The Fibonacci series 0,1,1,2,3,5,8,13,.... is well known because so many properties have been found for it and because there are many instances of it occurring both in nature and mathematics. It is of course formed from the recurrence relationship

The Tribonacci series 0,1,1,2,4,7,13,24,44,81,149,... is formed in a similar way but from the recurrence

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.

What a pleasure it was to read the note by Rev. L.M. Brown. It would be hard to resist exploring an article with a title like ‘An algorithm for square roots: an episode in the campaign against dotage’ under any circumstances. However, there are two reasons which made the exercise especially interesting for me.

The first reason is personal. My career as a university lecturer started in Session 1970-71 when I was appointed as a humble Demonstrator in the Technical Mathematics department at the University of Edinburgh, a group of which Dr Brown was for many years in charge. It is a delight to see him in great form at the age of 93 showing what can be done with paper, pencil and a mind still as sharp as a tack.

What is the area of the shaded region between the tyre tracks of a moving bicycle such as that depicted in Figure 1 ? If the tracks are specified, and equations for them are known, the area can be calculated using integral calculus. Surprisingly, the area can be obtained more easily without calculus, regardless of the bike’s path, using a dynamic visual approach called the method of sweeping tangents that does not require equations for the curves.

Recent Notes in the Mathematical Gazette have suggested directly or by implication the surprise felt by some students of mathematics when Σx3 turns out to be (Σx)2, Perhaps the results appear less surprising if they are seen, not in isolation, but as two particular cases of summation formulae for odd powers of the integers expressed in powers of {x(x + 1)}.

The fact that the sums of odd powers of the integers are expressible in powers of {x(x + 1)} can be used as a demonstration of the distinctive pattern of the difference tables of these sums, which facilitates their speedy calculation.

Mobius‘ function has the important property of inverting functional relations [1] of the type and giving in turn (Also relations (0.1) and (0.2) follow respectively from (0.11) and (0.21))

I establish in this paper that no other arithmetical function μ*(n) can perform the above inversion. In other words, I prove that Inversion is the characteristic property of Mobius‘ μ-function. Theorem 1 below asserts that if for one pair of functions g(n) and f(n) (subject to a special condition) relations (0.1) and (0.11) are true with an arithmetical function μ*(n) (possibly depending on g and f), then μ*(n) coincides with the Mobius‘-function. Theorem 2 below gives a similar result in connection with functions G(x) and F(x) defined for all real x ≥ 1.

The little paragraph with the above title on page 170 of the Mathematical Gazette has interested me, as I have been able to offer a career to quite a number of girls just leaving school with Higher School Certificate in Mathematics. It has been in mechanised scientific computing. I use the word mechanised rather than mechanical, because the latter may give the false impression of routine and drudgery, whereas the former correctly conveys the fact that mechanical aids are used to lighten the labour.

When can a triangle cover a square? More precisely, when can a square of side s fit into a triangle with sides a, b, c?

Questions about when, and whether, one shape can fit into another are fundamental to the elementary study of shapes in geometry, and yet such questions have only rarely been considered in the literature. In 1872 Cayley thought the question of determining precisely when a given triangle fits into a given circle sufficiently interesting that he posed it as a problem in the Educational Times. In 1956 Ford asked when one given rectangle fits in another given rectangle. A necessary and sufficient condition on the sides was soon supplied by Carver (see also Wetzel). The corresponding question for two triangles, asked in 1964 by Steinhaus, went unanswered for thirty years until Post supplied a set of 18 inequalities on the six sides whose disjunction is both necessary and sufficient.