The mathematical investigations of natural phenomena in the seventeenth century led to the inventions of calculus and probability. While we know the works of eminent natural philosophers and mathematicians such as Isaac Newton (1643-1727), we know little about the learned women who made important contributions in the seventeenth century. This article features Princess Elisabeth of Bohemia (1618-1680), whose intellectual ability and curiosity left a unique mark in the history of mathematics. While some of her family members were deeply involved in politics, Elisabeth led an independent, scholarly life, and she was a close correspondent of René Descartes (1596-1650) and Gottfried Leibniz (1646-1716).

A tournament is a series of games (matches) among competitors for an overall prize. Beyond sporting events, tournament structure can relate to elections, applicants competing for an employment position, etc. Tournament structure also exists within certain species of birds and mammals where dominance relations develop, forming a tournament-like pecking structure [1]. The format of the tournament depends on the tournament’s objective giving consideration to time, financial, geographical, and other constraints. The two most common formats where players compete pairwise are the single elimination (knockout, sudden death) tournament and the round robin tournament. Our discussion includes both. A third popular format is that of a contest, where all players perform simultaneously only once. Examples of this type include most golf tournaments as well as track and field athletic events. A detailed analysis of contests as they relate to topics discussed here is given in [2].

Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.

The world is amply supplied with probability teasers. These can mystify and intoxicate. Here are some classics.

This article begins with a new look at earlier work on this topic and continues with the presentation of a graphical technique for the determination of the months of a year for which the 13th is a Friday.

It might be thought that the likelihood of the 13 th day of a month being a Friday is the same as that of the 13th being any other day, but this is not so. The full day/date repeat cycle is 400 years, this being the interval between century years which are leap years. The outcome of an extraordinary counting exercise some 50 years ago by a 13-year-old Eton schoolboy, S. R. Baxter [1, 2], was to show that, over this period, the 13th of a month will be a Friday at least once more than any other day. Of necessity, Baxter’s calculation was intricate and an independent confirmation was undertaken as a matter of interest. The 400-year period beginning 1 March 2014 was considered. The work is described in four stages.

An important first step in understanding or solving a problem can be the selection of coordinates. Insight can be gained from finding invariants within a class of coordinate systems. For example, an important feature of rectangular coordinates is that the Euclidean distance between two points is an invariant of a change to another rectangular system by a rigid motion, which consists of translations, rotations and reflections. Indeed, the form of the distance function is an invariant. In calculus courses, students learn about polar coordinates, so that useful curves can be simply expressed and more easily studied.

A geometric dissection is a cutting of a geometric figure (or a finite set of figures) into pieces that we can rearrange to form another geometric figure (or finite set of figures). If our figures are required to be polygons, then there is always a dissection that has just a finite number of pieces. This was established by John Lowry [1], William Wallace [2], Farkas Bolyai [3], and Karl Gerwien [4]. The American Sam Loyd [5] and the Englishman Henry Ernest Dudeney [6, 7] emphasised the goal of minimising the number of pieces that resulted from such a standard dissection.

Figure 1 below shows a triangle ABC and three cevians AX, BY and CZ with intersection points

This Article explores how root multiplicity and polynomial degree influence the structure of the roots of a univariant polynomial. After setting up the notation, we draw upon a result derived in [1], and show that all polynomial roots have a common underlying structure comprising just five parameters. Finally we present some examples involving the lower polynomials.

In [1], the following conjecture was proposed concerning the distribution of ages in a closed interval [0, A]