Volume 59
, Issue 3

2023

Dear Reader, welcome to this issue of Parabola!

Let $T(P)$ be the number of non-congruent triangles with perimeter $P$ and integral side lengths. Alcuin’s Sequence is the infinite sequence $T(3)$, $T(4)$, $T(5)$, $\ldots$ . I present an algorithm and an elementary derivation of a formula for $T(P)$ .

Given a common 2-dimensional optimisation problem, it is natural to ask: Is there any simple way to instantaneously identify the correct half-plane without substitution? The answer is affirmative: The coefficients of the straight line tell us everything!

I investigate a $3\times3\times3$ and a $4\times4\times4$ Rubik's Cube to find the total number of configurations and thereby to understand patterns in these respective calculations.

Interesting iterative sequences are investigated, including the sequences related to the Collatz Conjecture, Kaprekar's Routine, the digits factorial process and the digits factorial power process.

In this article, elementary mathematics is used to provide a partial proof of Fermat's Last Theorem.

This paper explores how such a seemingly simple theorem as the Pigeonhole Principle has important applications in more difficult mathematics, ranging from geometry to number theory and algebra.

We study the Laplace transform and how it can be approximated for functions with no or complex such transform.

Malfatti's problem is that of finding three non-overlapping circles packed inside a given triangle that have maximal total area. This article explores and compares two potential solutions.

Is there a way to determine an order in which $n$ jobs should be processed at $m$ machines in the shortest possible time, without exhaustive checking? This is the flow-shop scheduling problem. This article surveys solutions that use matrices over the max-plus algebra.

Q1729 We have $n$ coins, all placed heads up on a table. It is permitted to select any $k$ of the coins and flip them; and to do a similar operation repeatedly. Here, $k$ is a fixed positive integer less than $n$. The aim is to get all of the coins facing tails up.

Problem A1: Alice plays the following game. She writes every number from 1 to 125 in her book. On every move, she replaces a couple of numbers with the remainder after dividing the sum of these numbers by 11. What number will be in the book after 124 moves?

The list of winners of the 61th UNSW School Mathematics Competition.
Congratulations, and well done!