### Issue 3

John Loxton

There are essentially only two operations involved in the encryption of a message: substitution and transposition.

Peter Petocz

Ring a-ring a roses,

A pocketful of posies,

Atishoo, Atishoo,

We all fall down.

Over fifteen years ago, mathematics students from different countries began meeting for an annual competition called the International Mathematical Olympiad.

There were once again some interesting problems in the 1982 H.S.C. Turning first to some extreme value problems, starting with the 3 unit paper.

**Q.551** Solve the system of equations

John Price and Matthew Price

The game Scissors, Paper and Rock (SPR) has a long history. One famous contest took place in the pages of *You Only Live Twice* by Ian Fleming and resulted in our hero James Bond defeating the inscrutable Tiger Tanka.

A while ago we received a short article from one of our readers, Tim Bedding about some features of this celebrated ratio.

6 positive whole numbers $A,B,C,D,E$ and $F$ satisfy $A+B=29,C+D=45,E+F=65,AC=36,BE=312$. Find $A,B,C,D,E$ and $F$. Show there is only one solution to the equations.

This term we will contentrate on complex numbers, and solve problems from the 1981 and 1982 4-unit papers.

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

**Q.539** From the set of whole numbers $\left\{0,1,2,\dots ,999999999\right\}$ two are selected at random. What is the probability that they differ by a multiple of 10000?

John Loxton

According to Francis Bacon, perfect ciphers may be recognised by the following virtues: "that they be not laborious to write and read; that they be impossible to decipher; and, in some cases, that they be without suspicion."

In 1981 and 1982 there were some 4 unit questions on the conical pendulum.

Geoff Ball

The 1983 International Mathematical Olympiad was held in Paris in July.

In the Junior Section of the competition there were three questions which were fairly easy, two rather difficult, and one impossibly so.

Scot McPhie of Kings School Paramatta, has written asking for help in writing a programme for the Apple computer.

Alex Reichel

There is much interest in the origins of science, with a view to understanding the creative process of the human mind.

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

**Q.551** Solve the system of equations