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 various $p>0$ 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 $k\cdot {2}^{n}+1$ 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 $a{x}^{2}+bx+c=0$ 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 $x\ge 0$ much smaller than 1, the linear function 𝑥 approximates $\mathrm{ln}(1+x)$. 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 ${a}_{1},{a}_{2},\dots ,{a}_{n}$ are equal to $1,2,\dots ,n$ but not necessarily in that order. Find the maximum possible value of

$S=\sum _{k=1}^{n}(k-{a}_{k}{)}^{2}$

David Angell, Sin Keong Tong and Mircea Voineagu

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 ${a}_{1},{a}_{2},\dots ,{a}_{n}$ are equal to $1,2,\dots ,n$ but not necessarily in that order. Find the maximum possible value of

$S=\sum _{k=1}^{n}(k-{a}_{k}{)}^{2}$