Thomas Britz
Dear Reader, I am proud and excited to announce that it is the 60th anniversary of Parabola!
Marcus Collins
The isoperimetric problem asks for the least-perimeter way to enclose a given volume. We numerically solve this problem for double, triple and quadruple bubbles in the plane with density \(r^p\) for \(p>0\) using Brakke’s Evolver.
Aadit Jain
What is an amplifier and how does it work? To answer these intriguing questions, I constructed a model of an amplifier.
Onur Kaan Genc
A ray of light emanates at some angle from a corner of a square region and follows a path determined by its reflections off the walls of the square. We determine when the ray’s path is finite, and we compute its length in this case.
Joseph Levine
How much do financial management fees cost investors? This article studies fees charged annually as a percentage of Assets Under Management (AUM).
Frédéric Beatrix
We will attempt to multiply like a Babylonian student and will derive beautiful sexagesimal approximations.
Jack W. Leventhal
Wacław Sierpiński proved that there exist infinitely many odd integers k𝑘 such that numbers of the form are never prime for any integer 𝑛. The values of 𝑘 with this property are called Sierpiński numbers. The Sierpiński Problem is to find the smallest Sierpiński number.
Jozef Doboš
The solution formula to the quadratic equation is usually derived in textbooks by completing the square. This is very unnatural and potentially confusing for students. A more appropriate approach is given here.
Kyle Wu
We describe Vieta Jumping, a technique that was used to solve the notorious 1988 International Mathematical Olympiad’s Problem 6. We provide explanations, examples and visual representations, as well as other problems that can be solved by this technique.
Robert Schneider
It is a well-known estimate that, for small values much smaller than 1, the linear function 𝑥 approximates . Alas, this easy approximation does not hold on all of the interval [0,1]. A far better almost-linear approximation is presented in this article.
Alaric Pow Ian-Jun
I consider primeless and single-prime intervals of any given length, and show easy ways in which to construct them.
David Angell, Sin Keong Tong and Mircea Voineagu
Q1732 Suppose that the numbers are equal to but not necessarily in that order. Find the maximum possible value of
David Angell, Sin Keong Tong and Mircea Voineagu
Q1721 Nine people are participating in a "secret Santa" at an office Christmas party.
Thomas Britz
Dear Readers, welcome to the second issue of this year’s 60th anniversary of Parabola! I am very excited to share two exciting celebrations with you!
Bernard Kachoyan
A little while ago, I was looking at a particular path-finding problem. This led me to rediscover the fascinating world of percolation theory.
Milan Pahor
The 100 Prisoner Problem is one of the most bewildering puzzles in the theory of probability. The solution is simple but wildly counter-intuitive.
Shoei Takahashi, Hikaru Manabe, Keita Mizugaki and Ryohei Miyadera
In this study, we investigate Nim games, which are combinatorial games. A combinatorial game is one of the best themes for high school mathematics research, because there are still many unstudied themes.
Thomas Britz
The book Vector can be seen as a history of the discovery of the perceived natural laws of physics during the past few thousands of years. The focus on vectors and tensors provides an interesting and well-chosen framework for this history, as does the focus on its pivotal figures.
Robyn Arianrhod
This article reproduces an extract from the recently published book Vector: A Surprising Story of Space, Time, and Mathematical Transformation (UNSW Press, July 2024).
Brendan Mabbutt
The mathematical relationships are derived between the volume of acid and base present, the concentration said acids and bases and the pH of the mixture for multiple types of titration. Equilibrium methods are employed to derive such relations for monoprotic and polyprotic acids along with strong and weak acids and bases.
Ajay Kumar K S
This article presents a new way to prove Pythagoras’ Theorem using geometry, trigonometry and algebra.
Friday Michael
We provide several equivalent statements that characterise the Kepler triangle. We then show that there is a non-right triangle that exhibits properties similar to those of the Kepler triangle.
Janelle Powell
This article gives a nice example of a function with both finite area and finite rotated volume.
Brian K. White and Edward T. Bednarz III
Society uses decimal (base 10) as the standard number system but we argue here that octal (base 8) is a preferable number system.
David Angell, Sin Keong Tong and Mircea Voineagu
Q1748 Three cyclists Andy, Bobby and Cassie ride around a circular track, all in the same direction, at respective speeds of 24, 40 and 50 kilometers per hour. At certain times, all three of them are together. In between two successive triple meetings, how many times are there when two of the cyclists meet?
David Angell, Sin Keong Tong and Mircea Voineagu
Q1732 Suppose that the numbers < are equal to but not necessarily in that order. Find the maximum possible value of
Thomas Britz
Dear Readers, welcome to the third and final of this year’s 60th anniversary of Parabola! Looking back at the year, many exciting things have happened.
Alaric Pow Ian-Jun
Whereas most of us are familiar with exponents, not much has been said about the tetration function. In this paper, I dive into the fascinating aspects of this function and its closed-form derivative.
Kyumin Nam
In this note, we evaluate definite integrals of the floor function and of more general functions.
Diya Gandhi
This paper provides different techniques for solving maths competition problems that involve sequences.
Peter Shan
The dots and lines method is a common technique that can be used to solve surprisingly many counting problems. This paper will extend the method to cases in which individual variables are bounded, before exploring various interesting problem solutions that involve the technique.
Revanth Sainath Killamsetty
In this article, linear optimisation is applied to the problem of setting up facility locations to optimise the service for pharmacies. The programming environment used is JuMP, a specialised modelling environment embedded in Julia, and the linear optimisation model used is HiGHS.
Benjamin Hogan
You are a surveyor trying to measure the dimensions of a small, convex island in the ocean from your ship. Besides the trivial solution of directly measuring the island’s dimensions by sailing the perimeter, how could one approximate the island’s area with just angle measurements and a log of the ship’s movement?
Benson Bens
The purpose of this article is to introduce a new and useful notation for an equivalence on the tangents of functions. Properties of the relation are given, and examples are presented to show how it can be used to useful effect.
Koshiro Fujiwara, Konoha Natori, Kosei Kaku, Manami Suwa, Nobuaki Komeda, Tomoya Kaneshiro, Yoshino Kasahara and Kohhei Yasuda
We derive the parametric solutions of Diophantine equations $a^2 + pb^2 = c^2$ in two different ways. The first way is based on elementary functions and the second way uses elementary geometry. As far as we know, the second way has not previously been published.
Fabian Dat Trinh and Christopher C. Tisdell
How can we join two given points when our physical straightedge isn’t long enough?
We present a solution to this problem that is more efficient than Yanagihara's classical solution.
Nishant Iyengar and Anju Iyengar
This article presents a new method to compute rational approximations for the positive square roots of non-negative real numbers. Using only addition, multiplication and division, this method has the speed and accuracy of top-down methods and the ease of use of bottom-up methods.
David Angell, Sin Keong Tong and Mircea Voineagu
Q1751 Show that it is possible to partition the set of unit fractions
into finite subsets in such a way that the sum of the fractions in each subset is 1.
David Angell, Sin Keong Tong and Mircea Voineagu
Q1742 Show that if the product of two, three or four consecutive positive integers is increased by 1, then the result is (respectively) never, sometimes or always a square.
Denis Potapov
Problem A1: Let \(x\) be a positive integer. The next-to-last digit of the square \(x^2\) is odd. Find the last digit of \(x^2\).
Denis Potapov
The list of winners of the 62nd UNSW School Mathematics Competition.