In his article on bingo in the June 1985 Gazette, M. G. Sykes makes the implicit assumption that the fifteen numbers on a bingo card may be chosen from the set {1, 2,…, 90} without restriction. This is not correct—there are severe constraints on the choices. He also assumes in his calculations that the cards are printed at random. Whether this is ever the case I do not know, but the bingo cards produced for various tabloid newspapers in recent years are produced in a highly systematic manner, some details of which will be given in this article.

I read the first page of John Hersee’s article in the June 1984 Gazette and sat down immediately, “stimulated to contribute comments” on his question. “What do we mean by ‘investigationary work’… in the context of A-level?”

The problem of finding a geometric construction in a finite number of steps to trisect an arbitrary angle using straightedge and compasses alone is a very old one, originally proposed by the mathematicians of ancient Greece. It was only with modern algebraic techniques in the nineteenth century that it was shown conclusively that no such construction can exist. This fact has in no way deterred a legion of crank angle-trisectors from presenting their alleged solutions to the problem! However some angles (even non-constructible ones) can be trisected: we’ll see later, for example, that given an angle π/7 it can be trisected. We can also consider the more general problem:

The two most common measures of central tendency and dispersion in statistics are the mean and standard deviation on the one hand, and the median and absolute deviation on the other. For most purposes the former measure is preferred for two very good reasons; the first is that the squares of quantities are easier to handle analytically than their moduli; and secondly for all the common symmetrical distributions, such as the normal, uniform and binomial distributions, for which the mean and median coincide, if a sample is taken to estimate the central value, then the mean of that sample has a smaller variance than the median, and is therefore relatively more efficient as an estimator of the central value of the parent population.

This article arises because of a lecture I attended where the speaker, who had been heavily involved in the new JMB A level syllabus, was saying that roots of unity had gone, “thank goodness.” Many A level teachers seemed to agree and I began wondering just how “traditional” this topic is. I came to the conclusion that it is the most revealing topic to show the meshing of traditional and modern ideas. I hope to illustrate this by the end of this article.

Padé approximations consist of one polynomial divided by another (analogous to the polynomial approximations obtained from Maclaurin series). They can serve, at an elementary level, as a corrective to the idea that a power series is the only form of useful approximation. Indeed, they are more powerful than Maclaurin approximations, since, in general, a suitably chosen Padé approximation of a given order gives greater accuracy and is useful over a wide range than the corresponding Maclaurin approximation.

In order to obtain the area of a curve by the method of the integral calculus the following two conditions should hold:

1. (a) the equation of the curve must be known,

2. (b) the function representing the equation of the curve must be integrable.