Year 2008 - Volume 44
2020 - Present

Welcome to this first issue for 2008. I hope you enjoy the articles and problems. The articles by Michael Deakin and Peter Brown in this issue both relate to proof: Dedkind’s proof that there are an infinite number of objects and the proof by mathematical induction.

Michael A. B. Deakin
During my student days at the University of Melbourne I first encountered the passage I want to share with you. It was brought to my attention by a fellow-student, who found it interesting and unusual, as I did then and still do today.

P.G. Brown
The method known as mathematical induction is generally thought to have been introduced by Pascal (circa 1654), although a contrapositive form called the 'method of infinite descent’ was used by Fermat a little earlier. The name ‘mathematical induction’ was first used by De Morgan.

Bill McLean
A lot of useful mathematical software is freely available via the internet. If you look hard enough you can find a code to solve just about any standard mathematical problem, and for many years professional scientists have downloaded programs and subroutine libraries from epositories like netlib.

Q1261 A hat contains N=2n tickets, n=2,3,4,…, each marked with a number from 1 to N . (Each ticket has a different number.) In a game, players are asked to draw from the hat two tickets, read them, and replace them. Prize winners are those who draw two numbers whose ratio is 2 .

Q1251 Show that the product of 4 consecutive integers is always one less than a perfect square.
ANS: We can denote the 4 consecutive integers by n−1,n,n+1 and n+2 . Then .

Welcome to this first issue for 2008. I hope you enjoy the articles and problems. The articles by Michael Deakin and Peter Brown in this issue both relate to proof: Dedkind’s proof that there are an infinite number of objects and the proof by mathematical induction.

Si Chun Choi
In my first year of teaching, I was given a formula summary sheet to be handed out to my year 11 general mathematics class. One formula was particularly appealing.
2(xy+yz+xz)

Michael A. B. Deakin
Euclid’s Elements [of Geometry] is one of the most influential books ever written. It was first compiled in about 300 BCE, and it reigned supreme in the classroom into living memory.

Lara Scharenguivel and Bin Deng
Mathematical and statistical methods are well known to underpin our highly technological society but it is perhaps less well known that mathematics and statistics are also being used as tools in the defence of human rights.

Problem 1
1. What is the angle between the long hand and the short hand of a clock at twenty minutes past four?
2. What is the next time, to the nearest second, at which the two hands of the clock have the same angle between them?

Competition Winners – Senior Division

Q1271 (suggested by Julius Guest, Victoria) Solve simultaneously
x2+xy+y2=189
x-(√xy)+y=9

Q1261 A hat contains N=2n tickets, n=2,3,4,…, each marked with a number from 1 to N . (Each ticket has a different number.) In a game, players are asked to draw from the hat two tickets, read them, and replace them. Prize winners are those who draw two numbers whose ratio is 2 .

Welcome to the final issue for 2008. I am delighted to let you know that the U-Committee of the University of New South Wales has generously provided us with funds to set up a website for online mathematics outreach to secondary school students throughout Australia.

Bill McKee In an earlier article in Parabola (Volume 42, Number 2, 2006), I showed how we could find a straight line which is drawn so as to approximately fit some data points via the process of least-squares fitting.

Michael A. B. Deakin The German publishing house Springer-Verlag puts out a continuing series of volumes on biomathematics, the application of mathematics in a biological context of one kind or another. Volume 22 of this series, published in 1978, was entitled The Golden Age of Theoretical Ecology: 1923-1940.

Q1281 Prove that for any real numbers a and b there holds
(1+|a|)/(1+|b|)≤(1+|a−b|).

Q1271 Solve simultaneously
x2+xy+y2=189
x-(√xy)+y=9.
ANS: (suggested by Julius Guest, Victoria, and Keith Anker, Victoria)