Volume 33
, Issue 2


You have probably all heard the expression “How long is a piece of string?”. It’s usually offered in rhetorical response to a question that has no sensible answer.

Some years ago I saw the following problem in a mathematics competition. Solve the simultaneous equations

\begin{align} x+y+z & =1 \\
x^2+y^2+z^2 & =29 \\
x^3+y^3+z^3 & =-29

Find the smallest positive integer $n$ such that $\frac{1}{3}n$ is a perfect cube, $\frac{1}{5}n$ a perfect fifth power and $\frac{1}{7}n$ a perfect seventh power.

Q.1001 On the island described in question 2 of the Junior Division for this year’s mathematics competition, each town entered one Australian Rules football team in the national championship.

Q.993 Consider

$$ p(n) = a_0 + a_1n + a_2n^2 + \cdots + a_kn^k $$