Thomas Britz

Dear Readers, welcome to this year’s first issue of Parabola! It features the most articles (10) by the most contributors (24) from the most countries (9), in Parabola’s history long. Enjoy!

Janelle Powell

The concept of integrals can seem paradoxical. It’s difficult to comprehend how the area underneath some curves that continue towards infinity could be finite.

Moosa Nasir

When representing numbers in number systems to a given base $b$ , such as $b=10$ (decimal numbers) or cases $b$ , such as $b=2$ (binary numbers), there is nothing to stop us from using non-integer bases. This article shows how arithmetic works when using reciprocal bases $1/b$ .

Frédéric Beatrix and Peter Katzlinger

The *doubling of the cube*, also known as the Delian problem, is one of the three ancient problems from the 5th century BC. In 1837, Pierre Wantzel proved that this problem is impossible to solve precisely. However, it is possible to solve approximately.

James Houghton and Kristin Osika

This paper investigates Lévy Stable distributions, how they can be fit to stock market return data, and the methods used to convert these distributions into predictive indicators of stock market crashes.

Trevor Chi-Yuen Tao

This article explains the concept of zero-knowledge proofs and offers an interesting application for detecting the use of engine-assistance in competitive chess.

Aadit Jain

20-year mechanical calendar is created using only elementary school mathematics.

Rikuto Tanaka, Jinya Miyamoto, Yuki Maruo, Keita Nakayama and Ryohei Miyadera

In this article, we present an elementary proof of the following fact:

A regular polygon has the largest area among all polygons inscribed in a circle.

Rizky Reza Fauzi, Steven and Jonathan Hoseana

You might have been told that subtraction is the inverse of addition. Strictly speaking, this is not true; subtraction is actually not the inverse of addition. This article aims to carefully explain this fact.

Enoch Suleiman and B. Sury

The article presents the beautiful binomial transform and several pretty identities arising from it, involving Fibonacci numbers, Catalan numbers and trigonometric sums.

Krish Mahtani

The Lotka-Volterra model is used together with Matlab to predict the dynamic behavior of COVID-19, and the model's merits and limitations are discussed.

David Angell, Sin Keong Tong and Mircea Voineagu

Q1701 A school class consists entirely of twins: 2n2+2n pairs of them, where n≥2. Including the teacher, there are 4n2+4n+1 people in the class, so they can stand in a 2n+1 by 2n+1 square array.

David Angell, Sin Keong Tong and Mircea Voineagu

Thomas Britz

Dear Reader, welcome to this issue of Parabola

Grant Blitz, Timothy Chen, Marcus Collins, Milan Duong-Gordley, Bennett Feng, William Gao, Xiaorui Hang, Summer Kang, Tina Lou, Pranav Mallina, Noah Mok, Ashwin Naren, Eve Parrott, Zubi Talwar, Aadi Upadhyayula and Edward Zhang

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.

Jason Zimba

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

John Winkelman and Mark Yeo

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.

Frédéric Beatrix

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.

Kyumin Nam

We present a number of elegant proofs of the inequality πe<eπ and of more general inequalities.

B. Sury

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.

Onur Kaan Genc

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

Shoei Takahashi, Aoi Murakami, Hikaru Manabe, Daisuke Ikeda and Ryohei Miyadera

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.

Peyman Fahimi and Ali Rastqar

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

Md Faiyaz Siddiquee

Let T(P) be the number of non-congruent triangles with perimeter P and integral side lengths. Alcuin’s Sequence is the infinite sequence T(3), T(4), T(5), …. I present an algorithm and an elementary derivation of a formula for T(P).

Hoover H. F. Yin

Given a common 2-dimensional optimisation problem, it is natural to ask: Is there any simple way to instantaneously identify the correct half-plane without substitution? The answer is affirmative: The coefficients of the straight line tell us everything!

Naren Ramesh

I investigate a 3×3×3 and a 4×4×4 Rubik's Cube to find the total number of configurations and thereby to understand patterns in these respective calculations.

Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori and Ryohei Miyadera

Interesting iterative sequences are investigated, including the sequences related to the Collatz Conjecture, Kaprekar's Routine, the digits factorial process and the digits factorial power process.

Kelvin Muzundu

In this article, elementary mathematics is used to provide a partial proof of Fermat's Last Theorem.

Sizhe Pan

This paper explores how such a seemingly simple theorem as the Pigeonhole Principle has important applications in more difficult mathematics, ranging from geometry to number theory and algebra.

Joshua Im and Jeonghyeon Seo

We study the Laplace transform and how it can be approximated for functions with no or complex such transform.

Martina Škorpilová

*Malfatti's problem *is that of finding three non-overlapping circles packed inside a given triangle that have maximal total area. This article explores and compares two potential solutions.

Ricky Wijaya, Jonathan Hoseana and Iwan Sugiarto

Is there a way to determine an order in which n jobs should be processed at m machines in the shortest possible time, without exhaustive checking? This is the flow-shop scheduling problem. This article surveys solutions that use matrices over the max-plus algebra.

David Angell, Sin Keong Tong and Mircea Voineagu

**Q1729 **We have n coins, all placed heads up on a table. It is permitted to select any k of the coins and flip them; and to do a similar operation repeatedly. Here, is a fixed positive integer less than n. The aim is to get all of the coins facing tails up.

David Angell, Sin Keong Tong and Mircea Voineagu

Denis Potapov

**Problem A1**: Alice plays the following game. She writes every number from 1 to 125 in her book. On every move, she replaces a couple of numbers with the remainder after dividing the sum of these numbers by 11. What number will be in the book after 124 moves?

Denis Potapov

The list of winners of the 61th UNSW School Mathematics Competition.

Congratulations, and well done!