We received an interesting letter from one of our readers, S.J. Cohen, who has found a way of generating sequences of Pythagorean Triangles by means of certain irrational square roots.

In the 1980 H.S.C paper Question 9 in the 3 unit paper and Question 2 part (ii) in the 4 unit paper concerned polynomials which seemed to prove somewhat troublesome for students.

J. W. Sanders

Pick a book containing lots of four digit numbers (for example a telephone directory or book of maths tables) and choose a number, let's call it $N$, from the book, at random.

Kieran Lim

Since the knight is the only piece that moves asymmetrically in chess, more problems have been based on the knight than on any other chess piece.

In the middle of the page you can see a sharply pointed solid, formed by congruent "kite" shaped rectangles.

**Q.479 $a679b$ is a five-digit number (in base 10) which is divisible by 72, determine $a$ and $b$ **

**Q.455** The rule for leap years runs as follows: A year which is divisible by 4 is a leap year except that years which are divisible by 100 are not leap years unless they are divisible by 400.

Suppose that our Government has decided to improve the capabilities of the Royal Australian Navy. It has resolved to buy one new destroyer for each 1000km of the Australian coast-line. The only problem left for the Navy is to measure the length of the coast-line.

The differential and integral Calculus has important applications in physics and/or mechanics.

Michael N. Barber

Summing series of the form ${u}_{1}+{u}_{2}+\cdots {u}_{N}$ is a common problem in mathematics.

It is well establihed fact that it is impossible to trisect a general angle using Euclidean tools, i.e. compass and straightedge only.

Andrew Jenkins

This Cross-Numbers was devised by Andrew Jenkins, Year 9, North Sydney Boys High School.

(The cube problem) Each face of a cube is marked with a square, a triangle or a line segment.

**Q.491** Find a four digit number which becomes nine times as large if the order of digits is reversed.

**Q.467** In a plane are 127 toothed cog-wheels, numbered $(1),(2),\cdots ,(127)$. The teeth of wheel$(1)$ engage those of wheel $(2)$, and similarly $(2)$ is engaged with $(3)$,

P. J. Blennerhassett

On several occasions I have been in the position of having to cook a roast dinner. My experience in this field is rather limited and so I have to rely on cookbooks to provide cooking temperature and time.

In the last issue, we posed the innocent question: "How long is the coast of Australia?" Some sceptical readers have queried our conclusions, which was, as you will recall, that the question is not as innocent as it seems.

Doug Mackenzie

In trying to deal with questions such as:

- In a new housing sub-division, how many letter boxes should Australia Post provide and where should they be placed?
- What is the most effective way to divide the fuel between the stages of a multistage rocket?

"Size of body is no mere accident. Man, respiring as he does, cannot be as small as an insect, nor vice versa... In fact each main group of animals has its mean and characteristic size..."

Let us pose the question: is it possible to form a rectangle (or even better, a square) by putting together smaller squares, all of which are different?

Andrew Jenkins, who devised the Cross-Numbers sent us not only the solution, but the detailed reasoning leading to it. We print it here in full.

**Q.503** A rectangle 11cms ×× 7cms is divided by ruled lines into 1cm ×× 1cm squares, each containing a button.

**Q.479 **If** **$a679b$ is a five-digit number (in base 10) which is divisible by 72, determine $a$ and $b$