bluuebunny/crossref_metadata_2025 huggingface dataset:
Transition
Authors: Tanner R.
Published: January 2007

This September the transition from Grammar to Senior High School will be virtually complete. The transfer age is 14, and is, at the moment, by parental choice. We still have all boys, but receive our first intake of girls this year. Our first non-selective entry was in 1964, and all members of the mathematics staff, only one of whom had met them previously, have now taught the lower ability groups. We have had our problems, none insurmountable, and more of administration than of adaptation. Anyone embarking on a transition to Comprehensive education and needing advice is recommended to read Canon Eperson’s admirable article in the recent Gazette. Much of what he says we have learned by experience, and if I requote from Newsom, it is because the quotations are particularly apt.
On Analytic Functions
Authors: Besicovitch A.S.
Published: January 2007

This note is due to the fact that the identity of two analytic functions, with the same values for every z, is not obvious.
Notes on Matrices
Authors: Sawyer W. W.
Published: January 2007

It is very desirable when teaching mathematics not merely to state definitions and prove theorems, but also to show the train of thought that would lead us naturally to these definitions and theorems. Students are thus enabled to see mathematics as an alive, continuously developing subject. They are reminded of the interdependence of the various branches of mathematics; the memory of each branch is kept alive in the students’ minds, because it is continually being revisited, and the student finds it easier to remember results when he understands the thinking that produced them. This approach also encourages a student to try to carry the train of thought further and to make his own discoveries.
A Geometric Duality for Two Metrics for the Co-Ordinate Plane
Authors: Rhodes Frank , Cox David , Dixon Patricia , Fever John
Published: January 2007

who arrived at the result through discussions in an undergraduate class at the University of Southampton.

Of the various metrics which can be placed on the co-ordinate plane R×R, the Euclidean metric is universally known, and two others are becoming more widely known in schools. With the co-ordinate notation for points x = (x1, x2), these metrics are
Function
Authors: Thornton E. B. C.
Published: January 2007

There seems to be doubt about the meaning of the word “function.” Roughly speaking applied mathematicians tend to use it in its old meaning as a synonym for “dependent variable”, while some pure mathematicians use it to mean a mapping in which no element of the domain can have more than one image.
The “Disaster Kit”
Authors: Banks Bertram
Published: January 2007

Professor Geoffrey Matthews has described the scheme emanating from Ridgewaye School, Southborough, Kent, as a “Disaster Kit” and has asked me to write a description of it. This I gladly do because I am not sure about the title he has bestowed on it and you are invited to title it yourself when you have read my notes.
Vectors and Relative Velocity
Authors: Rayner Margaret E.
Published: January 2007

In many books on elementary mechanics, it is assumed that velocity is a vector quantity on the grounds that it has magnitude and direction. It is, however, very far from obvious that these are sufficient reasons and a rather more thorough investigation of the nature of velocity may provide not only a clearer idea of what velocity is but may also clear up some of the difficulties associated with “relative” velocity.
“Some Inequalities for a Triangle”
Authors: Oppenheim A.
Published: January 2007

In a paper with the same title, Carlitz [1] proves two inequalities about a triangle ABC and an internal point O:
The Seven Point Plane
Authors: Burn R. P.
Published: January 2007

Finite projective planes are a lively field of research, particularly in America and Germany. SMP Further Mathematics gives them some attention, and recently two American text-books which would serve as suitable introductions to the subject at the beginning undergraduate level have been published [Albert and Sandler, Introduction to Finite Projective Planes, Holt, Binehart and Winston (1968); and Blattner, Projective Plane Geometry, Holden-Day (1968)].
A Comparison of Teachers’ and Employers’ Opinions Regarding the Content of Secondary School Mathematics
Authors: Bishop A. J. , McIntyre D. I.
Published: January 2007

The aims, methods and content of secondary school mathematics are undergoing quite searching examination at the present time to the dismay of some teachers and the satisfaction of others. One of the products of this examination has been a growing dissatisfaction with the content of the older textbooks and a consequent increase in the number of books containing so-called “modern” mathematics. Typically these textbooks attempt to provide a terminal mathematical education, to the extent that they say that they cater for pupils who are not necessarily going on to do more advanced maths. It is appropriate at this time to ask just how much the consumers of the educational product are in agreement with the producers of that product over the qualities that the product should have in order to satisfy future requirements. Accordingly, this investigation compared the opinions of employers and secondary teachers over the mathematical content of a terminal course for 15-16 year olds.
A Non-Commutative, Associative Operation on the Reals
Authors: Budden F. J.
Published: January 2007

At the Cambridge Conference, 1969, I proposed the following problem to several members:

Find a binary operation on the reals which is associative but not commutative.
Matrices and Sequences
Authors: Alway G.G.
Published: January 2007

The following is offered as an example of a proof in which the important part is most easily written out in the form of a block flow diagram familiar to programmers of universal computers.

We shall be concerned with matrices of only a particular kind, namely antisymmetric matrices whose non-diagonal elements are all plus or minus unity. Call such matrices, p-matrices.
On Geometric Problems of Erdös and Oppenheim
Authors: Mordell L.J
Published: January 2007

The American Mathematical Monthly in Vol. 68 (1961) page 380 gave a solution by Leon Bankoff of the following problem proposed by A. Oppenheim of the University of Malaya in Vol. 67 (1960) page 802.
A History of the Teaching of Modern Mathematics in England
Authors: Rollett A.P.
Published: January 2007

The Editor is under the impression that I am conversant with all the experimental work in this country that is directed towards modernising the syllabus in schools. This is a fallacy. A great deal is going on, but only a small part of the iceberg shows above the surface (an unsuitable analogy, however, for a topic so often associated with hot air!). But the genuine innovator is usually modest and self-critical; when discovered, and the question put: “Why don’t you send an account of this to The Gazette, it is just what is wanted” the invariable reply is : “Oh, this is a very amateurish effort, surely there are lots of others making a much better job of it”. There have always been experimenters, because of the nature of the English temperament and the absence of a central syllabus (though that would probably make little difference) ; and until the last War there was no lack of able mathematicians in the schools, as is shown by the large number of original (and consequently unsuccessful) textbooks published during the past hundred years. Forty years ago Mr. Hope-Jones was teaching probability and pleading for its adoption by all ; it has been available in several examination syllabuses since the War. Others advocated the earlier introduction of trigonometry or vectors, but use of these in a geometry paper could be dangerous ; nowadays an examiner would hesitate to mark-down a candidate who used vectors instead of congruent triangles in solving a problem. It is not uncommon, of course, for the unenterprising teacher to say that if such-and-such a modern topic were in the examination syllabus he would be glad to deal with it, but at present he dare not risk his pupils’ future careers, etc. But a new examination syllabus, though desirable, is neither a necessary nor a sufficient condition for the success of an experiment. There appears to be some acceptance of what might be called the First Law of Motivation, namely, that every boy (and master?) continues in his state of rest, or of uniform motion along the tramline, except insofaras he may be compelled by externally applied force or an external examination to change that state.
The Teaching of Mathematics in Schools: A Criticism of the English Educational System
Authors: Rosenhead L.
Published: January 2007

In opening this afternoon’s discussion I wish to put before you some reasoned criticisms of a number of attitudes and arrangements embedded in the English Educational System; I want to talk generally—not submit a detailed blue-print of some brave new system. During my talk I will ask you a number of questions and I would be grateful if some of you would respond and indicate your views. Although I feel deeply on some of the points I shall raise, I put forward my arguments with diffidence, for I realise that opinions may, and frequently do, differ.
An Experiment in the Teaching of ‘Modern Mathematics’ in Schools
Authors: Wilson Robin J.
Published: January 2007

There has been a great interest in the possibility of teaching ‘modern mathematics’ in schools ever since the Southampton Conference in 1961 when Professor Bryan Thwaites and his team put forward their plan for modernising school mathematics, especially for the period spanning the four years before ‘0’ level. Their reason for the change was that, since during that period the child’s mind is most intellectually alive, a lot of dull, heavy manipulative work would do far more harm than good, whereas they felt that ‘modern mathematics’ would be more intellectually stimulating and enjoyable. Since then, several Examination Boards have produced alternative syllabuses for ‘0’ level, containing set theory, group theory, etc. More and more schools have been doing the two-year course leading up to this examination, and the first papers were set last year.
On Some Properties of Elliptic Circles
Authors: Mittra S. S.
Published: January 2007

Goodner has studied the equations to conic sections in the elliptic plane by following the methods of Euclidean geometry and using Weierstrassian co-ordinates. In the present note the author has investigated several properties of the elliptic circles from the same analytic standpoint and has shown that some of the well-known properties of the Euclidean circles hold as well for the elliptic circles.
Illustrations of Simple Group Theory
Authors: Holland J.
Published: January 2007

It is well known that the six rotations of an equilateral triangle form a group; that is to say that, if an equilateral triangle ABC occupies a certain triangular space, there are six operations and six only which when performed either singly or in succession will leave the triangle still occupying the same space, but with vertices in different positions. The set of six possible orientations of the movable triangle are drawn below and numbered.
Calendar Reform
Authors: Bushell W. F.
Published: January 2007

The difficulty of a good calendar lies in the fact that the earth travels round the sun in 365 days, or to be precise in 365 days 5 hours, 48 mins., 46 secs, which, we should note, is less than 365 days by just over 11 minutes.
A Model of a Twisted Cubic
Authors: Bayes A. J.
Published: January 2007

The representation of a twisted cubic as the intersection of a hyperboloid and cylinder with common generator makes an interesting model. This note gives details of its construction.
A Generalisation of Simson’s Theorem
Authors: Zylbertrest S.
Published: January 2007

Let w be a point and ABCDEF a polygon, convex or not, which we call (P). Projecting w orthogonally on the sides AB, BC, CD, DE, EF, FA of (P), and joining the respective projections, we obtain the polygon A1B1C1D1E1F1, called (P1) [see Figure 1]. We call (P1) the first pedal of w with respect to (P).
A New Method of Calculating Roots of Algebraic Equations
Authors: Smith G.S.
Published: January 2007

A method of calculating real roots by a simple division process when an integral approximation is known is described. It is very powerful and economical in labour as well as being of extreme simplicity It surpasses Horner’s method in these respects and in many applications it is preferable to the Newton-Raphson method and others. A root is expressed directly in any scale of notation or, alternatively, as a continued fraction, and the processes involved would appear to be suitable for use in an automatic computer.
On Liking Mathematics
Authors: Coulson C. A.
Published: January 2007

The inspiration for this address comes from Little Tim—or should I call him Tiny Tim? And this is odd because I don’t really know his name. He belonged to the D-stream of a school, and he sat on the front row of the class to which I was speaking. I have no idea now what I was talking about: but I do know that to all intents and purposes Little Tim was unmoved by any of my eloquence. His round little face was immobile, and nothing that I could do seemed able to disturb its obvious complacency. And then—I do not know what caused it all to happen, and I like to think it was not because he had noticed that my shoelace was undone or my front eollar-stud missing—the trace of a smile began to appear upon his lips. It broadened to a horizontal line across his round and homely face; and then the vision faded. Little Tim returned once again to his former state.
The Effect of Atmospheric Drag on the Orbit of an Arfificial Satellite
Authors: Bramson Brian
Published: January 2007

The study of artificial satellites is today a subject of topical interest and naturally much work has been completed in the field of satellite orbits. Many aspects of this work involve the use of differential equations. Unfortunately very few of these may be solved by ordinary analytic techniques. One possibility is to define a new function to be the solution of the given equation, for example the logarithmic function. Nowadays we use computers. Nevertheless it is highly satisfying to produce an analytic solution to a problem in applied mathematics for in a sense we have succeeded in displaying a pattern in the physical world.
Why calculus cannot be made easy
Authors: Schwarzenberger R. L. E.
Published: January 2007

Calculus made easy by “F.R.S.” was published seventy years ago. Its anonymous author wrote explicitly for engineering students rather than for future mathematics teachers. He ended by asking those who had enjoyed the book “not to give the author away nor to tell the mathematicians what a fool he really is”.
Hooke’s joint and the Avignon sundial
Authors: Cundy H. Martyn , Hogg Richard W.
Published: January 2007

In the summer at Avignon a little Disneyland train conveys tourists from the Palais des Papes to the summit of a little hill behind the cathedral called the Rocher des Doms. In the garden on this hill there is a most remarkable sundial; the visitor himself provides the gnomon, and a linear scale on the flagstone pavement shows him where to stand at different dates. His shadow then falls towards one or other of a ring of fixed pyramidal markers which indicate the hours of the day.
The Sieve—again
Authors: Langford W. J.
Published: January 2007

Many teachers in junior schools will know the interest which children show in drawing patterns of multiples on a 10 × 10 lattice. If those for the primes 2, 3, 5, 7 and 11 are drawn on separate transparent sheets and are then superimposed, the picture produced by the overhead projector will have some of the appearances of a sieve. It will then be worthwhile to refer to an article by R. A. Fisher in Gazette XIV, 564–6 (No. 204, December 1929) which carries out the process in a slightly different way to reveal all the odd primes less than 1600, by drawing the factor lines of the primes up to p = 37; this same set will, of course, isolate all the primes less than 1681.
A new construction of the real numbers
Authors: Shiu P.
Published: January 2007

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.
The truth about Venn diagrams
Authors: Stewart Ian
Published: January 2007

Venn diagrams are usually considered to be a method for visualising set-theoretic identities and suggesting true theorems. It is frowned upon to prove a set-theoretic identity using them, such proofs being considered non-rigorous. One of the better kept secrets of the trade is that they can be made rigorous by proving the right kind of ‘metatheorem’. A proof is quite instructive, and might be of interest to set-theoretically minded sixth formers.
Is this a Recorde?
Authors: Cassels J. W. S.
Published: January 2007

At some time in the eighties or nineties of the last century the Rev. William Done Bushell, a mathematics master at Harrow School, bought a portrait painting at an auction sale at a house in Harrow. It does not seem to have had any pedigree but bore the inscription “Robt. Recorde M.D. 1556”. It was thus presumed to depict the Tudor physician and mathematician Robert Recorde (1510?–1558) who wrote some of the earliest books in English on mathematics and was the first to use the symbol “=” for equality. No other portrait was (or is) known.
Sums equal to products
Authors: MacKinnon Nick , Lord Nick
Published: January 2007

The aim of this short article is to produce a sequence of real numbers x1, x2, x3, … with the property that, for each n,

x1 + x2 + … + xn = x1x2 = xn.

This turns out to be an interesting classroom exercise involving simple equations and iteration.
A New Definition of Rounding
Authors: Ball A.A.
Published: January 2007

Decimal expansions

Every real number x has a decimal expansion of the form

x=±N.n1n2n3…,

where N is either zero or a positive integer and each ni is one of the digits from 0 to 9. The notation N.nln2n3… is an abbreviation for the infinite series
Three points for a win
Authors: Newson Graham
Published: January 2007

The rationale for the introduction of the system of three points for a win in the Football League in 1981-82 was, presumably, that more reward for winning games would encourage more positive attitudes from teams and that the consequent more attractive and attacking football would bring in bigger crowds. Certainly the philosophy of many away teams, or at least their managers, that they began a game at 0-0 with one point to hang on for
had become commonplace in recent seasons.
Space filling with identical symmetrical solids
Authors: Drummond J. E.
Published: January 2007

In one dimension, a set of rods of equal length may be used to cover the line. In two dimensions, an area may be covered by joining up sets of identical symmetrical squares, equilateral triangles or regular hexagons. In three dimensions, there are at least four different symmetrical solids which can be packed together to fill space. Of these, the cube is most obvious, two others are moderately well known (see [1] for example), while I have been unable to find any record of the fourth solid.
Natural philosophy
Authors: McCrea W. H.
Published: January 2007

In this lecture we shall see how natural philosophy—or what we traditionally call applied mathematics—is continually changing. I hope to show that the subject is far livelier and more adventurous than we sometimes suppose, and that each advance opens ever wider vistas of further progress.

We are concerned with the relation between mind and experience of the physical world. Our minds are such as to be most strongly persuaded that there is a significant relationship by their being able to predict experience that no one has ever had before.
Symmetrical intersections of right circular cylinders
Authors: Moore Moreton
Published: January 2007

The intersection of two circular cylinders of equal radius is not only of mathematical interest but also has application in both engineering and architecture. The joining of pipes of circular cross-section at a variety of given angles is an obvious example. The Romans and Normans, in using the barrel vault to span their buildings, were familiar with the geometry of intersecting cylinders where two such vaults crossed one another to form a cross vault. Larger numbers of equal intersecting cylinders arise in the following way.
The mathematics of population growth
Authors: Andrew A. L.
Published: January 2007

The word “lesson” came back to Pooh as one he had heard before somewhere. “There’s a thing called Twy-stymes”, he said. “Christopher Robin tried to teach it to me once, but it didn’t.” A. A. Milne.

In assessing the value of the mathematics taught in schools, considerable emphasis is normally given to the use which a minority will later make of it in their jobs. It is the purpose of this article to point out that even those who never use mathematics professionally could better understand important social issues if they had a good basic training in mathematics. This is illustrated by a set of mathematical models related to one such problem— the population explosion.
Transcendental numbers
Authors: Halberstam H.
Published: January 2007

1. You all remember that in that well-known play, Waiting for Godot, the eponymous hero never actually appears. In the same way my talk should perhaps have been entitled Waiting for transcendental numbers—for while transcendental numbers are at the back of everything that follows, they will not become, for reasons of technical difficulty, proper part of the action.
Dangerous loads and lattices
Authors: Lloyd E. Keith
Published: January 2007

There are probably few readers of the Gazette who also read the weekly periodical Commercial Motor, but it was a table appearing in an article in the latter which caught the author’s eye and provided the inspiration for the present work.

In recent years legislation has been introduced which requires that lorries carrying dangerous substances must also bear panels on which appear, inter alia, codes identifying the substances and giving an indication to the fire brigade of what action should be taken in the event that the vehicle is involved in a collision or other emergency. Examples of the codes are shown in Figs. 1 (a) and (b). The former is probably the most commonly seen code as it is that displayed on petrol tankers; the latter is a code seen on a vehicle carrying liquid nitrogen.
The log route to means and inequalities
Authors: Chorlton Frank
Published: January 2007

In this short article I consider the centre of mass of a collection of particles placed on the log curve, and deduce various inequalities concerning the arithmetic, harmonic and geometric means. I have seen the idea used for a pair of points on the log curve but thought that the general approach could be of interest to Gazette readers.
Elections and swings
Authors: Fletcher Colin R.
Published: January 2007

The change from essentially two party contests to three-, or even four-cornered fights, has provided the impetus for a fresh look at both diagrammatic representations of an election outcome, and the measure of swing from one election to the next. The representation of the votes in a three-cornered contest has been considered, amongst others, by Ibbetson [1], by Hollingdale [2] and by Miller in a series of papers and books (see for example [3] and [4]). Miller puts forward three ways of describing the outcome of such a contest, the ‘box-corner’ method (three-dimensional coordinate axes), homogeneous coordinates, and the ‘rectangular coordinate derivation’, but, as he notes, these three are fundamentally the same. Miller states that ‘mathematicians will instinctively go for the homogeneous coordinate representation’, and Hollingdale kindly obliges, although Miller himself plumps for the third representation, [3].
Plane surveying
Authors: Crilly Tony
Published: January 2007

Areal measurement of a two-dimensional plane figure is a problem of ancient origin and one which continues to hold both mathematical and practical interest. There are various approaches possible, ranging from counting squares to the more sophisticated techniques based on integration.
Roots of unity revisited!
Authors: Denton Brian
Published: January 2007

This article arises because of a lecture I attended where the speaker, who had been heavily involved in the new JMB A level syllabus, was saying that roots of unity had gone, “thank goodness.” Many A level teachers seemed to agree and I began wondering just how “traditional” this topic is. I came to the conclusion that it is the most revealing topic to show the meshing of traditional and modern ideas. I hope to illustrate this by the end of this article.
Digital Roots, Rings and Clock Arithmetic
Authors: Gardiner A.
Published: January 2007

The ‘digital root’ of 73219 is 4, since 7 + 3 + 2 +1 + 9 = 22 and 2 + 2 = 4. This idea features in some popular puzzles (and in a recent Gazette note 65.26). Section 1 of this article concerns some basic properties of digital roots, while Section 2 gives a characterisation of functions like the ‘digital root’ function. The former provides interesting exercises in modular arithmetic; the latter in elementary ring theory.
Random triangles
Authors: Easingwood Trevor
Published: January 2007

“What is the probability that a randomly drawn triangle is acute-angled?” The problem sounds simple enough. However, what is interesting about it is the interpretation given to “random”. We investigate three possibilities

(a) that the angles of the triangle are chosen;

(b) that the lengths of the sides are chosen; and

(c) that the coordinates of the vertices of the triangle are chosen.
How long is a game of Beetle?
Authors: Watson F. R.
Published: January 2007

Finding myself one day involved in a Beetle drive, and somewhat deficient in the “will to win” essential for success in such competitive circumstances, I began idly to speculate on some of the probability problems involved—they proved to be more absorbing, and more complicated, than I had originally envisaged.
Greenspan’s harmonic oscillator
Authors: Astin J.
Published: January 2007

In a recently published book by Greenspan classical dynamics is modelled by a set of difference equations. There are two distinct reasons why this approach could be useful. First, since all our measurements are made at specific times and are thus not continuous it may be that the model is a “better” representation than the usual one using the calculus. Secondly, since most practical problems are now solved by numerical techniques, and one of the most used of these is the method of finite difference, a formulation of the basic equations in difference form makes them directly amenable to numerical solution. There are numerous ways of modelling the derivatives as differences and Greenspan chooses a way which preserves the energy equation. This is aesthetically pleasing and also gives rise generally to stable numerical schemes.
Mathematical sailsmanship
Authors: Galbraith P. L. , Praegar D. A.
Published: January 2007

It has been the experience of one of the authors of this article that students in senior secondary and early tertiary courses tend to regard functions defined on split domains and concepts such as differentiability as somewhat pathological and of interest principally as traps for students in pure mathematics.
The effect on the solution of a problem of errors in the calculation
Authors: Bailey D. E.
Published: January 2007

There is a popular view among sixth form mathematicians, and therefore presumably among some who teach them, that evaluation of an integral by the approximate Trapezium Rule is made more accurate by increasing the value of n.
What the eye doesn't see, …
Authors: Quadling Douglas
Published: January 2007

We are just contemplating the purchase of a new car, and have been testdriving a number of competing models. A mirror fixed to the driver's door is nowadays standard equipment, and on most of the models this is attached near the front of the door, more or less in line with the hinges. But in the VW Golf it is set further back; a good feature, claimed the salesman, since it reduces the driver's “blind spot”.

I confess that it was not obvious to me that this is so. If true, then perhaps it would compensate for the inconvenience of having to turn one's head further round to look in the mirror. But mathematical investigation seemed to be called for.
The Stresses in a uniform circular ring subject to symmetric radial loading
Authors: Budden F.J.
Published: January 2007

In the general case suppose that the radial forces are of magnitude P and that the angle between adjacent forces is 2α (so that α = 20° in the above figure). Consider the portion YY′ of the ring, as in the left-hand figure below. From a consideration of the rotational symmetry of the ring, it is evident that the tensions at Yand Y′are equal, these being marked T in the figure. Similarly, if the shear force at Y is Stowards the centre O, then that at Y′ is S away from the centre. Moreover, the bending moments at Y and Y′ are equal and opposite. These are marked M in the left-hand figure.
Early Warning
Authors: Smithers Graham S.
Published: January 2007

There is an ever-increasing need for a collection of statistical experiments that can be used in the classroom. Ideally they should be linked to (and motivated by) attempts to model or simulate real situations. Mandel attempts to satisfy this need by studying some simple classroom experiments in physics from a statistical viewpoint. Unfortunately most of the experiments deal with artificial situations. Downton discusses the role of statistics in schools and lists a dozen suitable classroom experiments. This paper is strongly recommended and is full of interesting and controversial ideas.
Antiprism frameworks
Authors: Cundy H. Martyn
Published: January 2007

1. Introduction. Recently I was looking for a simple but interesting model which a class of students could complete in about twenty minutes, which would illustrate groups of symmetries in three-dimensional space. My thoughts went back to the lampshade described in the Gazette for June 1971 (No. 393, pp. 310–312) but I felt that it would take too long to make, even if each student merely cut out a strip which I had marked out beforehand. Forced in this way to reconsider the matter, I found that the last sentence of my article was false—there are in fact four other such structures, which I now describe.
A problem of unusual gravity
Authors: Chapman C. J.
Published: January 2007

One of the standard results of the theory of Newtonian gravitational attraction is that the potential outside of a spherical shell can be obtained by regarding all the mass as concentrated at the centre of the shell.
Some more connected problems in probability
Authors: Jones David Pierce
Published: January 2007

Frank Budden’s article in the March 1982 Gazette concerned various problems in probability about the formation of triangles from, for example, a glass rod broken into three. That article, and an Oxford Entrance Examination question, set me thinking about the following connected problems. The first is from the 1972 examination.
Mathematics and the less exact sciences
Authors: Conway Freda
Published: January 2007

School mathematics is customarily taught in three stages which can be described as practical, pure and applied respectively. They differ from each other in the way that they make use of scientific data; [1]. The first stage involves teaching mathematics through its applications; pure mathematics makes no use of applications and at the third stage known pure mathematics is applied to other scientific problems.
Symmetry and differential equations
Authors: Greenman J. V.
Published: January 2007

An interesting example of the usefulness of group theory occurs in the solution of first order differential equations. If such an equation retains its form under each member of a group of transformations on the plane then, under certain circumstances, we can determine from the structure of the group new variables in which the equation is separable and hence solvable by integration. In the first section we show how to construct the new variables and in the second we prove separability.
Notes: 61.20 Using graphs to count logic block chains
Authors: Ball Derek
Published: January 2007

Last summer a colleague of mine suggested a counting problem which I eventually solved by a curious method involving the counting of subgraphs. The problem was to find the number of different ways of arranging twelve Dienes logic blocks which differ only in shape and colour in a closed chain, in such a way that there is one difference in attribute between each pair of neighbouring pieces in the chain.
Notes: 61.24 How to serve in tennis
Authors: Prussing John E.
Published: January 2007

Most tennis players have two types of serve in their repertory. One is a safe serve, which is relatively accurate, but also relatively easy for the opponent to return effectively. The other is a risky serve, which is more likely to be a fault, but also more difficult to return if it is good.
Notes: 61.25 Prescribed numbers of tiles and tilings
Authors: Harborth Heiko
Published: January 2007

In their recent article [1] B. Grünbaum and G. C. Shephard asked whether, for every pair of natural numbers k, r, there exist sets of k tiles that admit precisely r distinct tilings of the plane. (Two tilings are said to be distinct if they cannot be brought into coincidence by a rigid motion, and it is to be understood that in the tilings each distinct tile must occur at least once.) They gave examples for k = 1, r = 1, 2 and for k = 2, r = 1, and referred in a footnote to a later discovery of a solution for k = 1, r = 3.
Five integers which sum in pairs to squares
Authors: Thatcher A. R.
Published: January 2007

I am most grateful to the Mathematical Gazette for publishing, in the March 1977 issue, my letter about the problem of finding five positive integers such that the sum of any pair is a perfect square. The response far exceeded my expectations and showed that the problem aroused great interest.
Factor groups
Authors: MacDonald I. D.
Published: January 2007

Most elementary texts on group theory define factor groups. Most beginners find factor groups difficult. The present exposition is based on the notion that it may be helpful to have a look at factor groups in a context slightly wider than that of “factoring out normal subgroups”.
Constructions with a rigid compass
Authors: Croft H. T. , Körner T. W.
Published: January 2007

This article is addressed to anyone who has ever doodled idly with a fixed compass on a blank sheet of paper. Perhaps he has drawn a circle centre O, picked a point P1 on it, drawn a circle centre P1 intersecting the original circle at P2 and P3, drawn a circle centre P2 producing two new points of intersection, slowly building up an interlacing hexagonal pattern II of points each a distance at least R from each other. Then he has picked some point P* not of II, and drawn a circle centre P*, and almost immediately the pretty pattern has been replaced by a jumble of intersections apparently covering the whole of the paper.
Sums of Squares and Pyramidal Numbers
Authors: Singmaster David
Published: January 2007

In the June 1981 Gazette, Ian Anderson discussed several ways of finding the sum of the first n squares:

His methods are mostly algebraic and I wondered if there were more geometric ways of obtaining the formula. After some time, I have found several methods which I present here. These methods are based on counting balls in piles and are suitable for exploratory activity at school level.
What is an Asymptote?
Authors: Giblin P. J.
Published: January 2007

Consider the hyperbola y = 1/x. (It will be convenient to restrict attention to that part for which x > 1.) Everyone will agree that the x-axis is an “asymptote” to this curve; however, different people are likely to advance different reasons for this fact.
Be careful when you dodge round Newton
Authors: Arscott F. M.
Published: January 2007

In the course of attacking a research problem recently, I came across a numerical dodge which, though really very simple, was new to me; in the course of using it I involuntarily gave myself an object lesson in the dangers of blind use of formulae. I hope there are readers of the Gazette who will appreciate learning of the dodge, and/or laughing at my misfortune.
Who gets good degrees?
Authors: Anderson Johnston
Published: January 2007

In a previous article in the March 1979 Gazette I looked at a population of 589 students entering a particular university department between 1971 and 1975 to read for a degree in mathematics, or in mathematics-with-another-subject. These students graduated between 1974 and 1978.

The present article provides updated information for students entering between 1976 and 1983 (graduating 1979 to 1986 respectively). Several factors have changed since the earlier survey. A significant change is the increase in the number of entrants offering single-subject mathematics at A-level. A second important factor has been the consequences flowing from the UGC cuts in universities in 1981; admission targets are now much more closely controlled and the level of offer to intending entrants has often tended to rise (certainly the case at this institution). There has been, as a result, a decrease in the number of students entering with poorish grades.
Paradigms of proof
Authors: Anderson Johnston , Austin Keith
Published: January 2007

First year mathematics undergraduates entering British universities are often very unprepared for the kind of proof activities that occur throughout undergraduate degree programmes. Their previous experiences of the idea of proof are frequently limited to situations where the ‘proof’ is simply an extended chain of calculations or algebraic manipulations. In fact, this is well-illustrated by their approach to proof by induction, where they can carry out the mechanical details of the inductive step, but have no understanding of why proof by induction works, nor what status the initialisation step and the inductive hypothesis have in the proof. There is no real grasp of what it means to prove that some assertion is true. Much time, in consequence, is spent in the first years of mathematics degree courses up and down the country attempting to bring about this understanding.
An interception problem
Authors: Lord Nick
Published: January 2007

The problem A balloon which ascends with constant vertical velocity—is released from ground level into a steady wind which blows it towards an archer who is situated at ground level and able to fire an arrow at any elevation. What is the minimum speed of release of the arrow needed to achieve a hit?

(He is only allowed to fire after the balloon is released (!) and we shall assume the flight of the arrow to be unaffected by the wind.)
Two problems in telegraphy
Authors: Hart Vincent G.
Published: January 2007

A recent excellent book by Paul J. Nahin on Oliver Heaviside, a self-taught genius of the Victorian age, contains several nice examples of problem solving by the master. Two of these, concerning the flow of electricity along telegraph cables, are suitable for sixth formers and are described here. In passing it may also be mentioned that these cables, used first for telegraphic signals and later for voice signals, were a triumph of both the practical and theoretical skills of the nineteenth century, and were in use for world-wide communications for over a hundred years up to the nineteen seventies.
Monochromatic and monotonic: A theorem of Ramsey and an application to analysis
Authors: Austin Keith
Published: January 2007

A familiar result is that in any group of six people there are three who either all know each other or are all strangers to each other. This can be reformulated to say that, if six points are marked on a piece of paper and every pair of points is joined by a red or blue line, then there will be a monochromatic triangle, i.e. one which is all red or all blue. This result is related to a theorem which the Cambridge philosopher Frank Ramsey proved in 1930.
An optimisation problem for triangles
Authors: Crilly Tony , Millward Stanley
Published: January 2007

Here is a problem which can be tackled at various levels. It has the merit of being straightforward and requiring little abstract mathematics, though it is far from trivial.

Our investigation began with the well known problem of fencing a field one of whose borders runs along a straight river. Given 100m of fencing, describe the rectangular field which has maximum area, where no fence is required along the riverbank.
Piecewise linear recurrence relations
Authors: Crampin Mike
Published: January 2007

The properties of linear, homogeneous, constant coefficient, second order recurrence relations such as

are well known. Consider however the recurrence relation

Such things are rarely studied; it may not be obvious that equation (2) generates a sequence which is periodic with period 9, which is to say that for any initial values x0, x1, the sequence x0, x1, x2, x3, … satisfies xn+9 = xn for every n. This article records some discoveries which resulted from an attempt to understand this interesting fact. These discoveries open up a field for exciting, elementary, but by no means trivial, investigations.
Five space-filling polyhedra
Authors: Inchbald Guy
Published: January 2007

Solid shapes which pack together to fill space cover a large and varied range. I recently found five such polyhedra which I have not seen described elsewhere. The names which I have used for them here are rather ferocious I am afraid, but that just happens to be the way the names of polyhedra work.
In Which Key did the Angels Sing?
Authors: Bowers John F.
Published: January 2007

Among the very few reported observations of singing angels, the account given by the shepherds of the angelic choir on Christmas night records some of the words but gives no account of the music, even though shepherds are much celebrated as pipe players and so form an unusually musical profession. But in which key did the angels sing? This question could only have been answered for us by a shepherd with perfect pitch, that is, an unusually good ear for pitch (how high or low the music is). However, the question also contains the hidden assumption that the angels based their music on the key system, which was not known on earth until the sixteenth century AD, so the question should be changed to “Which musical system did the angels use?”. Clearly, as angels are perfect, the system must be the best available, so we may be able to answer the question theoretically.
Rational approximations
Authors: Robin Anthony C.
Published: January 2007

Any positive fraction can be represented by a lattice point in the positive quadrant of the coordinate plane, (x, y) being used to represent the fraction y/x. The diagram shows the points representing 2/3,4/6,6/9 and it’s easy to see that equivalent fractions will in general lie on a line through the origin, the gradient of that line equalling the value of the fractions in question.
Mathematics and the motion of the human body
Authors: Roper Tom
Published: January 2007

A number of recently published articles have dealt with aspects of mathematics as applied to the motion of the human body, [1, 2, 3], some with special regard to the modelling process. There seems to be a growing interest in the application of mathematics to this area and the recent introduction of A-level “Sports Studies” into some schools and sixth form colleges must help this trend. At a talk I gave recently to a branch of the Mathematical Association on “Mechanics and gymnastics”, five out of an audience of twenty-five were P.E. specialists looking for ideas to enable them to teach the mathematical/mechanical part of this A-level syllabus.
A problem in squash strategy
Authors: Brooks W. J. , Hughston L. P.
Published: January 2007

Two players A and B have a game of squash. Player A serves at the start of the game. They play a sequence of rallies. If the player serving the rally wins the rally, he scores a point, and serves the next rally. If he loses the rally, then the other player serves the next rally, with the score unaltered.

Generally the first player to reach nine points wins the game. However, if the score becomes eight-all then the first player to reach eight has a choice: he specifies whether the game is to be played to nine points or to ten ponts.

What is the correct choice? We propose to analyze the problem by use of an elementary probabilistic model.
Quasi-primes and the Goldbach conjecture
Authors: MacKinnon Nick
Published: January 2007

In 1742 Goldbach sent a letter to Euler which contained the conjecture that:

Every even number greater than 2 is the sum of two prime numbers.

It is easy to verify Goldbach's conjecture numerically up to some quite high limits but no proof has yet been found. I have long had the uneasy feeling that the conjecture is not about the prime numbers at all but would equally apply to any ‘sufficiently dense’ sequence of odd numbers. However a recent attempt to generalise the sieve of Eratosthenes has led me to regard Goldbach's conjecture with a good deal more respect.
From quadrature to integration: thirteen years in the life of the cissoid
Authors: Van Maanen Jan
Published: January 2007

In this article I shall take a problem, the quadrature of the cissoid, and compare a hard (but brilliant!) pre-calculus solution of 1658 with the easy (and routine) calculus solution of only 13 years later. Afterwards we shall perhaps see the calculus through different eyes and maybe even teach it differently. Calculus is mechanical, but that it is mechanical is a marvel.
A not-so-flawed draw
Authors: Jones M. C.
Published: January 2007

People seem to regard the draw for the quarter-finals of what appears to have been the 1988 Benson & Hedges Cup in cricket as having been bungled: a quote from the relevant authority is that it was “an embarrassing cock-up”. Well, I'm not so sure! To start with, the draw as made—unlike some others!— was perfectly fair. The draw had to be restarted twice (this is what concerned observers) and so there is a question of efficiency of the chosen method of drawing. In fact, in seeking alternatives, there arise interesting questions of balancing efficiency, “excitement” and the (generally innumerate) public's perception of fairness. Let me explain!
An exploration of hyperpower equations nx = ny
Authors: Voles Roger
Published: January 2007

My interest in this fascinating area of mathematics was aroused by a recent article [1] in which it was shown that

has an infinity of non-trivial solutions (i.e. u ≠ v) in positive integers, rationals and algebraic irrationals. Replacing u and v by their reciprocals, x and y, permits the equation to be written as
How to lose at Tetris
Authors: Burgiel Heidi
Published: January 2007

Tetris is a computer game which has obsessed many computer users and attracted much attention, despite the simplicity of its rules. This paper addresses the question: ‘can you “win” the game Tetris?’

Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989. Versions of Tetris are sold for most personal computers. There are Tetris arcade games, Tetris Nintendo cartridges, and hand-held Tetris games; Tetris has been played on machines ranging from mainframes to calculators. The game's success has prompted the invention of several similar games, including Hextris, Welltris, and Wordtris.