bluuebunny/crossref_metadata_2025 huggingface dataset:
The Size of Elephant Herds
Authors: Holgate P.
Published: January 2007

Populations of large mammals such as elephants tend to group themselves into herds. As these herds wander about, it may happen that two of them meet and amalgamate, or that a large one may split into two smaller ones, (see John Hillaby, “Elephants as a pest control problem”, New Scientist, 12 (1961), 736-8). Suppose that a given area contains n animals, and let pk (t) denotes the probability that at time t they are grouped into k herds. Conditional on this, let x1(t), …, xk(t) denote the sizes of the herds arranged say, in non-increasing order. The chance that two herds will meet and amalgamate will increase with the number of herds, and it may, for instance be taken to be proportional to the excess of this number over unity. Thus the probability of an amalgamation in the interval (t, t + δt), given that there are k herds at time t, is
Markov Chains And Snakes And Ladders
Authors: Daykin D. E. , Jeacocke J. E. , Neal D. G.
Published: January 2007

Everybody is familiar with the game of “Snakes and Ladders”, but not so many realise that the game constitutes an interesting example of a Markov chain. We feel that a good way of introducing Markov chain theory is by solving the problem, How long does a game of Snakes and Ladders last? Doubtless the reader has found that the last game before sending a child to bed seems to drag on interminably, and we used a computer to determine the actual average number of moves when 2 people play on a typical board.
Convergence of the Newton-Raphson Method For Arbitary Polynomials
Authors: Lancaster P.
Published: January 2007

Readers may be interested in an elementary exposition of the convergence properties of the sequence of iterates obtained by applying the Newton-Raphson process to an arbitrary polynomial. The coefficients of the polynomial may be real or complex. By restricting attention to polynomials we will give some precise results using the least possible analysis, while proofs for more general functions would necessarily require more high-powered methods. The arguments we will use require only an elementary knowledge of limiting processes.
The Theory of the Centralizer or Transverse Coupling in Electric Locomotives
Authors: Jenkins R.B.M.
Published: January 2007

It has been noticed that heavy wear takes place on the wheel flanges of electric locomotives in countries where mountain railways exist and where curves are numerous and severe. Locomotives have to return to shops after a comparatively small mileage to have their wheels attended to unless steps are taken to reduce flangewear.
The Evaluation of Definite Integrals as the Limits of Sums
Authors: Flett T. M.
Published: January 2007

It is customary in an account of Riemann integration to give one or two examples of the evaluation of definite integrals directly from the definition of the integral in terms of Riemann (or Darboux) sums, but the method is used only to illustrate the definition, and is never fully exploited. The method is, however, capable of a more systematic treatment than one might at first expect, and in this note we show how it can be applied to some of the elementary functions of analysis.
Reducing Classical Axioms
Authors: Bryant Victor
Published: January 2007

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.
Rotation Groups and Permutation Groups
Authors: Fuller A. W
Published: January 2007

Several books on Group Theory show that the isomorphism between the group of proper rotations of a cube and P4, the group of 24 permutations on four arbitrary symbols, can be established by a consideration of the four diagonals of the cube. To each arrangement of these diagonals corresponds a unique position of the cube and therefore a unique rotation from one position as reference to that position. The extension of this result to other polyhedra does not appear in the books that I know.
On Polyhedra with Specified Types of Face
Authors: Crowe D. W. , Molnár J.
Published: January 2007

1. Introduction

In [1] the following three questions have been asked, and answered: For which values of f is there a convex polyhedron whose faces consist of

(i) f triangles? (Ans f = 4, 6, 8, 10, 12, 14, …)

(ii) f quadrilaterals? (Ans f = 6, 8, 9, 10, 11, …)

(iii) f pentagons? (Ans f = 12, 16, 18, 20, 22, …)
Paper Logic Machine
Authors: Holt Michael
Published: January 2007

Junior School children seem to be able to get the feel of the Klein group by using aptly-prepared paper sheets and wooden blocks. Perhaps, surprisingly, they can also handle such logical implications (when suitably embodied in devices) as:

A→B, ∼B→∼A, B→A, and ∼A→∼B

Here “→” means “implies” and “∼” means “not.”
‘An Easy Commerce of the and the New.’
Authors: Jeffreys Bertha
Published: January 2007

The Council has done me a great honour in making me your president, but has given me a difficult assignment in following our last president Professor Coulson, who achieved outstanding success in providing interest and stimulus for all of us. I played with the idea of the title “On not liking Mathematics” and there are indeed those who do not, but I felt that most of us here probably cannot remember a time when we did not like numbers and, moreover, my teaching life has been spent with those who have chosen Mathematics as their study at the University and this must mean that they like the subject at least better than others. A possible title would have been “Let ‘em be moderate as well as intelligent” which I found in a letter written in London by Sir Isaac Newton (1) (who represented the University of Cambridge in Parliament) on 30 April, 1689, to the Vice-Chancellor of Cambridge University concerning a Bill to settle ye Charters & Privileges of ye Universities: “And if at that distance you are from hence you cannot communicate your advice so easily as might be desired, you may perhaps do well to send up one or two intelligent persons (with such instructions as you shall see fit) for us to consult with here in drawing ye Bill. But if you send up anybody let ‘em be moderate as well as intelligent and let ‘em be sent as soon as may be”.
The Area of the Curved Surface of a Hemisphere in Ancient Egypt
Authors: Fletcher E. N. R.
Published: January 2007

An important contribution to the history of mathematics has appeared in the Australian Journal of Science, October 1967, Vol. 30 No. 4.
Stylistic Analysis
Authors: Harvey P.
Published: January 2007

A recent and fascinating application of statistics has been that of analysing the distribution of words in a written document in an attempt to determine the authorship. This article reviews some of the methods used and proposes a simplified test that could be performed using a desk calculator, although access to a digital computer would reduce the work required. An example of the test applied to a manuscript in the British Museum (Stowe 269) is given in detail to illustrate the method.
The Centenary of the Birth of W. H. Young (20th October 1863)
Authors: Sutton Graham
Published: January 2007

Tonight I propose to say something about a man who is now recognised as a pioneer mathematician of his time, but I do not intend to talk about his work as a creative mathematician. I shall leave that to others who are far better qualified to do so. As a young man I studied pure mathematics under Young and G. H. Hardy, but most of my work as a professional mathematician has been in mathematical physics, and I cannot say anything significant about subjects that I ceased to work on some thirty-five years ago.
Restricted Maxima and Minima
Authors: Smith R. C.
Published: January 2007

The method of finding the maxima and minima of a function of two variables f(x, y) in the unrestricted case is straightforward in theory One solves and then substitutes the roots in Hessian function
Some Potential Problems Involving Spheres
Authors: Chorlton F.
Published: January 2007

In many branches of mathematical physics it is of some importance to know how a given potential field is disturbed when a finite sphere is introduced into it. The new potential is usually obtained as the superposition of the former one and a suitable “perturbation potential” which becomes vanishingly small at large distances from the sphere. Such problems arise in electrostatics, hydrodynamics and steady heat flow. The nature of the perturbation potential depends on the boundary conditions on the surface of the sphere. In this note a method is given for finding this perturbation potential for several different types of boundary condition. The field need not be axially symmetric. The method is useful in several cases for determining image systems directly and without using special geometric properties of inverse points in a sphere.
Mid-Circles and Loxodromes
Authors: Coxeter H. S. M.
Published: January 2007

Any two distinct circles a and b (in the real inversive plane) have one or two mid-circles which may be described as the locus of the point of contact of a variable pair of tangent circles that touch a and b [8, p. 31; 4, p. 74]. We easily see by inversion that two non-intersecting or tangent circles have a unique mid-circle, but two intersecting circles have two orthogonal mid-circles, bisecting the angles of intersection. The circles a and b are interchanged by inversion in their mid-circle (or in either of their mid-circles). This subject has been surprisingly neglected, possibly because of the terrifying name “circles of antisimilitude” [1, p. 28].
The Changing Role of Mathematics in Education
Authors: Williams E. M.
Published: January 2007

The election of this year’s President has in some respects made history. There have been three previous women Presidents in the 95 years of the Association’s life and I recall with admiration the names of such eminent predecessors as Dr. Busbridge and Dr Cartwright, and with affectionate sadness at this time of loss, Miss L. D. Adams. But this is the first occasion on which one woman has succeeded another in the Presidency That there should be two women Presidents in sequence must be a significant event in the somewhat slow development of the recognition of women’s interest in mathematics. Moreover it is the first time that a woman who has had the ordinary responsibility of bringing up a family of children has also had the wonderful opportunity of continuing her mathematical work to the point of serving this Association as its President. This indeed makes me realise that mathematics is taking on a new role with a universal appeal.
An Early Nineteenth Century Arithmetic Exercise Book
Authors: Flett T. M.
Published: January 2007

School exercise books have usually a transitory existence. Even if we ourselves retain them as souvenirs of our schooldays—and we seldom do—then our descendants have no such sentimental regard for them, and consign them to the waste-paper basket when we depart this life. A school exercise book of 130 years ago is therefore remarkable for its longevity, as well as interesting for the light it throws on the teachings of that time.
Two Hexagonal Designs
Authors: Sharma Prakash Chandra
Published: January 2007

These designs were prompted by the “pursuit curves” of Mr. I. J. Good The first one consists of his six ‘hyperboloids’ and the second an equal number of ‘spearheads’ The former possesses what may be termed cyclical symmetry and the latter triple axial symmetry

There seems to be a technical improvement in these designs over those of Mr. Good’s insofar as the black areas at the centre of his triangular doodles are conspicuous by their absence over here.
A Slide-Rule Approximation to the Factorial Function of a Positive Fraction
Authors: Finucan H. M.
Published: January 2007

In a sense Stirling’s formula can be adapted to slide-rule computation of x! when a; is a fraction and tables are not at hand.
John Smith’s Problem
Authors: Chaundy T.W. , Bullard JE.
Published: January 2007

In the autumn of 1693 the Writing Master of Christ’s Hospital, one John Smith, asked Samuel Pepys the diarist, who was a Governor of Christ’s Hospital and eventually became its Vice-President, for the answer to the following question:

The Question.

A—has 6 dice in a box, with which he is to fling a 6.

B—has in another box 12 dice, with which he is to fling 2 sixes.

C—has in another box 18 dice, with which he is to fling 3 sixes.

Q—Whether B and C have not as easy a taske as A at even luck?
Antelope—An Amendment
Authors: Gillings A. M.
Published: January 2007

Correspondence on the ‘Antelope’ article has indicated that there is an error in the “Mr. Moneybags” argument on pages 234-235. The error lies in the translation of the argument from its English form to the corresponding Symbolic Logic form.
Down With the Mean
Authors: Clarke L. E.
Published: January 2007

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.
On the Advance of the Perihelion According to Special Relativity
Authors: Karlov L.
Published: January 2007

Undergraduate classes of physics and applied mathematics, in their studies of introductory relativity theory, are often presented with a special-relativistic calculation of the perihelion advance of planetary orbits. The result comes to be one-sixth of the gen-eral-relativistic value and serves as a demonstration of the inability of special relativity to account for the residual effect observed in the case of Mercury. As well, the calculation provides a rather pleasant example (perhaps more from the teacher’s viewpoint than the student’s) of two-dimensional relativistic particle dynamics.
Regular fault-free rectangles
Authors: Atkinson M. D. , Lunnon W. F.
Published: January 2007

Provided one of the integers m and n is even, it is obvious that an m × n rectangle can be covered with 1/2mn dominoes, each domino covering two unit squares. If some of the dominoes are oriented in each of the two possible ways, it is possible that there may be no continuous straight line of domino edges, either horizontal or vertical, across the rectangle: in other words, each of the (m — 1) + (n — 1) fault lines on the underlying grid is crossed by at least one domino. Such rectangles are called fault-free, and are mentioned by S. Golomb in [1] where the concept is attributed to R. I. Jewett. An example with m = 6, n = 8 is shown in Fig. 1.
Projects, problems and patience
Authors: Rymer N. W.
Published: January 2007

Project work has become a major constituent of Applied Mathematics courses at most British polytechnics and universities. There is a general consensus among academics as to what constitutes a project, how to discover and vet suitable projects, what their value is as a teaching aid, and how a student’s project work should be assessed.
Ambiguity in probability problems
Authors: Betteley I. G.
Published: January 2007

Of all mathematical problems, those concerned with probability seem to contain ambiguities more often than any others. In the majority of cases the reason is that too little thought has been given to the wording of the question, but occasionally the question sounds very sensible and it is only when a real attempt is made to solve the problem that difficulties in interpretation appear. I believe that this is mainly due to the loose way that the word ‘probability’ is used.
A combinatorial puzzle
Authors: Worboys Michael
Published: January 2007

A puzzle which introduces simple ideas relating to deduction within a mathematical system in a concrete and problem-oriented way is the following:

Find seven letters of the alphabet and seven three-letter words of the English language constructed from those letters such that each word contains three distinct letters and any two letters chosen from the seven letters belong to exactly one of the words.
A graph-colouring theorem
Authors: Walker R. C.
Published: January 2007

Imagine six drawing pins stuck in the wall. If we stretch threads, either red or blue, between every pair of pins, we inevitably create at least one triangle all of one colour. This we shall call Theorem 1: the proof follows.

Consider any pin. From this emanate five threads (since there are five other pins), of which three must be of one colour. We may without loss of generality assume that they are three red threads.
Sequences of averages
Authors: Gowers Jim , Davis Graham , Miles Elizabeth
Published: January 2007

A very simple way of generating a sequence is to take two starting values, a1 and a2, and construct a3 as their arithmetic mean, a4 as the mean of a2 and a3 and so on. This sequence always converges to the limit and seems to be fairly widely known as a topic for investigation in schools and colleges. It can immediately be generalised to the case of k starting values, and this forms a good subject for investigation using a micro-computer. It seems likely that such a simple set-up has arisen independently many times. One emergence occurred about ten years ago at a conference of the Association of Teachers of Mathematics, where the originator introduced it under the fictitious name ‘Littov sequences’. It has recently resurfaced at a Department of Education and Science conference.
The knight’s tour
Authors: Cannon Robert , Dolan Stan
Published: January 2007

The classical problem of the knight’s tour consists of moving a knight over a chess board in such a manner that it moves successively on to every possible square once and only once. If the initial and final squares of this tour are a knight’s move away from each other, then for obvious reasons the tour is termed re-entrant. The problem has a long and interesting history. Solutions due to De Moivre, Euler, Vandermonde, Warnsdorff and Roget, together with further references can be found in [1].
Quasilogarithms: an approach to the logarithm function
Authors: Bibby Neil
Published: January 2007

The decision by the GCE boards to ensure that all A-level Mathematics syllabuses contain a common core of pure mathematics is currently in course of implementation and near completion. In the wake of the syllabus changes a crop of new and revised A-level text books has appeared: many of these have a very familiar style and content, and have on the whole avoided any serious reappraisal of their subject matter. In particular most fail to exploit the calculator or microcomputer to any significant extent in the development of new concepts.
Pseudo-ordered fields
Authors: Csontos Peter , Morgan Christopher L. , Rebman Kenneth
Published: January 2007

A danger in teaching familiar material is oversimplification. In this article, we shall explore an example of this.

One of the authors observed a colleague (at another institution) introduce his students to the ordering of the real numbers by stating that < is a relation on ℝ which satisfies the following axiom system:

“For all x, y, z in ℝ:

1.

Exactly one of x = y, x < y, y < x holds.

2.

If x < y and 0 < z, then xz < yz.

3.

If x < y, then x + z < y + z.”
Counting Lines and Planes in generalised noughts and crosses
Authors: Taylor Paul
Published: January 2007

In the familiar [32] game of noughts and crosses or ticktacktoe, there are eight ‘winning’ Lines. These are the lines having the maximum number, 3, of Points in a 3 × 3 lattice, as in the figure.
Main mathematics in Colleges of Education
Authors: Flisher F. W.
Published: January 2007

Whatever may be the future of Colleges of Education after the re-organisation indicated in the White Paper Education: a framework for expansion, the great majority of teachers will continue to be trained in these colleges. In nursery, infant and junior schools most teachers are women, and the rate of turnover due to marriage is very high. In recent years, in order to staff the schools, allowing for this ‘wastage’, nearly one-third of the teaching profession has had to be in training at any given time, and rather more than two-thirds of these have been women. Clearly there are not enough young people leaving school with good academic qualifications to meet the needs of the profession, even if they wished to teach. In general, the more academically able choose not to teach, but rather think in terms of a career which is more remunerative and a Higher Education which is allegedly more ‘respectable’.
Some derivations from the marriage bureau problem
Authors: Brissenden T. H. F.
Published: January 2007

The following notes arose from a talk given by Professor Halberstam to the South West Wales branch of the Association. In the marriage bureau problem we imagine that n boys and n girls are to be paired off in marriage, that we are able to establish preference lists for each boy of all the girls and vice versa. These preference lists may be displayed in a table such as that given below, which shows the rankings for four boys α, β, γ and δ, and for four girls A, B, C, and D. The boys’ rankings are the first members of the number pairs, in rows, and the girls’ are the second members, in columns.
A plea for simpler theorems
Authors: Fletcher Colin R.
Published: January 2007

The best mathematicians, from the days of Euclid and before, have defined the terms they use. Now a definition may be a complete fraud, and Euclid was a case in that which hath no part; although to be fair to our greatest text book writer, I am sure he knew a fraud when he perpetrated one. On the other hand, a definition may be a perfect hidden gem. That mighty man Eudoxus was able to define what was meant by the equality of r1 and r2, without knowing what r1 and r2 were. A lesser man would have thought that impossible, but Eudoxus was not a lesser man, by definition. Euclid and Eudoxus, those great definers, would be proud of much of what we now do in the way of definition. There are still frauds, such as definition by example, so beloved of biologists and new mathematicians. (By the latter we mean those persons who consider their business to be new maths.) And there is that modern phenomenon, the railway definition, where properties are hitched together in series just like the carriages of a train, to create one monster definition, which, on the face of it, and sometimes not only superficially, has no justification and no relevance.
The historical development of complex numbers
Authors: Green D. R.
Published: January 2007

The Alexandrian Greek mathematician Heron—whom we associate with the formula √{s(s − a)(s − b)(s − c)} for the area of a triangle—got involved in a calculation about a pyramid design which led to the evaluation of √(81 − 144). This occurs in his book Stereometria (C. A.D. 75) and to ‘solve’ it the numbers are turned round thus: √(144 − 81), to give √63 which is taken to be . (Is this a reasonable approximation for √63?) It is not known whether Heron made this transpositional error or whether a copier of his work was responsible. This seems to be the first occasion in which the square root of a negative number was stumbled across—a concept not properly understood for another 1750 years!
Elastic impacts at a circular boundary
Authors: Chorlton F.
Published: January 2007

A particle is free to move on a smooth horizontal circular table having a circular insurmountable barrier Γ. The particle is projected from a point P of the table and it subsequently suffers reflections at the points P1, P2, P3, … on Γ due to elastic impact. Investigations are made of the possibility of its passing again through P at some stage.
Finite groups, wallpaper patterns and non-euclidean geometries
Authors: Beardon A. F.
Published: January 2007

Transformation geometry is now taught in many schools, and complex numbers are frequently used to describe the geometric transformations. For example, a translation is represented by (where b is a complex number), a rotation by a magnification by and a reflection in the x-axis by

The set of all transformations of the form forms a group and consists of precisely those transformations which preserve distance. Among the subgroups of this group there are the groups which give rise to the well known 17 distinct ‘wallpaper patterns’. These can be described in an elementary and geometric way and are (quite rightly) a popular topic in transformation geometry.
Mathematical modelling in the classroom
Authors: Nelson R. D.
Published: January 2007

A teacher can make a great deal out of some of the questions which pupils and friends put to him from time to time. In an article in Mathematics in School a few years ago [1] I listed a number of such questions and discussed one, which involved a geographical application of matrices, in detail. The present article continues this theme, but its main aim is to stimulate discussion of the place of “mathematical modelling” in the curriculum. For it is felt by some that, in spite of the opportunities offered by (say) linear programming, mechanics and probability, insufficient emphasis is placed on modelling in the secondary school. It has been said too that we have not been notably successful in finding good applications of topics new to our syllabuses and incorporating them into our teaching [2].
What is Geometry? The 1982 Presidential Address
Authors: Atiyah Michael
Published: January 2007

Of all the changes that have taken place in the mathematical curriculum, both in schools and universities, nothing is more striking than the decline in the central role of geometry. Euclidean geometry, together with the allied subject of projective geometry, has been dethroned and in some places almost banished from the scene. While educational reform was certainly needed there is always the danger that the pendulum may swing too far the other way and that insufficient attention may be paid to geometry in its various forms. Much of the difficulty here centres round the elusive nature of the subject: What is geometry? I would like to examine this question in a very general way in the hope that this may clarify the educational reasons for teaching geometry, and for deciding what is appropriate material at different levels.
Limits of Interest
Authors: Fletcher Colin R.
Published: January 2007

Interest rates are playing a large role in many people’s lives these days. The creation of a property owning democracy is part of the reason; another lies in their rapid increase. Yet in spite of this phenomenon the actual method of calculation of interest is almost wholly ignored by the mass of the population, and the fluctuations in the rate are looked upon in qualitative terms. In fact the formula for simple interest is easy to write down and, from this, compound interest can be deduced in a straight-forward fashion. That is, provided you remember what compound interest is. I am short enough of breath to have been at Grammar School when O-level Mathematics was divided into three parts, Arithmetic Algebra and Geometry; and the most boring of these was Arithmetic. So boring indeed thatcompound interest, stocks and shares and other useful tricks were retained until the examination and then promptly forgotten.
Does it matter?
Authors: Shuard H. B.
Published: January 2007

When I started teaching real analysis to first-year college of education students, I thought that at least I ought to be consistent in what I said. When I told the students what a word meant, I thought they ought to hear the same story as they heard from my colleagues who were teaching other branches of mathematics. I even thought I ought to tell them something like what they would find in a consensus of the real analysis books they might look at. I knew that I ought not to use different meanings for the same word on different occasions. Alas . . .
A square root algorithm
Authors: Planitz Max
Published: January 2007

Although the approximation of elementary functions is a well-researched area of mathematics, very little has been published on built-in routines in any particular calculator or computer. Computer manufacturers are, understandably perhaps, reluctant to publicise details about their algorithms. Hewlett-Packard have been less secretive than most and allowed an occasional glimpse behind the scene in their own Journal and various other sources. The following square root routine for one of their machines, the now extinct 2000F series, has appeared in an Open University text on numerical computation [1]. The process of evaluating √x is carried out in four steps:
The Fundamental Formula in the Algebra of Sets
Authors: Goodstein R. L.
Published: January 2007

The algebra of sets admits a quite elementary presentation in the form of an axiomatic system, which may well be suitable for sixth form school teaching. In this note, however, I am concerned not with axiomatics but with the development of set theory in terms of the membership relation; or to be more precise I am concerned with that part of set theory which deals only with the Boolean operations and the relations of equality and inclusion, but does not seek to generate sets.
Integer-sided triangles with an angle of 120°
Authors: Selkirk Keith
Published: January 2007

John Gilder’s article on integer-sided triangles in the December 1982 Gazette [1] immediately suggests the question of what other triangles with integer sides have an angle which is a whole number of degrees. This is very simple to answer, for if the sides are integers, the cosine formula implies that the cosines of all three angles must be rational. Only angles of 60°, 90° and 120° fulfil this condition for the appropriate range of values, and it follows that, apart from the familiar right-angled triangle and Gilder’s case of the 60° angled triangle, the only other possible case is a triangle with an angle of 120°.
Uniform distributions and random triangles
Authors: Griffiths David
Published: January 2007

“What is the probability that a randomly drawn triangle is acute-angled?” This question has been discussed in recent Gazettes (for example by Trevor Easingwood in [2], followed up by Stephen Ainley in [1]). As Easingwood pointed out, the answer to this question depends on what properties of triangles are assumed to be random and on what one means by random. And as Ainley pointed out some of these assumptions can lead to a contradiction. A statistical model is necessary to define the sense in which the chosen properties (lengths of sides, angles, co-ordinates of vertices,…) are assumed random. This model may (and preferably will) relate to a possible physical ‘random’ mechanism for generating the random triangle.
Repaying a loan by instalments
Authors: Puritz C. W.
Published: January 2007

A good deal of interesting mathematics at various levels arises from the situation where a loan is repaid by regular equal instalments, with interest being charged on the balance outstanding at each stage. The loan may be a mortgage, in which case the instalments are reckoned as annual (even though they are actually paid monthly), or a hire purchase debt or a personal loan of some sort with interest reckoned monthly. The following topics are in roughly increasing order of difficulty.
Classifying triangles and quadrilaterals
Authors: Robertson S. A.
Published: January 2007

Among the definitions at the beginning of Book I in Euclid’s Elements [1] there are several that pick out special kinds of triangles and quadrilaterals. In his commentary [2] on Book I, Proclus observes that Euclid classifies triangles in two ways: firstly ‘by sides’ into equilateral, isosceles and scalene triangles; and secondly ‘by angles’ into right-angled, obtuse-angled and acute-angled triangles. With regard to quadrilaterals, Proclus ([2], p. 134) attributes to Posidonius the classification scheme on p. 39, which is to be found in Heath’s edition of the Elements ([1], p. 189). Thus the ancient classification of triangles and quadrilaterals produces three (or six) species of triangles and seven species of quadrilaterals.
Continued radicals
Authors: Allen Edward J.
Published: January 2007

Expressions like

are known as continued radicals and they occur frequently in the mathematical literature, if only because of their orderly structure and their close links with continued fractions. But it’s worth bringing them out again here because they can make very interesting classroom exercises on sequences, and they lead to unusual ways of expressing π and the golden ratio, etc.
The regularity of randomness
Authors: Taylor S. J.
Published: January 2007

My title seems self contradictory, for surely random happenings are essentially unpredictable—so how can they be regular? And yet this apparent contradiction is the key which justifies the mathematical model of probability theory. For the regularity is the basic reason for our model providing a new mathematical tool with powerful applications to analysis, potential theory, number theory, quantum theory, statistical mechanics....

Let us start by asking, “What is probability?” What do I mean if I say “When you toss a coin there is a probability ½ that it turns up a head” or “When you pick a card from a well shuffled pack there is probability that it will be an ace”?
Investigating A Non-Euclidean Geometry
Authors: Cannizzaro Lucilla , Carosi Mauro
Published: January 2007

In this article we describe how a group of third year mathematics students of the University of Rome who were following a mathematics education option explored some of the properties of hyperbolic plane geometry before preparing a unit on non-Euclidean geometry to be used with sixth-formers at the Liceo Virgilio, Rome. We recall here that the Mathematical Association considered the possibility of such work in its 1923 report on geometry [1]. Nowadays the report is associated with the three ‘stages’ of geometry teaching, A, B and C, but in fact it proposed two further stages. The fifth stage included non-Euclidean geometry both ‘for a few gifted specialists’ and also because it might interest ‘able boys who are not mathematical specialists’.
An elementary method of calculating π
Authors: Puritz C. W.
Published: January 2007

Boys who have learned to find π using jam jars and string are often mystified to learn that the number has been calculated to thousands of decimal places: the infinite series and products usually quoted cannot be justified without analysis.

This article is about a method whose derivation involves only elementary algebra and trigonometry (including square roots and Pythagoras), and whose use is a feasible proposition even when only four-figure tables are available for the square roots.
Triangular Square Numbers: a Postscript
Authors: D. A. Q.
Published: January 2007

The article by Mr. D. C. D. Potter in Gazette LVI, No. 396 (May 1972), pp. 109-10 attracted a number of interesting letters. We cannot print all of these in full, but the main points are included in the following extracts.

First, Dr. K. Szymiczek of the Silesian University, Katowice, Poland (writing from Cambridge) offered some information on the history of the problem. “According to L. E. Dickson, L. Euler was the first to show that there are infinitely many square triangular numbers and S. Roberts proved in 1879 that Euler's formula gives all square triangular numbers ([1], pp. 13, 27).”
An introduction to the programmable calculator
Authors: Watson Hunter
Published: January 2007

Fifteen years ago one could purchase a hand operated calculator which, by much lever pulling and handle twirling, performed the four basic arithmetical operations. Such a calculator cost around £40. Today electronic programmable calculators can be bought for prices which are less, in real terms, than that of the old mechanical calculator. As will be illustrated by examples later, such calculators incorporate the major features of a full-scale computer, and by constructing programs for a hand-held electronic calculator one can learn the basic tactics and strategies of computer programming. There are definite advantages in starting programming with a hand-held calculator; it is inexpensive, much of the ‘language’ will be familiar because hand-held calculators are now so commonly used, and one can test and use programs immediately they are written.
Pyramids and pavements: some thoughts from Cairo
Authors: Macmillan R. H.
Published: January 2007

On a recent visit to Cairo I was struck by two matters which may be of interest to other members and their pupils.
Power versus exponential
Authors: Shiu P.
Published: January 2007

Let k be a positive integer and a > 1. It is well known that

.

The shortest proof seems to be the following. By estimating the exponential function with only one term from its series we see at once that

.
Two prime centenaries – a number theorem and a miscarriage of justice
Authors: Fletcher Colin R.
Published: January 2007

The prime numbers are often called the building blocks of number theory, a classic case of a ‘sine qua non’. If the corpus of the theory of numbers is looked upon as an architectural pile then the primes will be found amongst its foundations, and amongst its walls and buttresses, and indeed amongst the array of pinnacles and turrets which burst forth from it and stand proud, alone and magnificent. One such pinnacle had been discussed and planned for a hundred years before it was finally constructed in 1896 by two men working totally independently of each other (collapse of stout analogy). With this achievement, the prime numbers, those familiar yet raw beasts of mathematics, in one sense had been tamed. The long journey, which had begun with the tentative definitions of the ancient Greeks, was completed.
Some properties of Brocard points
Authors: Shail Ron
Published: January 2007

The Brocard points P and P' of the triangle ABC, illustrated in Figure 1, are such that ∠PAB = ∠PBC = ∠PCA = ω and ∠P'BA = ∠P'CB = ∠P'AC = ω. Problem 78.A in the March 1994 edition of the Mathematical Gazette required a proof that the maximum length of PP' is R/2, where R is the circumradius of triangle ABC. The solution given by various solvers in the November 1994 edition of the Gazette is essentially the same as appeared in a paper by H. E. Piggott in the Gazette in 1924. In the present paper various properties of the triangle and its Brocard points are established in terms of the lengths of the sides of the triangle. Some of the results may be new, but the methods used encompass ‘Mathematics Ancient and Modern’ in the guises of homogeneous areal coordinates and computer algebra!
Boole’s mathematical blindness
Authors: Deakin Michael A. B.
Published: January 2007

George Boole, who very clearly rates as a mathematician of the very first rank, set out to solve a mathematical problem and his efforts to do so continued throughout the whole of his professional life. He never succeeded, despite the fact that he held the key to the solution for a good deal of this time. How and why he came to fail in his attempt is the subject of this paper. The difficulties encountered by students may be better understood if we realise that even the very greatest of mathematicians experience the same limitations and that they arise from the same causes.
Vectors, centres and a touching theorem
Authors: Howlett Graham
Published: January 2007

Although it does not often feature in today’s syllabuses, the study of the incentre and circumcentre of a triangle can be a useful source of material for showing the applications of vector methods to geometry. This idea has featured in some recent Gazettes, but in this article I illustrate an alternative and essentially simpler vector approach.
Moving the first digit of a positive integer to the last
Authors: Braza Peter A. , Tong Jingcheng
Published: January 2007

Arithmetical properties of numbers have been a source of fascination for mathematics students and professionals throughout the ages. In this paper we consider a puzzle of sorts in which we move the first digit of a positive integer A to the last position and establish conditions under which A will be equal to a multiple of the new integer. For instance, if A equals 820512 then A = 4A’where A’ = 205128.
A simple proof of the Beijing theorem
Authors: Deakin Michael A.B.
Published: January 2007

If a square pyramid is cut obliquely by a plane as shown in figure 1, a pentagonal cross-section PQSTR is generated. The “Beijing theorem”
states that the only case in which this pentagon can be regular is that in which the pyramid is half an octahedron and (except for a complication to be discussed later) the coordinates of the points P, Q, R, S and T are uniquely determined.
Singular matrices applied to 3 × 3 magic squares
Authors: Gauthier N.
Published: January 2007

If you enjoy magic squares and their unusual and fascinating numerical properties, you are in good company. Indeed, many a professional mathematician, scientist and amateur has shared the same interest. Technical research work on the properties of magic squares is available aplenty in the literature or in books concerned with the recreational aspects of mathematics. Studies of magic squares naturally lead to some elements of group theory, of lattices, of Latin squares, of partitions, of matrices, of determinants, and so on …

By definition, the general n x n magic square is a square array of n2 numbers {aij; i, j = 1,2,... , n) whose n rows, n columns, and two main diagonals have the same sum s. It is therefore natural to think of a magic square as a matrix and we shall do this in the present note, simply referring to it as a magic matrix.
The harmonic hurdler
Authors: Crilly Tony
Published: January 2007

The harmonic series seems to be a topic of enduring mathematical interest. Readers will be aware of the many neat methods which have appeared in the Gazette for showing its divergence but let us here investigate one of its discrete properties.
Games—a look at strategies
Authors: Robin Anthony C.
Published: January 2007

The word “game” includes a variety of human activities, some like football or cricket, involve physical strength and skill, others like bridge, involve luck and skill, and some involve only mental skill, the best example perhaps being chess. This is the type of game which we are going to look at in this article, although to analyse a game as complex as chess by the methods below, (although theoretically possible) would be completely impractical—even with computer help.
On square-rooting matrices
Authors: Scott Nigel H.
Published: January 2007

Following on from Nick MacKinnon’s work in the June 1989 Gazette concerning four different methods of square-rooting 2 × 2 matrices, we may ask if it is possible to compute all the square roots of any given 2 × 2 matrix, and if so, how? From now on by a “matrix” we mean a real or complex 2 ×2 matrix. We know that every non-zero number has precisely two square roots, but the situation is rather more complicated for matrices. In fact we shall see that the zero matrix and any other multiple of the identity matrix has an infinite number of square roots. All other matrices have only a finite number of square roots, with some (rather surprisingly) having none at all! Our method of enquiry is based on Nick MacKinnon's first method, namely matrix diagonalisation.
The graph of xx: An A-level investigation
Authors: Boardman Sam , Harrison Chris , Webster Terry
Published: January 2007

With the arrival of calculators having graphical displays, students at Alevel have been able to investigate the graphs of various functions where previously the analysis involved would have prevented them from doing so. Unfortunately (or fortunately?) the technology on most hand-held calculators and even on many graph-plotting packages for standard micros is not capable of handling all functions which A-level candidates might like to investigate.
Mission possible: Computers in chess and A-level mathematics
Authors: Hope Linda
Published: January 2007

Twenty years ago one of the plots in the TV series Mission Impossible involved a chess tournament in which Leonard Nimoy (without the pointy ears, but in every other respect identical to his Star Trek persona) won a game by obeying the instructions of a computer relayed over an earphone. While many of the antics of the Impossibles remain … impossible, it is only be a matter of time until chess computers can defeat every human being on the planet. However a chess positioninvented by David Norwood [Daily Telegraph, Weekend p.28, 7 Aug 1993] shows that the game may not yet be over for homo sapiens. The position in figure 1 could occur in actual play, but is most unlikely to.
A-level mathematics: where next?
Authors: Bibby Neil
Published: January 2007

Since the introduction in 1978 of the idea of a “common-core” for A-level mathematics, there was been a continuing debate over the structure and content of the subject at this level. Much of this debate remains pertinent in 1989: see, for example, [10], [15] and [28]. However, the setting-up of the Higginson Committee to review A-levels in March 1987 had the effect of putting much of the discussion specific to mathematics on the back-boiler. It is well known that the committee’s report of May 1988 recommended a wider subject-spread, roughly speaking on the model of the International Baccalaureate, and that its immediate rejection by the government created considerable furore in educational circles.
The closed-form integration of arbitrary functions
Authors: Fitt A.D. , Hoare G.T.Q.
Published: January 2007

Consider the usual experience of a student who progresses far enough in school mathematics to begin to study the calculus. After motivation of the topic of “rates of change”, simple differentiation of polynomials is learned. More advanced functions are then considered, and eventually the student meets the product, quotient and chain rules. The result: with enough algebraic accuracy and persistence, the student can determine derived functions for virtually any sufficiently wellbehaved combination of standard functions.
Mathematical people: Cheryl Maiden
Authors: Maiden Cheryl
Published: January 2007

Cheryl Maiden is director of the Maths 100 Foundation, which is based in Henley-on-Thames.

The day begins before 9.00 a.m. and it’s a relaxed time opening mail over breakfast, punctuated by the occasional call. This morning it’s a query from the printers about the paper for Part 3, and the Yellow Pages again. I enjoy the post as it has become increasingly varied, bringing interesting new people into the Maths 100 project.
Filling holes in the real line
Authors: Burn Bob
Published: January 2007

Is there a largest number on your calculator? There is on mine; it is 9·9999999 x 1099. It is larger than any number I have ever wanted to calculate, though it puts some known Mersenne primes out of reach. An upper boundary to the numbers with which one can calculate gives one striking difference between our imagined number line and the numbers on a calculator.

A more serious difference between the numbers on a calculator and the real number line is the existence on the calculator of a smallest number, greater than 0. On my calculator, the smallest positive number is 10-99. So, if I imagine the numbers my calculator will display marked on a number line, and zoom in on the number line to magnify it, the calculator numbers will appear spaced out, rather like integers on a conventional number line. We will see shortly that the calculator numbers are not evenly spread out, but the existence of gaps on the line, in any form, is critical. The calculator numbers are not dense on the line.
An example of chaotic dynamics
Authors: Johnson R. C.
Published: January 2007

The article describes an accessible example of dynamical chaos which has been discussed recently in the Journal of Physics. It is simple enough to be explored by microcomputer at home or school. It concerns a point “billiard ball” bouncing round an ideal square “billiard table” that rotates steadily. Without rotation the ball's motion would be a perfectly regular sequence of straight-line transits; with rotation most of the motion is unpredictably jumbled—chaotic, as we'll see. First some simple theory is needed.
A beautiful new playground – further investigations into i to the power i
Authors: Parker Greg , Roberts Steve
Published: January 2007

In an earlier note the iterative behaviour of taking the unit imaginary, i, to higher and higher powers of i was described. This iterative power sequence converged to a limit. In order to make discussion of later developments simpler we need to introduce some definitions.