It has now been possible to study the many comments received since the publication of the “interim statement” in Gazette No. 407 (March 1975). The recommendations given below are based on the original suggestions, taking these comments into account. Sidelining indicates a change or modification of the original suggestion.

It is proposed also to publish these recommendations as a pamphlet, so that they may be widely available to authors and others.

Given a triangle with vertices P1, P2, P3 its Fermat point is defined as that point F which minimises the sum P1F + P2F + P3F. It is known that if each of the angles P1, P2, P3 is less than 120°, then F will be the point inside the triangle at which each pair of P1, P2, P3 subtend an angle of 120°, while if any of the angles at P1, P2, P3 exceed 120° then F will coincide with that vertex – see Figure (1).

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.

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.

In a recent paper, under the same title, I proved that inversion of certain functional relations is the characteristic property of the Mobius’ μ-function, by establishing the:

THEOREM I′: f(n), g(n) and μ*(n) (possibly depending on f) are three arithmetical functions with