Pi (π) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. Although for calculations using the number π usually one only needs a few decimal places, mathematicians have devoted much effort to obtain as many decimal places as they have been able to calculate. For a general description of the methods used to approximate the value of π, see e.g. [1, 2].

A couple of years ago, a non-mathematician friend of mine who takes an interest in mathematics mainly as puzzles mentioned a problem of Euler.

It is well known that the real geometric series ${\sum\limits_{n = 0}^\infty{a{k^n} = a + ak + a{k^2} + }}$ … converges to a definite sum if the common ratio, k, is such that |k| < 1, the sum being ${a}\over{1-k}$. For example, if a = 1 and ${k ={{1}\over{2}}}$ we obtain the series ${{1} + {{1}\over{2}} + ... + {{1}\over{2}^n} + ...}$, whose partial sums are ${{1}, {{3}\over{2}}, {{7}\over{4}}, ..., 2 -{{1}\over{2}^n}}$, …, and these are clearly approaching the value 2 as n becomes larger and larger. As n → ∞, ${2} - {{1}\over{2}^n}{\unicode{x2192}} \,\,2$, in agreement with the formula ${{a}\over{1-k}} {=} {{1}\over{1}-{1\over2}}=2$.

A series of consecutive odd numbers has interesting properties useful for classroom investigation [1]. This Article was inspired by the series of fractions for ${{1}\over{2}}$ using only sums of consecutive odd numbers

$$\frac{1}{3} = \frac{{1 + 3}}{{5 + 7}} = \frac{{1 + 3 + 5}}{{7 + 9 + 11}} = {\text{ }} \ldots {\text{ }} = \frac{{1 + 3 + {\text{ }} \ldots {\text{ }} + (2n - 3) + (2n - 1)}}{{(2n + 1) + (2n + 3) + {\text{ }} \ldots {\text{ }} + (2n + (2n - 1))}}.$$

When first learning about geometric series, students often wonder why these series are termed 'geometric'. The geometry of some geometric series is readily apparent for some common series such as

$$\sum\limits_{n = 1}^\infty{\left( {\frac{1}{2}} \right)}^n = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...{\text{ }}.$$

This Article is motivated by the observation that the median triangle theorem can be thought of as a dual to a theorem of Pompeiu. It treats questions that arise from the pursual of this duality, and especially of certain imperfections in this duality.

Computing mathematical expectation for an experiment involving a finite number of numerical outcomes is straightforward. Let X denote the random variable having n possible values x1, x2, x3,…, xn. Letting pk denote the probability of xk, the expected value of X is

$$E\,(X)\, = \,\sum\limits_{k\, = \,1}^n {{x_k}{p_k}} ,$$

which can be interpreted as a weighted average of all xk, where the weight of each outcome is represented by its probability. But caution is required when interpreting the sum if there are infinitely many outcomes and the series fails to converge absolutely.

For integers n, , let Sk (n) denote the power sum 1k + 2k +… + nk, with S0 (n) = n. As is well known, Sk (n) can be represented by a polynomial in n of degree k + 1 with constant term equal to zero, i.e. ${S_k}\,(n)\, = \,\mathop \sum \nolimits_{j\, = \,1}^{k\, + \,1} \,{a_{k,j}}{n^j}$, for certain rational coefficients ak,j. In 1973, in the popular science magazine Kvant*, appeared an article by Vladimir Abramovich [1] in which, among other things, the author derived the following minimal recurrence relation which expresses in a very compact way the connection between the power sums Sk (n) and Sk-1 (n):

(1)

where the polynomial $S_{k\, - \,1}^*\,(n)$ is constructed by replacing in Sk-1 (n) the term nj with the expression ${{{n^{j + 1}} - n} \over {j + 1}}$ for each j = 1, 2, … , k. Note that $S_{k\, - \,1}^*\,(n)$ has degree k + 1, as it should be.

The Euler numbers are the coefficients En in the expansion

(1)

Since sec x is an even function and tan x an odd one, this decouples into

(2)

and

(3)

where all three series converge if and E2n + 1 are sometimes called the secant numbers (or secant coefficients) and tangent numbers (or tangent coefficients) respectively.

We present some less known variations of the the Vecten configuration and give purely geometric proofs for them. It is unlikely that these variations (and even proofs?) are new, probably just well-hidden in the literature. If a reader happens to know references for the variations discussed (or other geometric proofs), please let the authors know. At [1] the reader can find a dynamic webpage on our topic.

According to Mitchelmore [1], generalisations are the cornerstone of school mathematics, covering various aspects like numerical generalisation in algebra, spatial generalisation in geometry and measurement, as well as logical generalisations in diverse contexts. The process of generalising lies at the heart of mathematical activity, serving as the fundamental method for constructing new knowledge [2, 3]. In this paper we will generalise an interesting geometry problem that appeared in the 1995 edition of the International Mathematical Olympiad (IMO) [4].

Let ABC be a triangle with sides a, b, c, semiperimeter s, circumradius R, inradius r and area Δ. We introduce

$$Q\, = \,{(a - b)^2}\, + \,{(b - c)^2}\, + \,{(c - a)^2}$$

$$M\, = \,{(\left| {a - b} \right|\, + \left| {b - c} \right|\, + \,\left| {c - a} \right|)^2}.$$

At university I was intrigued by a historian friend and his constant quest for gobbets – short, apposite quotations that he avidly collected for possible use in future essays. It strikes me that there might be analogous nuggets of mathematics and I offer the following five short, self-contained items both for amusement and for possible classroom use. Each contains a surprising element and I have deliberately left a few loose ends to encourage further exploration.

It is a well known and easily verifiable fact that not all integer triangles have integer areas. Consider the triangles with sides {9, 10, 17}, {13, 14, 15}, {5, 7, 8} and {6, 7, 9} with respective areas 36, 84, and . The first two, whose areas are integers, are called Heronian triangles. The second triangle also has the additional property that its sides are consecutive integers and is an example of a Brahmagupta triangle, named after an Indian mathematician, born in AD 598. These are called Super-Heronian triangles in [1] and a method is developed there for generating examples of such triangles.

Self-working card tricks have been a staple for budding magicians and entire books have been written about them, e.g. Fulves [1]. Lately selfworking card tricks have become common fodder for social media. In this note, we generalise two such card tricks (from [2] and [3]) that are based on the Australian shuffle.

The centre of mass of a plane, uniform, lamina in the shape of a regular n-gon coincides with the centre of mass of n unit masses, one at each of its vertices. As many n-gons fail to have this property it is remarkable that all triangular laminas, whether regular (i.e. equilateral) or not, do have this property. Here we discuss this difference and we assume throughout that each lamina has unit density (so, disregarding units, its mass is the same as its area). We shall refer to the centre of mass of the lamina as its centroid, and the centre of mass of unit point masses, one at each vertex, as its vertex-centroid. Also, we shall say that a polgonal lamina is vertex-equivalent if its centroid coincides with its vertex-centroid, i.e. if, and only if, .