Year 2018 - Volume 54
2020 - Present

Thomas Britz

Dear Readers, welcome to this year’s first issue of Parabola!
In this issue, you can find three excellent articles, David Angell’s beautifully set problems, and more of Robert Schneider’s hilarious comics strips 2Z2𝑍 Or Not 2Z2𝑍 which also offer the occasional cryptic puzzle.

Jeremy Muskat

Communication is no longer private, but rather a publicly broadcast signal for the entire world to overhear. Cryptography has taken on the responsibility of securing our private information.

Michael Barz

This paper describes when given polynomials zn+zk+zj−1 have a unimodular root.

Sin Keong Tong

The terminating sum T(n) of a positive integer n is obtained by repeatedly adding the digits of n until a single digit number is obtained.

Robert Schneider

An odd comic about even numbers.

David Angell

Q1551 We have a pattern of 34 dots arranged and may remove any three dots, provided that one of them is exactly midway between the other two; then to remove another three dots under the same condition; and so on. If we remove 33 dots, which are the possibilities for the remaining dot?

David Angell

Q1541 Consider 29x+30y+31z=366 where x<y<z are positive integers with x

(a) Without writing or using a computer, find such x,y,z
(b) Prove that there is only one solution.

Thomas Britz

Dear Readers, welcome to this issue of Parabola!
Here, you can find three excellent articles that each showcase aspects of maths: maths as art; maths as sport; and maths as science.

David Angell

Choose any function f(n) and, starting at any point on a sheet of paper, draw a line interval at an angle f(1) anticlockwise from the horizontal. By continuing in this way with f(2) , f(3) and so on, we might be lucky to end up with an interesting and pretty pattern.

Sin Keong Tong

Given an equilateral triangle Δ , what is the radius of the biggest circle contained in Δ ? What is the radius of the smallest circle containing Δ ? This article shows how to determine these radii, also for the higher dimensions.

James O. Hortle

This article serves as an introduction to the basics of what is commonly known as machine learning, focusing on support vector machines and applying them to data from a study on Alzheimer's disease.

Robert Schneider

An odd comic about even numbers.

David Angell

Q1561 Let a,b,c be positive numbers for which a+b/c=2018 and b+c/a=2019 . Evaluate a+c/b.

David Angell

We have a pattern of 34 dots arranged and may remove any three dots, provided that one of them is exactly midway between the other two; then to remove another three dots under the same condition; and so on. If we remove 33 dots, which are the possibilities for the remaining dot?

Denis Potapov

The problems and solutions from the 57th UNSW School Mathematics Competition.

Thomas Britz

The winners of the 57th UNSW School Mathematics Competition. Congratulations, all of you!

Thomas Britz

Dear Readers, welcome to this year-ending issue of Parabola!
It offers excellent problems as well as four articles that each look at numbers: primes, integers and reals. Enjoy!

Liangyi Zhao

Using only simple combinatorial arguments and binomial coefﬁcients, this paper proves upper and lower bounds for the number of primes up to x.

Michael Kielstra

Let us play Factorial Countdown, a game in which you are given numbers 1,2,…,n , a set of arithmetic operations and a target number N . The aim of the game if to use all of the numbers, and some of the operations, to reach your target N . Can you win this game?

Marian Maciocha

This article presents a simple inequality that relates arithmetic means to root mean square. It is simple to prove, has nice visual proofs in the two-variable case, and has equally nice applications.

Benny Lim

Highly composite numbers, denoted by Hn , are positive integers with more factors than any smaller positive integer. Based on an investigation of numeric data, this paper shows that there seem to be unusually many primes among numbers of the form Hn±1.

Robert Schneider

A new pun comic by the author of 2ℤ Or Not 2ℤ.

David Angell

Q1571 Find positive integers a,b,c such that a≤b≤c and 1/a+1/ab+1/abc=526.

David Angell

Q1561 Let a,b,c be positive numbers for which a+b/c = 2018 and b+c/a = 2019.

Evaluate a+c/b.