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.
Aliyah Maxwell-Abrams and Robert Schneider
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.
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
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 of the results of Collins from \(\mathbb{R}^2\) to \(\mathbb{R}^3\).
Kyler L. Beaumont and Harold J. Smith
“Exploding dice” is a mechanic used in many popular tabletop games. The idea is simple. Whenever dice are rolled, some specific outcome on any of the dice (usually the highest possible value) allows you to continue rolling those dice and adding to your total, thereby making arbitrarily large sums possible.
Mahmoud M. Ayesh
We consider sequences that have a fixed linear relation between their terms and the previous terms and we investigate the polynomial structures underlying them, characterize the structure of such recurrences, and explain their relationship to polynomials and solutions to polynomial equations, by analysing their algebraic and structural properties.
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
We show that Pythagorean triples can be used to generate matrices that have integer eigenvalues for all permutations of their coefficients, via simple formulas. Further, while integer multiples of such matrices trivially generate additional examples, we have also shown that just a single Pythagorean triple can generate a countable infinity of nontrivially related matrices.
Ryohei Miyadera, Mika Akaeda, Risana Arai, Yuga Fujiwara, Kota Inahara, Seimu Isoda, Iroha Oka, Riona Sakai, Yuito Shirataka and Hikaru Manabe
The theory of probability is a very attractive field of mathematics. It has beautiful theories and vast fields of applications. 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
Factorials are fundamental operations in mathematics, and they appear in many fields, from combinatorics to physics. However, the factorial as traditionally defined has a limitation: it is only valid for integers. A natural question arises: Can factorials be extended applied to non-integer values, such as (1.5)! ?
Bernard Kachoyan and Marc West
This article considers the question of what increased risk can be tolerated in the pursuit of more rapid scoring in cricket.
Samantha Blair
The objective of this note is to present four properties highlighting relationships between quadratic triangles and quadratic polynomials.
Jathan Austin
Unlike whole-number digit sums, which reveal properties such as divisibility by 3 or 9, digit products are not particularly well understood. A lot of interesting patterns arise among the digit products of a whole number and one of its multiples. This article presents such interesting patterns and examples that arise when exploring digit products.
Shoya Kise, Takesa Uehara and Takashi Shinzato
Historically, because the trigonometric ratios \(\sin\theta\) and \(\cos\theta\) are defined by side-length ratios in a right triangle, any proof of the Pythagorean Theorem based on these ratios has been deemed circular. Jackson and Johnson bypassed this obstacle by combining a geometric construction with an infinite geometric series and the Double-Angle Identity, marking a breakthrough in the field. This raises the question: is the novelty of their approach attributable primarily to trigonometric identities or to their use of infinite series? We answer this question and provide new trigonometric proofs of the Pythagorean Theorem.
David Angell, Mikhail Isaev and Sin Keong Tong
Q1781 Suppose that \(n\) positive integers \(a_1,a_2,\ldots,a_n\), all less than 2026, satisfy the equation
\[\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_n}=\frac{2025}{2026}\,.\]
Find the smallest possible value of \(n\).
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
Problem A1: You have all the whole numbers from 1 up to 2025: 1, 2, 3, ... , 2025.
You can put a plus (+) or minus (–) sign in front of every number. If you do this, will their total add up to zero exactly? For example, if the numbers were just 1, 2, 3, 4, you might try +1 − 2 − 3 + 4; but does this work for 1, 2, ... , 2025?
Denis Potapov
The list of winners of the 63rd UNSW School Mathematics Competition