N. V. Williams

It has been said that "behind every great computer there is a great memory". As with most sayings, of course, this is not the whole story.

M. Greening

As will be known, when we write$12\equiv 7\text{(modulo)}5$ we are expressing the fact that both 12 and 7 have the same remainder when divided by 5.

**J141** The real numbers $a,b$ and $c$ are such that

Solve for $x,y$ and $z$ the simultaneous system of equations

**J131** (i) How many lines equidistant from three given points can be drawn in the plane?

J. Blatt

Early in the 17th century, Johannes Kepler established, from actual observations of the positions of the planets in the sky, three laws of planetary motion.

G. Szekeres

A Mersenne number is an integer of the form ${2}^{p}-1$ where $p$ is a prime.

x and y are unequal positive integers. Prove that xy does not divide x^2 +y^2$.

"Murgatroyd's Mind-stretchers" by J. and F. Pinkney

Suppose you have two glasses; one contains water and the other contains the same amount of cordial.

**J151** (i) Prove that if

**J141** The real numbers $a,b$ and

Peter Donovan

This is an account of some elementary aspects of the subject known as "algebraic topology". It investigates the placing of nets on surfaces and Euler characteristics.

Richard Jermyn

This machine is easily constructed from scrap material, using simple tools.

Despite appearances, the following subtraction is wrong:

Peter Donovan

The following problem was considered for publication in *Parabola*, but rejected as the editorial committee could find no reasonable way of solving it.

Malcolm D. Temperly

The following is a poem which describes how to solve the problem of these instantly insane blocks.

M. G. Greening

Man seems to have known of Pythagoras' Theorem since the early days of civilisation, although the Greek geometers were the first to provide a logical proof.

R. James

Since the last *Parabola* went to print, the following people have submitted answers.

In a certain country the number of boys born is approximately equal to the number of girls born.

While participating in a census, a census-taker arrived at a certain house in a certain street and proceeded to question the woman who answered, as to the number and ages of the occupants.

Essentially this reduces to proving that

"Men of Mathematics Vols 1 & 2" by E.T. Bell.

**J161** Find a whole number, N, satisfying the following conditions:

(a) N is the product of exactly four distinct prime numbers.

**J151** (i) Prove that if