The problem is to join two given straight pipes by a curved pipe made up of two circular arcs of equal radii. The case when the two given pipes are coplanar has been discussed by E. Kosko in the Gazette of May 1964, pp. 192-196. Purely geometrical constructions for this case have been provided by S. N. Collings in the Gazette of October 1965, p. 302, and by me in the Gazette of February 1966, pp. 37-39. (Unfortunately Mr. Collings’s construction does not always work. On his diagram one can put in the two straight pipes in four different ways. For two of these ways the construction gives valid solutions, but for the other two ways the joining curve has a discontinuity of sense.)

The study of mechanics and the solution of problems in mechanics are an intrinsic part of the subject of mathematics as taught in British schools and universities; the subject is associated with such names as those of Routh, Lamb, Loney, Love. It is perhaps a peculiarly British tradition, well worth preserving as a discipline of the first order, comparable with Latin and Euclidean geometry; many examination candidates, even perhaps theoretical physicists of distinction, can write learnedly on atomic physics, and yet find the greatest difficulty in solving dynamical problems requiring a firm grasp of principle together with facility in applying it. The present logical unsatisfactoriness of quantum mechanics in certain aspects may be due to want of interest in the foundations of mechanics amongst its exponents.

1. Introduction

In [1] the following three questions have been asked, and answered: For which values of f is there a convex polyhedron whose faces consist of

• (i) f triangles? (Ans f = 4, 6, 8, 10, 12, 14, …)

• (ii) f quadrilaterals? (Ans f = 6, 8, 9, 10, 11, …)

• (iii) f pentagons? (Ans f = 12, 16, 18, 20, 22, …)

Take any positive integer written in the usual base 10 notation, reverse the digits and add to the original number. If the result is a palindrome, stop. If not, continue. Will a palindrome always occur at some point? Let us do a fe examples to see the sort of thing that happens.