Take a look at the ‘string’ (or finite sequence)

6,2,1,0,0,0,1,0,0,0.

Count the occurrences of the numbers:

Notice that the bottom row is the string we started with! In fact the above string is the unique answer to the popular question (posed, for example, in a recent edition of the Trans World Airlines magazine, The Ambassador, August 1978) to find a string

The purpose of this article is threefold. First to suggest other approaches to the ‘Glass rod problem’ (Gazette note 65.6, March 1981). Second, to show the unexpected connection between that and the article ‘Triangle of triangles’ by H. B. Griffiths on page 10 of the same issue and with the article on ‘Random triangles’ by T. Easingwood in the December 1981 edition. And third, to explore further problems which arise from a consideration of the above.

One of the problems of the first introduction to the calculus and the subsequent mental imagery developed by the student is that the functions involved are usually given by simple formulae such as f(x) = xn and the derivatives calculated by formulae crunching: f′(x) = nxn−1. The fundamental ideas of the calculus and any relational understanding recede into the background. Getting out of the strait-jacket and considering more general functions at some stage is rarely considered. When it is, it is usually performed in the context of university analysis where pictures are banned because they are claimed to mislead the intuition.

The lack of any obvious pattern amongst the Bernoulli numbers is one of the shocks of analysis which subsequent familiarity with the many beautiful and simple means of deriving them does not altogether assuage. Historically, they first arose in connection with the sums of the rth powers of the first n integers

Given a straight line L in the plane, how can we find all lines parallel to L? Here is one way: at each point P of L draw a straight line perpendicular to L. Now choose a number d and measure a distance d down these lines, always in the same direction. Joining up the points P′ so obtained gives the parallel to L at distance d.

The problem of expressing numbers as the sum of two cubes of non-zero rational numbers has a long history, some results being given by Diophantus. Several mathematicians have since contributed to our knowledge of the problem, in particular in the discovery of numbers which cannot be expressed as the sum of two cubes. Fermat discovered that 1, 18 and 36 belong to this class of numbers, Euler and Legendre demonstrated that so do 3, 4 and 5 and some infinite sets of such numbers were discovered by Sylvester. However, much of interest still remains to be discovered in this area. In this article I hope to both summarise the existing results on this problem and to produce a new class of numbers which cannot be expressed as the sum of two cubes. Most importantly, a general method of tackling the problem will be described which should stimulate further investigation.

It is now widely accepted that the first or “kinematic” part of Einstein’s 1905 paper on Relativity [1] can be made accessible to interested non-specialist students. This part is mathematically straightforward, although its first readers found it conceptually puzzling. “From this”, remarks Einstein, as he follows one of the simpler results of applying the Lorentz transformation to our measures of space and time, “there ensues the following peculiar consequence…” It was these “peculiar consequences” that repelled many physicists of his own day. Later, the challenge to intuition caught public imagination and led to the many popular and semi-popular introductions to the Special Theory. Today there are also several accounts available at a mathematical level appropriate to sixth form or college students.

Your problems are always welcome (together with solutions if you have them). Solutions are invited to the following problems, to arrive by April 15th. They will not be acknowledged but the best will be quoted in the October edition. Send them to the editor of the Gazette, not to the Problem Bureau.