Thomas Britz
Dear Readers, welcome to the first issue of the 61st volume of Parabola!
Mohammad Samiul Haque
Using the Shoelace Formula, we regenerate the formula for regular polygons.
Carlo A. Z. Pece, Emilly P. Marconcin and Renan S. Saraiva de Queiroz
A simple derivation of the algebraic expression for \(\sin(18^\circ)\). We hope you enjoy it.
Alaric Pow Ian-Jun
In this article, I derive various properties of the beautiful mathematical curve known as the Folium of Descartes, which came about from an honest challenge between Fermat and Descartes.
Martina Škorpilová
The most commonly used and well-known means are the arithmetic mean, the geometric mean, the harmonic mean, and the quadratic mean. The point of this article is to present and elegantly prove the relationships between them.
Sin Keong Tong
This article develops three methods of calculating the expected distribution of the completion length for the set of all \(k\)-digit strings.
Magdalena Stay
My aim is to explore Zipf’s Law by analyzing public domain fantasy and science fiction literature.
Marvin Hicke
This paper presents a concise yet intuitive introduction to Taylor series.
Camden W. Hulse
The purpose of this article is to introduce a new mathematical series called the composite function series.
Mika Akaeda, Yuga Fujiwara, Wakana Imura, Seimu Isoda, Mao Nishikawa, Iroha Oka, Riona Sakai, Rei Sakamoto, Shoei Takahashi and Ryohei Miyadera
In 1984, a mathematical formula was presented as crucial evidence at a murder trial. The formula expressed the probability of having a specific blood type, and the defendant served a ten-year prison term. Since his release from prison, the true culprit admitted to the crime. We study the formula presented at the trial and check its validity.
Yin Zhao and Salmiza Saleh
A natural question arises: Is the product of two numbers of the form \(a^2 + kab + b^2\) still of this form? We answer this question in the affirmative.
David Angell, Mikhail Isaev and Sin Keong Tong
Q1761 Let \(a\) be an integer. Find the number of integers \(b\) such that \((x + a)(x + b) + 2025\) can be factorised as the product of two linear factors with integer coefficients.
David Angell, Mikhail Isaev and Sin Keong Tong
Q1751 Show that it is possible to partition the set of unit fractions \(\Big\{\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots\Big\}\) into finite subsets in such a way that the sum of the fractions in each subset is 1.
Thomas Britz
Dear Reader, I hope that you are well, and welcome to the second issue of Parabola Volume 61!
John Winkelman and Paul Pontikis
Sprouts is an enjoyable topological pencil and paper game for two people invented by the mathematicians John Conway and Michael Patterson.
<p>Aliyah Maxwell-Abrams and Robert Schneider </p>
<p>In this paper we prove identities for six infinite series whose values involve linear combinations of \(\pi\) and \(\ln 2\), that do not appear in standard infinite series references.</p>
<p style="margin-bottom:11px"><span class="bg-unsw-yellow"><a class="btn btn-primary" href="https://www.parabola.unsw.edu.au/sites/default/files/2025-08/vol61_no2_2_0.pdf" rel="noopener" target="_blank"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Aptos,sans-serif">Read the article in PDF</span></span></span></a></span></p>
Md Faiyaz Siddiquee
Being one of the oldest and certainly one of the most well-known theorems in mathematics, the Pythagorean Theorem has a large number of proofs. This paper presents two new proofs, one geometrical and one mechanical, of this theorem.
Sawyer Jacobson and William Balz
This paper provides proof of Holditch’s Theorem in an intuitive sense, relying upon the concrete geometry of Green’s and Stewart’s Theorems.
Aaliya Syal
One day, while doing a square root question, I noticed something very strange and intriguing! This article is the story of my discovery.
David Michael Indraputra and Benny Yong
In the present paper, we will discuss a clever integration by parts trick which can make calculations much easier.
Philipp Guse and Arvin Lamando
This article aims to provide an exposition on tight frames in \(\mathbb{R}^2\), as well as new extensions to their characterization.
Andrew Jackson
This article presents a base theory, called GCSE+, and proves that many of the major theorems in A-level mathematics are equivalent under GCSE+.
Leonardo Gallego Hu
We use matrices to solve algebraic equations and thereby to study the matrix representations of the complex numbers, the bi-complex algebra and the quaternion algebra.
Aaron Manning
When choosing between two options, most of us would select the option that the majority of people prefer. This way of counting votes is called the majority vote. Is this really the “best” way to count votes though? Are there any other reasonable alternatives? What trade-offs must we accept if we determine a winner using an alternative method? What should we do when there is an even number of voters?
David Angell, Mikhail Isaev and Sin Keong Tong
Q1771 Consider sequences of positive integers \(x_1,\ldots,x_{100}\) with the property that either \(x_{k+1} = 4x_ k\) or \(x_k = 5x_k+1\) for each \(k = 1,\ldots,99\). What is the smallest possible sum \(x_1 + \cdots + x_{100}\) for such a sequence?
David Angell, Mikhail Isaev and Sin Keong Tong
Q1761 Let \(a\) be an integer. Find the number of integers \(b\) such that \((x + a)(x + b) + 2025\) can be factorised as the product of two linear factors with integer coefficients.
Thomas Britz
Dear Reader, I hope that you are well, and welcome to this year’s final issue of Parabola!
Eve Parrott
The Isoperimetric Problem is one of the oldest in mathematics. It asks for the least-perimeter way to enclose given volume. In this paper, we use Brakke’s Evolver to numerically compute double and triple bubbles in \(\mathbb{R}^3\) with density \(r^p\) for various \(p > 0\), extending many results in the literature.
Kyler L. Beaumont and Harold J. Smith
In this article, we will seek an answer to the following question: in how many ways can one roll \(n\) ``exploding dice" and get a sum of \(k\)? We will use some very basic techniques of enumerative combinatorics to find an answer.
Mahmoud M. Ayesh
We explore constant-recursive equations, which often appear in mathematical analysis, combinatorics and differential equations.
Swastika Dey and Mrittika Dey
Amicable numbers are two positive integers where each number is the sum of the proper divisors of the other. The abundancy index of a number is the ratio of the sum of all its divisors to the number itself. This article looks at the connection between amicable numbers and the abundancy index.
Z M Tanver Mahmud Anindya
This article provides a historical introduction to the Riemann Hypothesis.
Michael J.W. Hall
It is shown that Pythagorean triples can be used to generate matrices that have integer eigenvalues for all permutations of their coefficients, via simple formulas.
Ryohei Miyadera, Mika Akaeda, Risana Arai, Yuga Fujiwara, Kota Inahara, Seimu Isoda, Iroha Oka, Riona Sakai, Yuito Shirataka and Hikaru Manabe
This article studies various phenomena that deal with probabilities. You can read each section independently, so you are free to skip a section and read the next section.
Josephine Miller
The factorial as traditionally defined has a limitation: it is only valid for integers. A natural question arises: can factorials be applied to non-integer values while maintaining consistency with the traditional definition for integer values?
Bernard Kachoyan and Marc West
It is a reasonable assumption that a cricket team or an individual batter would prefer their expected score not to decrease by being more attacking. This issue will be explored here.
Samantha Blair
The objective of this note is to present four properties highlighting relationships between quadratic triangles and quadratic polynomials.
Jathan Austin
This article presents examples and interesting patterns that arise when exploring digit products of numbers.
Shoya Kise, Takesa Uehara and Takashi Shinzato
We investigate whether the Pythagorean Theorem can be derived solely from the Double-Angle Identity, without invoking infinite series. Furthermore, we attempt to prove the Pythagorean Theorem by employing the Angle-Bisector Theorem in conjunction with the double-angle approach. In addition, we derive a new relational formula for trigonometric ratios, and use it to prove the Pythagorean Theorem.
David Angell, Mikhail Isaev and Sin Keong Tong
Q1783 Let \(x\) be a positive integer. Prove that if \(x^2\) is the difference of two consecutive (integer) cubes, then \(x\) is the sum of two consecutive (integer) squares.
David Angell, Mikhail Isaev and Sin Keong Tong
Q1771 Consider sequences of positive integers \(x_1,\ldots,x_{100}\) with the property that either \(x_{k+1} = 4x_ k\) or \(x_k = 5x_k+1\) for each \(k = 1,\ldots,99\). What is the smallest possible sum \(x_1 + \cdots + x_{100}\) for such a sequence?
Denis Potapov
The problems and solutions from the 63rd Annual UNSW School Mathematics Competition.
Denis Potapov
The list of winners of the 63rd Annual UNSW School Mathematics Competition.