In 1979 Philip W. Kuchel published a paper [1] in the Mathematical Gazette on using curved mirrors as a means of demonstrating the transformation known as inversion in a circle. He called the mirrors ‘anamorphoscopes’ since he came to the idea as a special case of the conical mirror anamorphosis which was a popular optical toy from the seventeenth century onwards [2]. In this paper we revisit his ideas with current technology and provide some extensions to Kuchel's derivation.

Mathematics is full of wonder and surprise, and I had my first big mathematical surprise at the age of 10, in 1956.

I was keen on conjuring at the time, and came across the following ‘mind-reading’ trick un a book.

Write down a 3-figure number.

Any such number will do, as long as the first and last figures differ by 2 or more.

Now reverse your number, and subtract the smaller 3-figure number from the larger.

Snakes and ladders is an ancient Indian game of chance that offers amusement as well as a metaphor for life's many ups and downs. Games offer useful and fun ways of conveying ideas as well as solution techniques and this game has considerable mathematical tractability.

This note shows how snakes and ladders can be used to represent the ups and downs of share ownership and determine fair values of a multistage project that pays fixed dividends at uncertain completion times and has random returns

The following striking identities

are the cases n = 1,2,3,4 of a remarkable family given by G.J. Dostor [1]:

where m = n(2n + 1), and n = 1, 2, … The case m = −n is trivial. If m ≠ −n there are n + 1 squares of consecutive integers on the left and n on the right. We will treat the last term (m + n)2 on the left differently, and refer to it as a transition term relating two sums of squares of n consecutive integers.

Three of the oldest and most celebrated formulae for π are:

The first is Vieta's product of nested radicals from 1592 [1]. The second is Wallis's product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2], also from 1656. (In the remainder of the paper, for continued fractions we will use the more convenient notation

Over two thousand years ago, Archimedes discovered a remarkable result concerning two circles drawn with reference to a configuration of three circles and a straight line. Figure 1 displays this result.

In the figure, A, B, C are three collinear points, with B between A and C; circles ω1, ω2, ω3 are drawn on AB, BC, AC as diameters, respectively; a line l is drawn through B, perpendicular to AC; a circle ω4 is inscribed in the region bounded by {ωl, ω3, l}; and a circle ω5 is inscribed in the region bounded by {ω2, ω3, l}. The Archimedean property is that ω4 and ω5 have equal radii.

Following a recent revival of interest in both Thomae's function and digit functions (see [1] and [2] respectively) we present here a function providing an appealing link between the two. The former, nowadays often cited in courses on real analysis, was given by Thomae in 1875; see also [3], [4] and [5]. This function, which we denote by g(x), has the following definition:

where it is so be assumed that gcd (p, q) = 1 when x is rational.

In [1] there is a rather nice story regarding the coming into being of the hook-length formula. The year was 1953, and the Canadian mathematician Gilbert Robinson was visiting a fellow mathematician, James Frame, at Michigan State University. One of their discussions concerned the work of Ralph Staal [2], an ex-student of Robinson, and this led to Frame conjecturing the formula. Apparently, Robinson was not at all convinced initially that the formula could be as simple as the one Frame was proposing. He was, however, eventually won over, and the combined efforts of these two mathematicians soon elicited a proof.

In Figure 1, triangle ABC has circumcentre O and X, Y, Z are the midpoints of the sides BC, CA, AB respectively. P is a point in the plane of ABC not lying on any of its sides. Σ is the circumcircle and S is the circle centre O passing through P.

Points on S are defined as follows. Points U, V, W are the points of intersection of AP, BP, CP with S. Points L, M, N are the reflections of P in the lines XO, YO, ZO respectively.

Computer programs for dynamic geometry provide a very effective tool for motivating students. It is my experience that a certain effort in preparing an adequate applet and spending some time on a careful presentation can result in a great change of the atmosphere in the classroom. On the other hand, these programs also open new perspectives in geometry exploration and can provoke new interesting research questions related to the already known topics. The possibility of experimenting and the chance of visual examination of the results add new dimensions to research and can also be appealing to non-mathematical audiences.

‘There is a simple argument which shows that, for any positive real number x, the digit 7 appears in the decimal representation of one of the numbers x, 2x, … , 79x. I spent a wet Saturday afternoon showing that 79 can be reduced to 42, which is best possible.’

Some thirty years ago this interesting problem was given to me verbally by David Masser, who was a reader in mathematics at Nottingham University. I solved the problem, over several days, and forgot about it. While tidying up papers in retirement I found the notes on the problem, and when I mentioned this to David he too had forgotten about it until he searched through his notes. He now encourages me to publicise the problem, which is presented here with a small extension.

In [1] Niven used the integral

to give a well-known proof of the irrationality of π. Recently Zhou and Markov [2] used a recurrence relation satisfied by this integral to present an alternative proof which may be more direct than Niven's.

Niven did not cite any references in [1] and thus the origin or Hn seems rather mysterious and ingenious. However if we heed Abel's advice to ‘study the masters’, we find that Hn emerged much more naturally from the great works of Lambert [3] and Hermite [4].

In the paper [1], Euler was examining properties of the conic sections that could be shared by more general curves. Most of the paper is concerned with ‘oblique-angle diameters’, a concept that seems to have been familiar to his readers in the eighteenth century, but has been ignored today. In this paper we will explain this concept and, led by Euler, develop some of its consequences.

Without doing the morally unspeakable or the medically infeasible, can a preference for daughters rather than sons increase their relative number? If, to be more precise, the only variable over which you have control is your number of children, can you increase the ratio.

Expected value (no. of daughters) : Expected value (no. of sons) ? Naïvely, you might think so. If for example you adopt the policy ‘stop procreating as soon as a girl is born’ won't you bear more girls compared to boys than you would otherwise? No, in fact.

One day the Tortoise was discussing with mighty Achilles the behaviour of infinite series. Achilles boasted that his mental powers were so fast that he could exhaust any series summation problem by simply adding up enough terms. But Tortoise was unimpressed:

‘Is that so? What's the sum of this series, then?’ With his tail he drew a line in the sand, and wrote

which is the same as

‘I'll give you until tomorrow to find K to four significant digits!’

Many mathematicians' experiences with distances in the settings of geometry, analysis, and topology can lead to the impression that the only worthwhile or ‘reasonable’ distance functions are metrics. We hope to convince the reader otherwise.

Recall that a metric for a set X is a function d: X × X → [0, ∞) satisfying all of the following metric axioms:

Morley's theorem states that the points of intersection of adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle, known as Morley's triangle. See Figure 1.

Concise proofs of the theorem are given in recent papers [1, 2]. A good picture of important previous work can be obtained by looking at [3, p. 1999], and examining references cited there.