A famous unsolved problem in mathematics is the 3x + 1 conjecture. An excellent discussion of its origin and history is found in [2]. It is deceptively simple to state. Many are tempted to try it because of its enticing appearance. One aim of this note is to warn the uninitiated that a person can grow old and infirm by trying to solve the problem. There are probably fine mathematicians who have frittered away what might have been productive periods of their lives working on this devilish puzzle. The other aim is to say something at least mildly interesting about a question that I can’t answer. These are the writings of a person who has been frustrated, but not defeated, by the 3x + 1 conjecture.

The theorems that follow are mostly well known; the object of this note is to show that they all follow simply from a single argument, whereas they are usually reached from different starting-points.

As recently as ten years ago, professional mathematicians did most of their work at a desk, using paper and pencil. Today’s mathematician still sits at a desk, but facing a cathode-ray screen. The paper and pencil are still there, but a lot of the mathematician’s activities now involve use of the computer. In particular, powerful computer packages like Mathematica and Maple advertise themselves as systems for ‘doing mathematics on a computer’. This rapid transformation has changed the nature of doing mathematics in a fundamental way. The computer does not simply ‘assist’ the mathematician in doing business as usual; rather, it changes the nature of what is done. In particular, the logical structure of mathematical reasoning carried out with the aid of an interactive computer system is different from the structure of the more traditional form of mathematical reasoning. This article is a logician’s view of the ‘new maths’ of the computer age.

Cyclotomic polynomials, (xp - l)/(x-1), have (p-1) zeros exp(2πiq/p) (q = 1, 2, ... ,p - 1). We consider related polynomials whose zeros are given by cos(qπ/p) (q = 1, 2, ... ,p - 1) and indicate their relation to polynomials Pn (ζ) that satisfy the ‘double angle’ condition Pn(2ζ2 - l) = Pn(ζ)Pn(-ζ). The double angle condition is generalised to a chain of double angle conditions Qi(2ζ2 - 1) = Qi+1(ζ)Qi+1(-ζ), i = 1, 2, ...,n(Q1 ≡ Qn+1) For purposes of illustration, we will focus on P4(ζ), demonstrating how construction of the fourth order polynomial subject to the double angle condition yields an equation for cos . We also discuss more general features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple geometrical construction can be used to derive a polynomial for cos .

The first published description of the geoboard appears to have been given as an incidental illustration to Denis Diderot’s Letter on the blind (1774).

The savants of the eighteenth century were interested in blindness as a problem for speculative philosophy. If the mind is a tabula rasa at birth, how will it develop if the sovereign sense is absent? Diderot held that the blind man would not only form different physical concepts, but different aesthetic and even different moral values. In the end he would inevitably come to a different set of religious beliefs and hence, Diderot argued, the concept of God was relative to our senses.

Operational methods in pure mathematics, e.g. the work of Boole, Graves and Murphy, existed many years before Oliver Heaviside introduced new ideas into mathematical physics. His first paper using such ideas was published in the Proceedings of the Royal Society in 1893. Remarks therein indicate his unfamiliarity with some of the literature of the subject, owing to lack of reference facilities, for example, Liouville’s work on fractional differentiation.