Thomas Britz
Dear Readers, welcome to the first issue of the 62nd volume of Parabola!
Ian Jung
We derive a recurrence relation of the Riemann zeta function at even natural numbers by first establishing a recurrence for \(F_n\).
Soham Dutta and Nandita Sen
We provide a surprising spin-off of the AM-GM-HM Inequality and other inequalities for complex numbers with a fixed modulus.
Artem E. Atamanchuk and Zdravko F. Starc
This article examines the fundamental principles of the method of mathematical induction and its application to mathematical problems.
Fabian Schmitt and Christopher C. Tisdell
We explore the possibilities for using the finite double-edged straightedge at its full potential, and explore several different opportunities to resolve some of the tension between theory and practice in the learning, teaching and practising of constructive geometry. In particular, this extra feature will mean that we can remove the use of a compass.
Seung Won (Daniel) Oh
This article proposes an alternative to the usual approach for reducing the cubic equation to a quadratic equation.
Sawyer Jacobson and Jeffery Zhang
The paper presents a derived algorithm for estimating graphical minimal surfaces through the use of mean curvature flow. Simulations are provided, demonstrating the effectiveness of the numerical approach.
Onur Örnek
We discuss two fundamental questions that often arise when considering random binary strings, namely the waiting time and location until a fixed pattern appears.
Md Sadikur Rahman and Tandra Sarkar
We establish an alternative proof of the sum and square sum identities for consecutive tetrahedral numbers using the Cauchy Residue Theorem.
Marc Jornet Sanz
We study the mental-calculation theory behind square roots and its deep connection with mathematics, giving later relations with other tasks such as division or cube roots.
Mohammed Ilyas Dehbi
This note establishes a novel, non-differential approach for finding the local maxima and minima of polynomials.
Pranav Kudumula
Where are my manners? Let me introduce myself. I am Pranav, a 12-year-old eighth grader who seems to spend most of his time doing recreational and contest mathematics instead of going to school. In this note, I'm going to share a short result in number theory that I found while goofing around (hey, that rhymes!).
Md Sadikur Rahman
We present simple geometric proofs of the limits \[
\lim_{x \to 0} \frac{\sin x}{x} = 1\qquad\text{and}\qquad
\lim_{x \to 0} \frac{1 - \cos x}{x} = 0
\]
Andi Du
We give a new proof of the Soddy-Gosset Theorem by using linear algebra, and we give some applications of the theorem.
Artem E. Atamanchuk
A new family of weighted integral inequalities generated by logarithmic derivatives is constructed and investigated in this work. The obtained results substantially extend classical estimates by allowing one to account for the influence of a weight function on the nature of integral relations.
David Angell, Mikhail Isaev and Sin Keong Tong
Q1791 Consider an ``old--fashioned'' 12-hour clock face having an hour hand, minute hand and second hand. Is there any time at which the three hands are equally spaced around the clock face? If so, then find all such times.
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.
Thomas Britz
Dear Reader, it is with great pleasure that I welcome you to this second issue of Parabola Volume 62!
Ryan Bansal
We shall see how a simple combinatorial question regarding longest increasing subsequences leads to nontrivial connections across mathematics.
David Nachuan Chen
We examine the "scuffed badminton" scoring process: a two-player game in which the score \((a,b)\) is reduced to \((a/\textrm{gcd}(a,b),b/\textrm{gcd}(a,b))\) after each rally.
Advik Sakhare
This article develops an analytical approximation for the radial probability density function of s-orbital electrons by synthesising de Broglie's standing wave picture with probabilistic constraints.
Hyunsik Shin
I derive a mathematical model of planetary motion solely from Kepler's three laws of movement, by transforming the elliptical trajectory nature of planetary motion to a circular orbit by a suitable orthographic projection, simplifying analysis and computation.
Advik Sakhare
We formulate three mathematical models of increasing complexity to describe how the economic productivity and total monetary output of a region evolves over time.
Takashi Shinzato
New proofs without words for sine and cosine triple-angle formulae.
Anaya Kale and Brian Rushton
Subdivision rules are ways of creating fractals. They have been studied extensively in higher dimensions but have largely been overlooked in one dimension. We provide definitions, examples and fundamental results on these one-dimensional subdivision rules.
Songjia Ren
Gothic architecture originated in France in the 12th century and turns mathematical and physical understandings into beautiful order expressed in stone.
Wai-Yiu Keung
This note presents a simple and intuitive introduction to the famous and beautiful Nyquist-Shannon Sampling Theorem.
Ryohei Miyadera, Mika Akaeda, Wakana Imura, Kahori Komaki, Iroha Oka, Kikuno Ooyagi, Ryunosuke Shimada, Hikaru Manabe and Kazuaki Fujii
In this article, we study various aspects of exponential functions. Many interesting stories and a discovery are presented.
Jason Deng
In the famous Two Envelopes Problem, a player is told that one envelope contains twice as much money as in another envelope. The player opens one envelope, sees an amount, and is offered a choice: switch or stay. A common (but wrong) argument claims that switching is always better.
Candace Shi
Elliptic curves are beautifully puzzling yet practical mathematical objects that, from afar, hold an illusion of simplicity with their equation: \(y^2 = x^3 + Ax + B\).
Christopher C. Tisdell
An interesting part of the history of geometry is the gradual refinement of the "starter pack" sufficient to perform the standard geometric constructions. For instance, Euclid's starter pack involved a straightedge and compass together with two given points. This article summarises the evolution of geometric construction starter packs and highlights one of the most recent, from a Master's thesis by Yuyan Cai.
Arjun Sarma
Famous Scottish magician Alex Elmsley often began a trick with a deck of ordered cards and shuffled it multiple times. To the audience's amazement, the deck would magically be returned to its original configuration. This paper will teach you how use permutation graphs to recreate this trick and solve other fun puzzles.
Larsson A. Laldingliana
In the late 19th century, the statistician Karl Pearson observed that, in many moderately skewed unimodal distributions, the mean, median and mode satisfy an approximate relationship.
Anna McGann
There is something to be gained from this book for everyone. For a young mathematics enthusiast, it opens up new ways of thinking about mathematical ideas, and for mathematicians with more experience, it recaptures the joy of mathematics.
David Angell, Misha Isaev and Sin Keong Tong
Q1807 Let \(N = 1^1 + 2^2 + \cdots + 2026^{2026}\). Is \(N\) a perfect square?
David Angell, Misha Isaev and Sin Keong Tong
Q1791 Consider an ``old--fashioned'' 12-hour clock face having an hour hand, minute hand and second hand. Is there any time at which the three hands are equally spaced around the clock face? If so, then find all such times.
Denis Potapov
Denis Potapov