Volume 59
, Issue 2


Dear Readers, welcome to this issue of Parabola, and please enjoy!

We provide a standard list of the 3000 most common English words with a notion of distance, and seek subsets of small volumes of minimum or maximum perimeter.

Recently, I fell asleep having derived the quadratic formula in a way that I thought elegant enough to share with Parabola readers

A herd of goats has the odd habit of dividing into new groups every so often, by one goat leaving from each group to form a new group. The mathematics of this process involves a fascinating mix of partitions, triangle numbers and necklaces.

Welcome, fellow math enthusiasts! Today, I am excited to share with you a disruptive discovery in the world of geometry. By using sequences of fractions arising from periodic continued fractions as seeds to generate Pythagorean triples, I have unlocked a new realm of Pythagorean triples.

We present a number of elegant proofs of the inequality $\pi^e < e^\pi$ and of more general inequalities.

We use Hilbert’s Theorem 90 to parametrise the side lengths of all integer-sided triangles with an angle with a rational cosine value. Our discussion is elementary, self-contained and, hopefully, different and interesting.

A method based on number theory is developed to find the number of special right triangles with a fixed leg of prespecified length.

We study the problem of turning a square sheet of paper into a cup of maximal volume. When the cup has the shape of a rectangular parallelepiped, this problem is an easy exercise of elementary calculus. When the cup has the shape of a square frustum, however, this problem becomes difficult to solve.

We use optimisation theory to determine the factors that govern victory in a game of tug of war.

A regular polygon has the largest area among all polygons inscribed in a circle. In the last issue of Parabola, R. Tanaka and others gave an interesting proof of this fact. In this note, we present a simpler proof which requires only elementary geometry.

We have recently been looking through problems which have been posed and solved in the nearly 60–year lifespan of Parabola. Doing so suggested ideas for a number of new problems, some of which we publish this issue.