John Price and Matthew Price

In 1733 the French biologist G.L.L. Buffon (1707-1788) proposed an interesting experiment for obtaining estimates of the constant π.

D. R. Green

It is a remarkable fact that quite ordinary families can suddenly throw up quite extraordinary mathematical genius.

J. H. Clarke and A. G. Shannon

In 1742 Goldbach suggested in a letter to Euler that every even integer greater than 4 is the sum of two odd primes.

A cube, as shown below, can be made up from 6 square pyramids.

In the 1983 HSC 4 unit (Unique) paper there were some interesting problems on functions and their graphs.

**Q.588** We are given a set of 201 different numbers with the property that the sum of any 100 of them is less than the sum of the remaining 101. Prove that all of the numbers are positive.

**Q.563** In the figure, $X$ and $Y$ are the centres of the circular arcs $\mathrm{AB}$ and $\mathrm{AC}$ respectively, and $A$ is the centre of the circular arc $\mathrm{BC}$.

Cyril Isenberg

One of the mathematical results that one encounters early in life is that concerning the shortest path joining two points.

A diagonal polygon is a line joining two non-adjacent vertices. How many vertices has a polygon which has 152 diagonals?

This issue we concentrate on some problems involving trigonometric functions and complex numbers.

**Q.600** $P$ is a point inside a convex polygon all of whose sides are of equal length.

**Q.576** The game of Yellow Pigs is a favorite pastime at Hampshire College's Summer Science Training Program.

David Sharpe

Most of us will have a tree in our homes over Christmas. To the mathematician, a tree is a special type of *graph.*

A. G. Shannon

Suppose you were required to find the minimum value of

$$f(x)={x}^{4}-15{x}^{3}+72{x}^{2}-1135x\text{for}1\le x\le 15.$$

N. Ormerod

Mathematicians love formulae. Nothing pleases a mathematician more than to come up witha nice simple formula which solves a problem or summarises a result.

Doug Mackenzie

People came from all over the world to Australia to talk together and learn from each other about how best people from age 4 to infinity can learn mathematics and enjoy using it.

**Q.612** From a point $P$ inside a cube, line segments are drawn to each of the eight vertices of the cube, forming six pyramids each having $P$ as the apex and a face of the cube as the base.

**Q.588** We are given a set of 201 different numbers with the property that the sum of any 100 of them is less than the sum of the remaining 101. Prove that all of the numbers are positive.