Volume 44
, Issue 1

2008

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.

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.

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.

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, $n = 2, 3, 4,\ldots,$ 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