Despite its title, this article is not intended as a clarion call against miserliness, but rather as a plea for a greater use of non-parametric methods in the teaching of statistics to sixth-forms.

Numbers of the form n(n + 1)/2 are called triangular numbers. This is because and the left-hand side of this equation can be represented by a triangle as in Fig. (a). Numbers of the form n are called square numbers, or simply squares, for obvious similar reasons. We can also have rectangular numbers, but there is little point in defining them as numbers of the form m . n, since then they will include all the numbers. Instead we define a rectangular number as one which is of the form n(n + 1) —that is, the rectangle concerned must have one side exceeding the other by exactly one unit.

I read the article by G. Merlane on the Use of Matrix Methods When Solving Simultaneous Linear Equations, (The Mathematical Gazette, LIV, 390, December 1970, pp. 337-41), with considerable interest, but as an outsider (with only vague recollections of traditional mathematics in his own schooldays) who has taken a rather casual interest in the developments of school mathematics syllabuses in the last ten years or so.

In a well-known theory of infinite series, if a series converges it is taken to represent the number to which it converges, while if a series does not converge it is not thought to represent any number. Hidden behind this well-known theory, there are some intriguing examples of the representation of numbers by series which puzzled mathematicians of the 18 c. and 19 c. When taken individually the examples might be dismissed as being distinctly odd, yet because of the substantial degree of consistency which exists between them they have been studied extensively. Early disputes about the examples centred on the question of whether there is any sense in which the series could be regarded as representing numbers. Yet those who accepted such a series and used it in their work tended to ask the simpler question, what number does this series represent?

After fifteen years of using and teaching applied mathematics, I was called on to teach a course in elementary differential equations. To refresh my memory I consulted several recent textbooks and observed three major changes in their content over the intervening period, namely a clear description of the domain in which solutions are valid, extensive use of single valued and bounded functions and the avoidance at all costs of infinity.

For 34 years (1869-1903) Rawdon Levett served as chief mathematical master at King Edward’s School, Birmingham. He will be remembered as one of the founders of the A.I.G.T.

In studying an axiomatic subject, such as Group Theory, it is desirable to use the smallest set of axioms possible. In this short article we give three examples in which sets of classical axioms can be reduced.

A practical method is described which uses the result

a series which we shall denote by S8 . The approach follows the same general lines as that in a previous note (The Mathematical Gazette, Vol. LIV, October 1970, p. 261) though at a more sophisticated level. We carry out the process first and discuss error analysis, as well as the series itself, afterwards.

In a raffle with replacement a number of prizes are available, and each winning ticket is replaced in the “hat” for inclusion in the draw for the next prize. Thus any ticket may win more than one prize.

This study was suggested by C. S. Ogilvy’s “Tomorrow’s Math”, which poses: “A weighing problem which has been going the rounds of some mathematical communities is the problem of the balls. Given n distinguishable balls, of which it is known that no two have the same weight, it is required that they be arranged in order of magnitude by weighings of one against another in a pan balance scale (no weights). What is the smallest number of weighings, as a function of n, which will always suffice? What strategy (procedure) does one adopt to achieve this minimum? Some assortments (depending on luck) will be more quickly ordered than others. Will the best strategy automatically minimize the expected number of weighings if the original ordering is random?”

Two years ago I was in Paris for the New Year. The shops were gay with decorations, and polyhedra were obviously in fashion. I was particularly impressed with a lampshade which was a feature of many of the decorations; it was in the form of an icosidodecahedron, the pentagonal faces being dimpled, and with “petals” added, so that the whole resembled the flower of a Golden Chalice (Solandra grandifloral). Eventually I found a little shop by Notre Dame which would sell me one; I found myself the purchaser of a small box containing twenty strips of card, about 35 cm by 7 cm, which I was assured would make up into the lampshade I required.

The field of tape-recorders is ripe for some form of standardization. Except that speeds are supposed to be standardized (though a tape recorded on one model of a well-known make plays a semitone sharp on a newer model of the same make) most other things seem to be totally variable. This particular article was inspired by the frustration arising from the fact that some counters count the turns of the run-off spool, while others count the turns of the take-up spool. (Some counters only count every other turn; we shall ignore this wholly unnecessary complication.) The question is: how does one convert a reading from one type of counter to that from the other? This produces an unexpected graph in the form of a circular arc—a rarity which perhaps justifies a note.

In a previous issue of the Gazette J. G. Colman gives a formula for the nth derivative of the composite function f[g(x)], states that this is proved by induction and asks for a more direct and illuminating proof.

The standard tape–recorder consists outwardly of two spools on which the magnetic tape is wound, from the left to the right, between least and greatest radii of a and b. On my machine, a is about 1.05 in. and b is about 2.75 in. Apart from what is on the tape, the only indication at any intermediate stage as to how much playing time is still left is provided by a revolution counter geared to the left hand spool, the one which starts full and finishes empty. The numbers registered are about half the actual number of revolutions made on the left: this arrangement allows some 1400 revolutions to be registered by only three digits. It can be assumed that the tape is fed out at a constant length per unit of time: so the time still to run is proportional to the length still to run. If the total number of revolutions to run is N, how much of the tape ha3 been run off when the reading is n?