J. W. Sanders

Once upon a time, in a kingdom beset by evil influences, a benevolent king condemed his court sorcerer to death for casting evil spells and being in league with the devil.

J. Abel

If you look at Figure 1, you will see stars. Stars come in various sizes: the figure shows the five-pointed star, or pentagram, which was the secret symbol of Pythagoras, the six-pointed star, or star of David, the seven-pointed star and the eight pointed star, or octopus.

R. Baldick

What is a parabola? According to one small dictionary (brand-name suppressed), it is "the curve formed by the intersection of a cone with a plane parallel to its side."

Here, at last, are the results of the palindrome competition announced in *Parabola*, Volume 13, Number 3.

*Thou shalt not covet, but tradition
Approves all forms of competition*

Michael Hirshhorn

In Volume 14, Number 2, under this title, we gave you some deliberately faulty "answers" to a number of standard problems.

**Q.405** If $k$ and $N$ are positive integers with $k>1$ , show that it is possible to find $N$ consecutive odd integers whose sum is ${N}^{k}$

**Q.381** A square cake has frosting on its top and all four sides. Show how to cut it to serve nine people so that each one gets exactly the same amount of cake and exactly the same amount of frosting.

M. D. Hirschhorn

The following question appeared in the 1978 Higher School Certificate 2-unit and 3-unit Mathematics exams:

Take two twenty cent coins A and B. If the coin B is kept fixed and A is rolled round B without slipping, how many revolutions will A make about its centre before it returns to its original position?

V. Paul

In the great temple of Benares, beneath the dome which marks the centre of the world, rests a brass plate in which are fixed three diamond needles, each a cubit high and as thick as the body of a bee.

L. Freeman

The Poisson distribution often gives a useful statistical model to describe the occurrence of isolated events in an interval of time.

M. K. Vagholkar

If examinations do nothing else, they at least produce vast quantities of numerical data.

π's the limit

Our story begins in Ancient Greece at the start of the quest for perfection.

G. Szekeres

One of the great discoveries of the Pythagorean era was the fact that not all positive numbers are "commensurable", that is, expressible as a fraction a/b where a and b are natural numbers.

We start with 24 sheets of paper. Some are selected and cut up into 10 pieces each.

In Parabola, Volume 14, Number 3, Brendan Joyce described how to build a pile of bricks so that the top brick completely overhangs the bottom one.

**Q.417** Let a and b be integers. Show that 10a+b is a multiple of 7 if and only if a−2b is also.

**Q.393** Show that if $n$ is any integer greater than 2, of the fractions $\frac{1}{n},\frac{2}{n},\frac{3}{n},\cdots ,\frac{n-1}{n}$ an even number are in lowest terms.

J. Gani

The mathematics taught in schools consists mainly of arithmetic, elementary algebra (linear and quadratic equations), some plane geometry, the elements of calculus (simple differentiation and integration), and possibly some statistics.

Here is a way to bamboozle your friends with your powers of clairvoyance.

N. G. Barton

All cricketers and cricket followers know that a medium pace bowler can swing a new or well-preserved cricket ball in flight.

Many people have said... Well, a few people have said... Someone once said that the trouble with the Government of this Fair land is that it is run by politicians, lawyers, trade unionists, nincompoops, ... (delete according to prejudice).

J. H. Pollard

Let me begin by describing some of the basic properties of matrices.

Raymond Soo

This year marks the centenary of the birth of Albert Einstein, the most famous scientist of recent times.

In answer to the question asked by Julian Abel in *Parabola*, Volume 15, Number 1, I am sending you a four-pointed star.

**Q.429** Let $a$ be a positive integer. Prove that the fraction $(a^3 + 2a)/a^4 + 3a^2 + 1) is in its lowest terms.

**Q.405** If $k$ and $N$ are positive integers with $k>1$ , show that it is possible to find $N$ consecutive odd integers whose sum is ${N}^{k}$