Dr. C. D. Langford recently sent me some numerical work he had done on automorphic numbers. He found automorphic numbers in the scales of 6, 10, 12, 14, and 15 and he discovered what proved to be a general method of finding an automorphic number of r + 1 digits from one of r digits. Dr. Langford’s notes stimulated me to look for explanations of his results.

The method of solving linear differential equations with constant coefficients by using the operator D, which is given in elementary textbooks, was introduced in 1841 by Boole. In spite of the developments summarised below it remains the best method available for certain types of problem. As usually presented it is a tentative process, justified only by substitution in the original equation of the result produced. It is used only to obtain particular integrals, although it can be adapted to deal with complementary functions. This is not done in practice, however, since these can usually be written down directly.

In her Gazette article on “Roll-a-penny probabilities”, Ann Hirst sets out to find the probability that a player will lose after a certain number of rolls. She considers only the simplest case where the player starts with one penny, and solves the problem by setting up a recurrence relation and using induction. The problem of what happens if the player starts with more than one penny is left for the further consideration of the reader.

1. Introduction. While the operational calculus for the solution o initial value problems for linear differential equations with constam coefficients is now commonly introduced in a thoroughly satisfactory manner—either by means of the Laplace transform (e.g. ref. 1) of by the method of J- Mikusinski (ref. 2)—the same cannot be said of the simpler method for the general integration of such equations which is known as the D calculus. In fact, I cannot recall any text which gives a consistent definition of the meaning of expressions of the type [F(D)/G(D)]y, where F and G are polynomials. The reason for this state of affairs seems to be that the scope of the D calculus is rather limited so that the validity of the result can be verified in each case (e.g. ref. 3). In particular, this applies to the decomposition of an operator F-1(D) into partial fractions, which is the central step: in the solution of an equation with nonvanishing right hand side of general form. Nevertheless a rational approach to the entire problem is perhaps not out of place. This is attempted in the present paper.

When a parent sees a little girl sitting on the floor cutting paper dolls, many thoughts may come to mind: ‘She’s keeping out of trouble’ or ‘She’s making a mess’ or even ‘There go my tax returns’. The thought that should have come to my parent’s mind, however, was ‘One day she’ll be a mathematician’. My grandmother, who worked as a dressmaker, often allowed my sister and me to use her razor sharp scissors on the strips of leftover tracing paper. This paper is inspired by a notebook that I kept in grade school when I ‘studied’ paper dolls, and the figures are based on dolls found pressed between the pages.

In a Gazette article some years ago [1] Derek G. Ball proved the theorem that the only regular polygon which can be constructed on a square pinboard is a square. This note gives two more proofs of this result.

The other day I received a long tube from a Canadian stranger containing a large poster featuring over a hundred polyhedra, including ‘all 92 Johnson polyhedra’. This term, though probably unfamiliar this side of the pond, was not completely unknown to me; it means convex polyhedra, excluding the regular and Archimedean ones, all of whose faces are regular polygons. Of course, as usual, we have to exclude those naughty polyhedra whose faces go around in pairs collecting squares (prisms) or equilateral triangles (antiprisms) and don’t know when to stop. The word convex is vital, otherwise there would be another infinite set. A lot of them are rather trivial, like sticking pyramids on the faces of a dodecahedron, but they include the deltahedra and many other interesting members. But they have at least one imitator who didn't quite make the grade. Trying to discover why, and how to coach him so that he would, I found that my spherical trigonometry was getting rather rusty so I set out to make one and see what was happening. I thought perhaps other readers would like to share this piece of antiresearch.

Those who have the privilege of contributing new proofs of old theorems to the columns of the Gazette do so, normally, with complete confidence in the validity of the new proofs but possibly with some doubts as to whether the new proofs will be regarded as better than the old ones. The following attempt to set out new proofs for certain properties of the triangle differs from the norm in that there is no doubt whatever that the new proofs will be regarded as worse than the old ones and the only doubt is whether they will be regarded as valid. The doubt may arise because the aim is to deduce the truth of certain general theorems from the proved truth of one or two particular cases of the theorems. This is far from being the usual procedure in mathematics, though it is not without precedent; for example, if we show that ax2 + bx + c is zero for three values of x, it follows that it is zero for any and all values of x.

In the years 1603–1867, known as the Edo period, when Japan isolated itself from the western world, the country developed its own style of mathematics, especially geometry. Results and theorems of traditional Japanese mathematics, known as ‘Wasan’, were usually stated in the form of problems; these were originally displayed on wooden boards (‘sangaku’) hung in shrines and temples, but many later appeared in books, either handwritten with a brush or printed from wood blocks. (See [1, 2, 3] for more details.) Solutions to the problems were not given, but numerical answers were sometimes provided. The problems were written in a form of language called ‘Kanbun’, based on Chinese, which cannot be readily understood by modern Japanese readers even though written Japanese makes extensive use of Chinese characters.