W. J. Ryan

Most of you know what is meant by a Pythagorean triple. It is a set of three natural numbers $\left\{a,b,c\right\}$ such that ${a}^{2}+{b}^{2}={c}^{2}$ .

This crossnumber may have been hard - some people got it wrong!

Among the geometrical problems which have intrigued amateurs throughout the ages, the geometrical trisection (i.e. division in three equal parts) of an angle ranks high in popularity.

David Crocker

In an appendix to a Physics book, I found the derivation on infinite integrals of the type.

$${\int}_{0}^{\mathrm{\infty}}\frac{{x}^{n}}{{e}^{x}-1}dx.$$

During an investigation of polynomials, my class came across a problem which required finding a polynomial with roots that were reciprocals of the original polynomial's roots.

K. J. Wilkins

The game chosen for this issue of *Parabola* is quite a recent game. It was thought of by an American university student, William Black, in 1960.

"Coin Games and Puzzles" by Maxey Brooke

A. White

Once upon a time Liethagoras, jealous of Pythagoras' fame, proposed a theorem.

**Q.261** In a right-angled triangle, the shortest side is $a$ cm long, the longest side is $c$ cm long and the other side is $b$ cm long. If $a,b,c$ are all integers, when does ${a}^{2}=b+c$ ?

**J251** Framer Jones grew a square number of cabbages last year. This year he grew 41 more cabbages than last year and still grew a square number of cabbages. How many did he grow this year?

Glen Reeves

A perfect number is a number which is equal to the sum of all its factors except itself.

**Successful Solvers:**

Mark Hartley, Castle Hill High

John Rogers, Knox Grammar

J. McMullen

Everybody knows that, because of Pythagoras' theorem, the diagonal of a unit square must have irrational length.

I have a problem regarding periodic decimals. It concerns a decimal such as 0.49˙

(i.e. 0.4999…0.4999…).

For a change this time, we present to you something which is strictly not a game (but is an interesting challenge).

The sum of two positive integers is 10000000000.

"Understanding the Computer" by Michael Overman

**Q.273** What is the smallest and largest possilbe number of Fridays that can occur on the 13th of a month in any calendar year?

**Q.261** In a right-angled triangle, the shortest side is $a$ cm long, the longest side is $c$ cm long and the other side is $b$ cm long. If $a,b,c$ are all integers, when does ${a}^{2}=b+c$ ?

T. Kelley

In Vol. 11 No. 1, W.J. Ryan asked several questions about Pythagorean triples, which I will try to answer.

A chain is as strong as its weakest link, and this applies particularly to the chain of mathematical argument.

Greg Middleton

One of the interesting applications of arithmetic series is the result that

12

Recently I found the following relationship between $n!$ and ${n}^{n}$ while working through sequences and series:

Mastermind is a game which goes by several different names, and you make already know it, perhaps as "bulls and hits".

Question 1: In fact, the two numbers must each end in zero (see solution in last issue of *Parabola*)

"Mathematical Puzzles and Perplexities" by Claude Birthwhistle

**Q.285** C.F. Gauss was given the problem of summing the numbers from 1 to 100 when he was a student. He did it this way:

**Q.273** What is the smallest and largest possilbe number of Fridays that can occur on the 13th of a month in any calendar year?