The purpose of this note is to draw attention to two infinite families of antiprisms that differ slightly from the usual Keplerian kind. These figures have surely been noticed earlier by geometers and philomorphs, but I was unable to track down a reference to them. My attention was drawn to these figures by a Christmas gift that my family received recently. The gift was a cake in a box of a rather unusual shape (see Figure 1). On examining the box, I saw that it could be characterized variously as a deformed antiprism, an unfolded pyramid or a truncated prism. The box had two square faces parallel to each other, with the line joining the centres of the squares being perpendicular to the plane of either square. The squares were rotated 45° relative to each other, exactly as in a square antiprism, but one square had only half the area of the other. The separation of the squares was such that, of the eight triangles linking them, four were identical equilateral triangles while the rest were identical isosceles triangles.

Solid shapes which pack together to fill space cover a large and varied range. I recently found five such polyhedra which I have not seen described elsewhere. The names which I have used for them here are rather ferocious I am afraid, but that just happens to be the way the names of polyhedra work.

The prime numbers are often called the building blocks of number theory, a classic case of a ‘sine qua non’. If the corpus of the theory of numbers is looked upon as an architectural pile then the primes will be found amongst its foundations, and amongst its walls and buttresses, and indeed amongst the array of pinnacles and turrets which burst forth from it and stand proud, alone and magnificent. One such pinnacle had been discussed and planned for a hundred years before it was finally constructed in 1896 by two men working totally independently of each other (collapse of stout analogy). With this achievement, the prime numbers, those familiar yet raw beasts of mathematics, in one sense had been tamed. The long journey, which had begun with the tentative definitions of the ancient Greeks, was completed.

The Brocard points P and P' of the triangle ABC, illustrated in Figure 1, are such that ∠PAB = ∠PBC = ∠PCA = ω and ∠P'BA = ∠P'CB = ∠P'AC = ω. Problem 78.A in the March 1994 edition of the Mathematical Gazette required a proof that the maximum length of PP' is R/2, where R is the circumradius of triangle ABC. The solution given by various solvers in the November 1994 edition of the Gazette is essentially the same as appeared in a paper by H. E. Piggott in the Gazette in 1924. In the present paper various properties of the triangle and its Brocard points are established in terms of the lengths of the sides of the triangle. Some of the results may be new, but the methods used encompass ‘Mathematics Ancient and Modern’ in the guises of homogeneous areal coordinates and computer algebra!

The classical theorems of Ceva and Menelaus make assertions about the value of certain products of ratios of lengths in configurations in the affine plane. We shall use the term Ceva-type to describe any result of this general kind: one that specifies a configuration in affine space of n dimensions, defined only by incidences, about which one can make an assertion about a product of ratios of lengths, areas, etc. Several results of this kind are known. Apart from the classical results there are, for example, Ceva's and Menelaus' Theorems for n-gons, Hoehn's Theorem for pentagrams [1], and the Selftransitivity Theorem of [2].

The motion of a bowl on a grass green may be investigated mathematically, using the principles of dynamics established by Sir Isaac Newton (1642 – 1727). The analysis gives a clear picture of how the bias operates, and reveals some surprising and little-known facts which are important from the bowler's viewpoint.

George Boole, who very clearly rates as a mathematician of the very first rank, set out to solve a mathematical problem and his efforts to do so continued throughout the whole of his professional life. He never succeeded, despite the fact that he held the key to the solution for a good deal of this time. How and why he came to fail in his attempt is the subject of this paper. The difficulties encountered by students may be better understood if we realise that even the very greatest of mathematicians experience the same limitations and that they arise from the same causes.

As an elementary programming exercise I often ask students to find all three digit numbers equal to the sum of the cubes of their digits, for example 153 = 13 + 53 + 33. This is the usual solution:

Polynomials like (x + l)x(x - 1) or 5 (x - 2)(x - 4)(x - 6)(x - 8) are particularly easy to sketch because their zeros are real, distinct and equally spaced along the x-axis. As a simple exercise, please picture in your mind the graph of

y=(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)

Lagrange's inversion formula is usually presented in the following form. Let f be a regular (= analytic or holomorphic) complex function with the properties

Then it is a standard theorem that it has a regular inverse function g, such that g (f (z)) = z, with similar properties. (I assume here standard results to be found in appropriate text books.