic on the one hand and convenience on the other hand.

The most important properties of any distribution are its moments. The first four moments, for example, can be used describe any data set fairly well. Moments can also be used for estimation.

Given positive numbers xj, yj such that $\sum\limits_{i = 1}^n {{x_j}} = \sum\limits_{j = 1}^n {{y_j}} $ , it can happen that $\sum\limits_{j = 1}^n {x_j^2} = \sum\limits_{j = 1}^n {y_j^2} $ : for example, $({x_j}) = (7,\,3,\,2),\,({y_j}) = (6,\,5,\,1)$ . However, such cases are exceptional.