bluuebunny/crossref_metadata_2025 huggingface dataset:
Co-descriptive strings
Authors: Sallows Lee , Eijkhout Victor L.
Published: January 2007

What sentence in this Mathematical Gazette contains seven a’s, four c’s, thirty-two e’s, eight f’s, five g’s, ten h’s, twelve i’s, three l’s, three m’s, fifteen n’s, eight o’s, one q, seven r’s, twenty-seven s’s, twenty-five t’s, four u’s, eight v’s, seven w’s, four y’s & two z’s?
A-level mathematics problems
Authors: Hersee John
Published: January 2007

A-level courses and examinations are with us for some time to come. Even if Mevel is introduced, for most pupils who spend two years in the I sixth form A-levels will remain the basis of their studies. It seems to me that we need a major debate about A-level mathematics; we need to review it as it is and try to decide what developments are necessary. The discussion I paper ‘Mathematics at A-level’ has already raised a number of points, some of which are touched on here. This article is not a statement of official SMP policy; any opinions expressed in it are my own. The aim is to provoke discussion. I shall consider a number of issues which seem to me to be important, hoping that others will be stimulated to contribute comments and suggest other topics which need attention.
The mathematical boarding schools of the USSR
Authors: Owen A. , Watson F. R.
Published: January 2007

During the spring of 1973 we were able to visit the Soviet Union for 14 days to study mathematical education. Our visit was arranged by the British Council under the Anglo/Soviet cultural agreement. One of the most interesting features of the Soviet system which we hoped to study during our short tour was the specialised boarding school. Accounts of some of these have appeared already (for example, Bryan Thwaites’ article in the Gazette for December 1968) and it was intersting to compare our impressions with those which had been recorded by others.
The place of geometry in a mathematical education
Authors: Armitage J. V.
Published: January 2007

It is a commonplace that pictures leave a deeper impress on the human mind than abstractions and, in spite of its abstract nature, that is also true of many part? of mathematics. A proof of a theorem or the construction of a counter-example is often suggested by pictorial considerations, and that is true of algebra and analysis as well as of obviously geometrical subjects like topology. (For entertaining examples see Professor J. E. Littlewood’s book [1].) There is obviously a connection between a developed geometrical intuition of that kind and the kind of spatial experience which ought to be furnished by a school mathematics course. No one would deny its importance and the writers of textbooks on ‘modern’ mathematics are well aware of the need to stimulate geometrical intuition. What has been called in question is the relation between that kind of geometrical intuition and the formal treatment of the euclidean plane based, albeit insecurely, on Euclid’s Elements. Indeed, the traditional treatment has been abandoned with indecent haste and without serious question. In Part I of this paper we make a plea for a re-consideration of the case for some kind of formal geometry as an indispensable part of the education of the more mathematically able children.
Knots—a practical application of group theory
Authors: Baylis E. R.
Published: January 2007

The definition of a group normally used in the classroom is that of a set of a finite or an infinite number of elements, for which an operation is defined, subject to the four conditions of closure, associativity, identity and inverse. The group is usually specified by giving its group table. However, this is not very practical for a finite group of large order and is never possible for an infinite group. Also, the group table often contains redundant information and so is not very efficient.
Non-associative algebraic structures arising in genetics
Authors: Towers David
Published: January 2007

One of the difficulties encountered by students when they are first introduced to axiomatic algebra is that the axioms, such as commutativity, associativity, and distributivity, seem self-evident in the algebraic situations with which they are familiar. Indeed, the axioms are so useful, of course, because of their wide applicability. However, it can be instructive to study situations in which these axioms break down, particularly if the situations are not too contrived.
Who’s afraid of the measuring numbers?
Authors: Cable John
Published: January 2007

The term measuring number is acquiring currency as a popular synonym for non-negative real number. This is good. Yet the belief persists that the real numbers, by whatever name, must remain fundamentally inaccessible to school children, obscured behind a veil of Dedekind cuts or Cauchy sequences. (Thus Griffiths and Howson, in what is generally a humane book [1], devote several pages to an ingenious, but ultimately condescending, attempt to foreshadow the mysteries of real numbers in terms of points of a line.)
A circular vector space
Authors: Crilly Tony
Published: January 2007

This note describes an example which I have found useful for introducing ideas in linear algebra and group theory. It is simple and yet it is neither trivial nor too abstract.
The distribution of l.s.d. and its implications for computer design
Authors: Turner Peter R.
Published: January 2007

I was interested to see in a recent issue of the Gazette a note from Hoare and Wright [1] about the distribution of leading significant digits, l.s.d., and “Benford’s Law” that this distribution is logarithmic. This distribution has fascinated many people since Benford’s original observation [2] in 1938 of the dirty pages at the beginning of a book of log tables.
Symmetries of surfaces
Authors: Kosniowski Czes
Published: January 2007

Most school mathematics courses nowadays include a study of symmetry in the euclidean sense—isometries in 2- or 3-dimensional euclidean space which map a figure to itself. This article explores the theme of symmetry in a more general, topological sense. It could form the basis of a short self-contained unit in topology with a worthwhile end-point, which teachers might care to try out with students at sixth form level or perhaps in a college of education course.
A modelling approach to traditional applied mathematics
Authors: d’Inverno R. A. , McLone R. R.
Published: January 2007

There seems little doubt that the proportion of the mathematics sixth form who include traditional applied mathematics as part of their studies is steadily decreasing, and that this decline in its turn affects both the popularity of, and the ability of students to cope with, applied mathematics in universities. It is evident that fewer sixth formers are taking the ‘double mathematics plus physics’ route to reading mathematics at degree level; the combinations taking its place are many and varied, but often include statistics, for example, instead of applied mathematics, and economics or geography or a language instead of physics. Even where some students take so-called double mathematics, the syllabus followed often gives statistics as an alternative to applied mathematics, and an increasing number of schools are teaching this option.
Introducing probability generating functions
Authors: Ling J. F.
Published: January 2007

The idea of a probability generating function seems to be a difficult one for sixth formers, and this may well be because the formalism is introduced too abruptly. The t or z or whatever letter is used in a p.g.f. has to be explained as having “no physical significance”, to quote from a recent M.A. booklet on sixth form probability and statistics. The following gentler approach I have found to lead to better understanding on the part of ‘lower fliers’, and it has the advantage of drawing attention to an underlying structure.
How many bingo cards are there?
Authors: Lloyd E. Keith
Published: January 2007

In his article on bingo in the June 1985 Gazette, M. G. Sykes makes the implicit assumption that the fifteen numbers on a bingo card may be chosen from the set {1, 2,…, 90} without restriction. This is not correct—there are severe constraints on the choices. He also assumes in his calculations that the cards are printed at random. Whether this is ever the case I do not know, but the bingo cards produced for various tabloid newspapers in recent years are produced in a highly systematic manner, some details of which will be given in this article.
Forbidden marriages from a woman’s angle
Authors: Cashdan J. , Stern M. D.
Published: January 2007

A persistent problem of the mathematics teacher is the motivation of students especially those not particularly interested in numerical work. In this paper we present a notational device for analysing relationships based on binary notation and show its application to the problem of incest. The authors developed this theory in the course of their study of the rabbinic law on this subject and present this application here, though it could be applied to any legal system. It is hoped that the use of this approach may be useful to introduce mathematical modes of thought to previously uninterested pupils.
A locus problem
Authors: Grant M. A.
Published: January 2007

This article outlines a simple locus problem which leads to some interesting conclusions. It enables three elementary loci to be seen as special cases of a more general locus.

The problem might present a sixth-former with a challenging, open-ended investigation enabling him or her to practise some newly-acquired techniques in seeking the answer to an apparently simple question.
Tile factory
Authors: Crilly Tony
Published: January 2007

In The bequest of the Greeks Tobias Dantzig illustrates the Euclidean division algorithm using squares and rectangles. Although he dwells on the continued fraction expansion for a/b, where a and b are relatively prime, it is only a small step to use the Greek imagery of representing numbers by geometric figures for illustrating the algorithm in finding gcd(a,b), the greatest common divisor (or highest common factor) of two numbers a and b. Indeed, this geometric line of thinking leads to an easily programmable algorithm for directly finding lcm(a,b), the least common multiple of a and b, without first finding gcd(a,b).
61.14 A regular pentagon construction
Authors: Nelson David
Published: January 2007

An artist friend, Mr Graham Day of Corsham, Wiltshire, recently sent me the following, asking whether he had accidentally discovered a ruler-and-compass construction for a regular pentagon. His own main interest in the configuration stemmed from a similarity to one in Indian mystical art.
61.16 The vector triple product
Authors: Pamphilon J. A.
Published: January 2007

Let π be the plane spanned by the vectors b and c. The product vector b × c is normal to this plane and hence a × (b × c), being perpendicular to this vector, will be in π.
Two surprising limits revisited—(1)
Authors: Webb J. H.
Published: January 2007

The ‘Two surprising limits’ (note 66.31, October 1982) described by William A. Leonard and Harris S. Schultz are in fact special cases of two general limit problems which provide interesting applications of Taylor’s theorem and L’Hôpital’s rule.
Notes: 61.22 Pythagoras in three dimensions
Authors: Bunday B. D.
Published: January 2007

It appears not to be widely known that there is an analogy of the theorem of Pythagoras for a ‘right-angled tetrahedron’, i.e. a tetrahedron which can be formed by slicing off a vertex from a cuboid. The result is: the square of the area of the face opposite the vertex containing the three 90° dihedral angles is equal to the sum of the squares of the areas of the other three faces.
Graphs and Markov chains
Authors: Greenman J. V.
Published: January 2007

In its simplest form the theory of Markov chains states that, if the (i,j) element of matrix M is the probability pij of a one-step transition from state i to state j, then the (ij) element of the matrix Mk gives the probability of transition from state i to state j in k steps. Since the probabilities of transition to the different states add to 1, the sum of the elements in any row of M is necessarily equal to 1. (Readers are warned that in some elementary treatments the alternative convention is adopted of writing pij in the (j,i) position, so that the column sums are 1, in order that the probability vectors should be column matrices, which are more familiar in school mathematics.)
61.3 Another proof of the “φ sequence” property
Authors: Jackson T. H.
Published: January 2007

The property of “φ sequences” which D. A. Barwick describes in Note 60.31 was proved by reference to the theory of Farey sequences. When the author first showed me the sequences I set about trying to obtain a proof from first principles. This note is the result of that investigation.
Vectors and the geometry of a triangle
Authors: Chatwin Philip
Published: January 2007

MacNab [1] has recently considered the Euler line of a triangle, and derived expressions for the position vectors of certain important points like the orthocentre, the circumcentre and the incentre. In this note I show how these results, and some other ones, can be derived by a more direct use of vector methods than MacNab employed. I believe that such applications provide excellent material for use in a first course on vectors since only elementary properties are needed. Nevertheless, the power of vector methods is well illustrated and, in addition, the results are interesting and the symmetry of the algebra is very appealing. While I suspect that much of the material is not original–for example some of the results are in the well-known book by Weatherburn [2]–I am not aware of the existence of a single concise treatment, and I believe this has potential value.
Generating and enumerating magic squares
Authors: Fitzpatrick Patrick
Published: January 2007

Magic squares have long provided a fruitful source of classroom examples and stimulating exercises in recreational mathematics. Attention is often devoted to the so-called normal magic squares of a given order n in which the entries are the integers from 1 to n2. In these the size must be at least 4 for examples to be sufficiently plentiful: there is essentially only one normal magic square of order 3 (apart from rotations and reflections), while there are 880 of order 4 and several millions of order 5. However, if we drop the requirement of normality, the 3 × 3 square becomes a surprisingly rich subject for investigation, which can challenge pupils at all levels. In this note we discuss the following problems: (1) find a formula to generate all 3 × 3 magic squares with a given magic number m (the sum of entries in any row, column or diagonal), with no restriction on the entries—they can be any real numbers; (2) find conditions under which the square has all entries positive integers; (3) determine how many essentially distinct positive integer squares there are for a given m.
Modelling in the home
Authors: Ellis L. E.
Published: January 2007

My family enjoys doing large jigsaw puzzles. We find a large table, lay out all the pieces face upwards, with the edge pieces separate, and build up the outside first. Then we usually try to follow obvious boundaries—between sky and trees, or shore and water, for example, though sometimes a single brightly coloured object catches our eyes and gets built up in isolation.
The independence of “Huntington’s axioms” for boolean algebra
Authors: Gerrish F.
Published: January 2007

Among the various logically equivalent sets of axioms for boolean algebra ([1] discusses four of them), that referred to in the title of this article is popular because of its symmetry (self-duality) and convenience in application. The textbook presentation is essentially as follows; the qualifiers ‘left’, ‘right’ are often omitted, although the symbolic version indicates that they are intended.
The limitations of group multiplication tables: Group structure through extension theory
Authors: Liebeck Hans
Published: January 2007

In Mathematical note 3356 (Gazette No. 405, October 1974) George Berzsenyi produces new groups from old by redefining multiplication in an appropriate fashion. Though the group axioms are easily checked relative to the new multiplication, the structures of the new groups are not immediately clear. This is a common situation when groups are presented by their multiplication tables which, in the present author’s view, are of little use. They do not give a clear picture of how the groups ‘work’, as anyone who has tried to verify the associative law from a table knows to his cost. Indeed, a multiplication table of a group is usually constructed only after its structure has been worked out by some other means. Take for example the non-commutative group of order 6. Much more informative than a multiplication table are its abstract presentation in terms of generators and defining relations, and its concrete realisations as the group of all permutations on three symbols and the group of isometries of the plane that map an equilateral triangle onto itself.
Some thoughts on the use of computer symbols in mathematics
Authors: Brown Margaret
Published: January 2007

The increasing use of simple computing languages in schools gives an opportunity for rationalisation of our existing mathematical notation as follows:

It is time we abandoned “÷”, which is used in primary schools and causes great difficulties later on when children have to cope with the fact that or, worse still, If we used “|” from the start, what could be easier than (Nor is there a case for restricting ÷ to ‘sharing’ situations, since + and − are used for both natural numbers and rationals.)
A 24-Point Sphere for the Orthocentric Tetrahedron
Authors: O’Hara F.
Published: January 2007

Readers familiar with the nine-point circle property of a triangle (Figure 1) may have wondered whether there is a similar result for a tetrahedron.

In general the answer is “No”: there is no equivalent of the orthocentre for a tetrahedron, since the altitudes from the vertices onto the opposite faces do not intersect. There is, however, a special kind of tetrahedron for which these four lines do have a point H in common.
Linear Recurrences and Maximal Length Sequences
Authors: Laxton R. R. , Anderson J. A.
Published: January 2007

In this article, we are going to be looking at certain finite sequences composed of zeros and ones.
A visual approach to some elementary number theory
Authors: Bruckheimer Maxim , Arcavi Abraham
Published: January 2007

Euclidean geometry and fractions not only make unlikely bedfellows, but would seem to be a sure recipe for boredom. However, in this paper, with the help of a little history, we hope to prove the opposite.

Given a (rectangular) lattice, then a lattice polygon is a polygon whose vertices are lattice points. Pick's theorem [1] - the area of a simple lattice polygon is given by ½b + i – 1 where b is the number of lattice points on the boundary of the polygon, and i is the number in its interior – is well known and has been proved many times and in many ways.
The number of colourings of a polyhedron and the effect of indirect symmetries
Authors: Watson Rex
Published: January 2007

A fairly well-known problem is that of counting the number of different colourings of the set of faces of a polyhedron, using n colours. Of course it depends on what is meant by ‘different’. As perhaps the simplest non-trivial example, let n = 2 (red and blue say) and consider the regular tetrahedron. If we regard the tetrahedron as static, then perhaps the answer is 24 = 16, since each of the 4 faces can be red or blue. However we are in the habit of picking up objects and rotating them around, so maybe a better (and more interesting) interpretation of ‘different’ is to regard two paintings as distinct if one cannot be rotated into the other. We are left with the following 5 distinct possibilities :

all red or all blue ; 3 red, 1 blue, or vice-versa ; 2 red, 2 blue.
Unexpected groups
Authors: Brakes W. R.
Published: January 2007

The reader should first try the following exercise, which motivates the later work.

Exercise: Prove that the set of real matrices

forms a group under the usual matrix multiplication.

This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?
The World Cup draw’s flaws
Authors: Jones M. C.
Published: January 2007

I don’t know how many readers of this journal would have been watching the draw for the opening matches of the 1990 football World Cup, made in Rome on Saturday, 9 December 1989, and televised worldwide. But I do hope that most of you that did—and who were paying attention!—at least thought “Hey, wait a minute, that can’t be right!” Sure enough, the draw was flawed: it was not as fair to all concerned as it might have been. This was a great pity given that the world football authority, FIFA, were at pains to stress how fair these proceedings would be, and the agonies of FIFA trying to be seen to live up to their “fair play” motto were indeed prolonged. The errors lay not in the process of drawing balls from urns itself, but in the algorithm used to define the drawing procedure, as I shall explain below. The problem was caused by FIFA’s decision to keep the four South American countries involved apart and, in particular, by their inability to implement this decision in the most fair manner. There is no allegation of anything being “fixed” here, just one of incompetence on behalf of football’s governing body. That there should be errors of this kind in such an overtly public affair is, however, yet another sad indictment of the general lack of understanding of matters numerical and probabilistic in the public at large.
Sequences which omit powers
Authors: Nelson R. D.
Published: January 2007

Here is an interesting result on sequences amenable to investigation and proof in the classroom. Consider the sequence

where, as usual, [x] denotes the integer part of x. You will find that you get the sequence

2, 3, 5, 6, 7, 8, 10, . . .

which consists of the natural numbers with the perfect squares omitted. This result is due to H. Halberstam [1] and in this article we verify it and look for generalisations to sequences which omit mth powers.
Computer aided research into triangle geometry
Authors: Oldknow Adrian
Published: January 2007

This article considers the use of a range of computer tools for aiding geometric exploration, together with a suitable algebraic representation for objects connected with the triangle. The techniques are applied to produce new results in the geometry of the triangle. This is an extended version of part of the contribution: How do computers change the way we do mathematics? given at the Association’s 1994 Easter Conference.
Hidden ratios
Authors: Tahta Dick
Published: January 2007

This year marks the quincentenary of the death in 1492 of Piero della Francesca whose paintings have always seemed to me to have some special attraction for anyone who is mathematically inclined. I would like to celebrate this anniversary by describing a brief geometrical exploration – and some consequent speculations – prompted by looking at one of his paintings. It is not known for certain when Piero della Francesca painted his strange, enigmatic picture of the flagellation of Christ , now to be seen in the Palazzo Ducale, Urbino, which is on the cover of this edition. It is usually held to be one of his later works, painted in his late forties before that last period of his life which he seems to have devoted exclusively to mathematics. The experts disagree not only about the date of the painting, but also about its meaning and purpose. In particular, it is not known, though it is much discussed, which contemporary persons might have been portrayed as the three curiously aloof figures in the foreground.
Cracking the knapsack code
Authors: Mackinnon Nick
Published: January 2007

The Entrepreneur competition in the Sunday Times (and the Portfolio
competition in the Times) have the following structure. A list of (say 10) integers a(1),a(2),…,a(10) is given in the paper (in the form of share prices) together with a target integer T. Readers have a card on which is printed a second list of (say 4) integers b(1),b(2),b(3),b(4), this second list being different for each reader.
Expert predictions of the lifetimes of non-renewable resources
Authors: Bartlett Albert A.
Published: January 2007

The editor of this journal recently deplored the inability of members of the government to make correct use of the simple mathematics that they encounter in their various capacities. The purpose of this article is to call attention to a common mathematical failing of those who are “authorities” in questions such as the life expectancies of non-renewable resources. My particular example comes from Britain, although failures such as this one are common among “experts” in the U.S.
A family of golden triangle tile patterns
Authors: Clason Robert G.
Published: January 2007

An intriguing family of tile patterns can be generated by applying the self-replicating geometry of a pair of isosceles triangles called the golden triangles. The patterns have an aesthetic appeal and have geometric properties which might be investigated further. The family includes patterns which were discovered by Roger Penrose in the 1970s, and which received much attention when they were proposed as a model for quasicrystals. The use of golden triangles to obtain tile patterns stems from the work of R.M. Robinson cited in [2, p540] and extends the investigations of [3]. The method employed in obtaining the patterns is experimental mathematics; geometrically defined experiments are performed using a computer and the results examined for their properties.
Runs of composite integers and the Chinese Remainder Theorem
Authors: Watson Rex
Published: January 2007

The distribution of the prime numbers is of course a well-trodden path. The casual enquirer, working with small numbers, soon finds that the distribution has no obvious regularities, and sees that the primes thin out. The Prime number theorem (which is not elementary) helps to firm up this observation:

Prime number theorem: If π(n) is the number of primes no greater than n, then i.e. π(n)/(n/ln(n)) → 1 as n → ∞. Thus for the proportion of prime numbers upto to n, π(n)/n, we have that which tends slowly to 0.
Fair means or foul?
Authors: Stanley P.
Published: January 2007

The possible dangers and anomalies arising from the casual interpretation of “mean values” or “averages” are well known. The “velocity problem” is often cited as an example (see e.g. [1,2]). A typical version of this problem is as follows: A cyclist goes from A to B at 20 mph and returns from B back to A at 10 mph. What is the average speed? The casual answer is 15 mph, the arithmetic mean of the two speeds; this is wrong. The correct answer, obtained by dividing the total distance travelled by the total time taken, is obtained as 13.3 mph, and this latter figure is readily shown to be the harmonic mean of the two speeds. Langley goes on to say that “the harmonic mean is often appropriate when the data consists of ratios (miles per hour, . . .)”; this is true but not specifically informative. Reichmann comments that “the arithmetic mean cannot properly be used to average rates of speed” and “the harmonic mean is always the right one to use to average rates . .-.”; these generalisations are incorrect.
The geometry of folding paper dolls
Authors: Servatius Brigitte
Published: January 2007

When a parent sees a little girl sitting on the floor cutting paper dolls, many thoughts may come to mind: ‘She’s keeping out of trouble’ or ‘She’s making a mess’ or even ‘There go my tax returns’. The thought that should have come to my parent’s mind, however, was ‘One day she’ll be a mathematician’. My grandmother, who worked as a dressmaker, often allowed my sister and me to use her razor sharp scissors on the strips of leftover tracing paper. This paper is inspired by a notebook that I kept in grade school when I ‘studied’ paper dolls, and the figures are based on dolls found pressed between the pages.
Computations of the partition function
Authors: Shiu P.
Published: January 2007

How many ways are there of making up £1 using coins? Omitting the use of the £1 coin, we are asking for the number of ways of ‘partitioning’ 100 using parts of 1, 2, 5, 10, 20, 50. For n ≥ 0, let f (n) denote the number of solutions to the equation

in non-negative integers xi, so that our required number is f (100). It is not difficult to see that f(n) is the coefficient of tn in the product Π(1 + tk + t2k +… ) over the parts k = 1, 2, 5, 10, 20, 50.
On a driving hazard problem
Authors: Wang Yongji , Cai Kaiyuan
Published: January 2007

A driving hazard problem was proposed by Baylis [1] and revised by Bender [2]. The problem can be described as follows: when making a right turn on British roadways, one moves to the right as far as possible on one's own side of the roadway and then turns. Unfortunately, the rear of the vehicle swings leftward as the right turn is begun - toward the unsuspecting driver passing on the left (Figure 1). This can be quite noticeable if the turning vehicle is a long bus. Hence the driving hazard problem - just how far will the back corner of the bus (P in Figure 1) swing toward the left as the bus driver negotiates the right turn - was proposed.
Modelling irrational numbers in analysis using elementary programming
Authors: Tall David , Mills John
Published: January 2007

In recent years we have been teaching mathematical analysis using the computer to give experiences to help students understand the concepts [1]. It has proved to be a valuable aid in many ways, but the method depends on giving a suitably rich environment enabling students to experience the full range of possibilities. A great weakness of the use of regular computer languages is that they hold numbers in memory as a finite expansion in base two, so, effectively, all numbers in such a system are rational. This meant that we were unable to demonstrate any ideas that depend on the difference between rationals and irrationals; for instance, we could not plot graphs of the kind
Pop maths postscript : when is a vacation a holiday ?
Authors: Rowland Tim
Published: January 2007

July 1990. The Pop Maths Roadshow comes to town. “Chaos” Rules OK.

28 July. Receive invitation to post-Pop Maths party in September. One of the conditions of eligibility to attend is to tackle four problems listed on the invitation. Surely only mathematicians do this sort of thing? I glance at the problems. Two of them I know how to tackle, and a third I haven’t a clue how to begin; but it is the first problem that intrigues me most.
Making the grade
Authors: Quadling Douglas
Published: January 2007

The candidates have sat the examination. The examiners have marked the scripts. Everyone has gone on holiday … except the chief examiners, for whom one more task remains. Before the Board’s computer can start to churn out the results slips, they must decide the formula for converting the marks into grades.

When there is just a single paper, and no course work, as with many AS examinations, this is straightforward enough. The chief examiner, making a judgement of the quality of the scripts against grade descriptions, backed up by past experience, will propose a set of grade boundaries – that is, the lowest marks which should achieve each of the available grades – and the conversion is then automatic. For example, if the grade boundaries are A 76, B 68, C 59, D 50, E 41 and N 32, then candidates with marks of 61, 99 and 30 will be graded C, A and U respectively.
Hidden strengths of dimensional analysis
Authors: Ong P. P.
Published: January 2007

Perhaps the most extensive use of dimensional analysis in applied mathematics is in the application of the principle of similitude to obtain equivalence between different physical systems of unequal scales occurring in widely different branches of science and engineering. However, there is a widespread misconception that dimensional analysis can only yield the functional dependence of a physical quantity on various other parameters. In this article I shall therefore demonstrate some hidden strengths of dimensional analysis.
Virtual mathematics
Authors: Cochrane Peter
Published: January 2007

The single biggest problem we face is that of visualisation

Richard P. Feynman, Los Alamos 1945

Given the critical dependence of our civilisation on mathematics, it is paradoxical that it lacks any general acclaim or popularity, and usually invokes fear and mysticism. Perhaps this is because, for most of us, the process of acquiring mathematical skills is somewhat protracted and painful. It is often perceived as a nightmare topic, which engenders a defence mechanism of open ignorance that is often lauded. In truth, most people can become reasonably competent. However, the time and effort needed to acquire the formal rigour for even a modest capability is often seen as excessive by modern standards.
Planar graphs via squashing polyhedra and dynamic geometry
Authors: Vile Adam
Published: January 2007

Planarity is a concept that appears in most discrete mathematics courses (in particular, it is a syllabus item in all A level Discrete mathematics syllabuses). I have recently taught a unit in graph theory to a group of BEd students and, when I came to the concept of planarity, I decide to approach it in a different way – through the context of Cabri Géomètre, a dynamic geometry package. In the article I would like to share some of my ideas and reflections on this approach to teaching this topic, paying particular attention to the role of Cabri Géomètre in scaffolding the important ideas involved in an understanding of planarity.
Full circle
Authors: Reynolds Peter
Published: January 2007

Welcome to Leicester, our first Annual Conference here since it became the home of the Association in 1975 as the result of the centenary appeal in 1971 to buy a permanent headquarters building. The city is, of course, also home of the Association's library. Welcome to the University and to the Annual Conference, one of the many manifestations in which our Association appears. It is a great pleasure and honour to be here tonight to speak on the opening day of our 1990 Conference.
Sums of squares of digits
Authors: Beardon A. F.
Published: January 2007

This paper contains many opportunities to investigate, conjecture and verify results by writing simple computer programs, and the reader is strongly urged to participate in this way throughout.

For each positive integer n, let F(n) be the sum of the squares of the digits of n; for example,
Bordered magic squares of odd order
Authors: Vaithianathan S.
Published: January 2007

A normal n × n magic square consists of the numbers 1, 2, … , n2 arranged in a square so that the sum of each column, row and the two diagonals is In an odd order normal magic square , the central number is This article concerns bordered magic squares of odd order, which are squares that remain magic when each outer row and column is peeled off.
Probability, pi, and the primes: Serendipity and experimentation in elementary calculus
Authors: Young Robert M.
Published: January 2007

Many years ago, long before it had become fashionable for so many research mathematicians to concern themselves with pedagogy, the renowned topologist Bill Thurston appeared as the keynote speaker at a symposium held at Harvard University. As he approached the podium, he could be seen carrying a set of toy building blocks, the sort designed for children. He was going to use them to construct models for his talk. But before he began, he held up the box and said, ‘You know what, kids love this stuff. What is it that we do to them?’ Of course, Thurston was right. Children have a natural curiosity, an innate sense of wonder about the world. Picasso said that children were natural born painters. Thurston said that they were natural born geometers. How do we help them to recapture that child-like sense of wonder after so many years of neglect?
Tendril of the Hop and Tendril of the Vine: Peter Guthrie Tait and the Promotion of Quaternions, Part I
Authors: Pritchard Chris
Published: January 2007

Two days before his death in the summer of 1901 Peter Guthrie Tait gave his son some handwritten notes on quaternions for safekeeping. Though he had distinguished himself in many areas of mathematical physics and had influenced the work of Thomson and Maxwell it was his evangelical promotion of quaternions which would be remembered in the years to follow and it was fitting that his last energies be devoted to the cause.
Progress in Number Theory
Authors: Shiu P.
Published: January 2007

For this ‘centenary issue’, the editor has invited me to write a survey on progress in number theory over the last 100 years. He did not, of course, ask for a proper technical report, but just a short article suitable for Gazette readers. Fortunately number theory is one of those subjects in which one can speak, albeit superficially, on the problems and results without having to explain much of the technical details associated with them.
A hundred years of calculating aids
Authors: Macmillan Robert
Published: January 2007

Until the First World War the only effective aids to calculation were the slide-rule, for 1% accuracy, or books of tables of logarithms, roots and trigonometric functions. These were 4 or 5-figure for normal use and 6 or 7- figure for those who needed the extra accuracy. Many ingenious methods were devised for increasing the the length of the slide-rule scale to improve its accuracy: some scales were made spiral and others were wound round a cylinder or printed in multiple parallel lines. But higher accuracy always meant slower calculation.
Curves and conjugacy or a tour of some interesting geometry
Authors: Cundy H. Martyn
Published: January 2007

Since there is far more mathematics even quite elementary mathematics than anyone can carry in their working memory, it must follow that results which are unused will be forgotten, and are then often rediscovered after many years. A tour of some old haunts which are now being opened up again would therefore seem not out of place in a centenary number of the Gazette. Tourists, however, will need some preliminary briefing if they are to enjoy the tour fully.
A final-year university course on the history of mathematics: actively confronting the past
Authors: Fowler D.H.
Published: January 2007

Let me emphasise: I shall be describing the ideals and aims that have evolved out of a final-year undergraduate course on the history of mathematics that I have run, off and on, for several years. But some past participants may scarcely recognise what I am writing about!

One skill that historians need but mathematicians tend to disdain is the ability to read: to scan rapidly material of peripheral importance, to read closely what is relevant, and to retain an accurate summary. So, as an introduction, at the end of their second year, I circulate all students with an outline of the course and a short reading list: they are very strongly recommended to read, right through, some book that gives a panoramic view of the subject. I suggest Struik, as the shortest, Boyer, for more detail, and the superb but demanding and ridiculously expensive Open University Course for the energetic who can get their hands on it. Also I assign a preliminary selection of texts for study (see (a) below) and I point to a box of relevant articles that I maintain in our university library. Some students do prepare for the course in advance, but the majority only register a mental note to come to the first class then forget about it, even losing their piece of paper!
Desert Island Theorems: my magnificent seven
Authors: Crilly Tony
Published: January 2007

Choosing seven mathematical theorems or ideas to take to a desert island is
more difficult than I first thought. In an earlier article (with Colin Fletcher) I
discussed other people’s choices. Now comes the time to bite the bullet and make
my own. I was reminded that the choices do not have to be ‘great’ or
‘state-of-the-art’ or involve any crystal ball gazing. They do not have to be
among the hundreds of fundamental theorems that abound in mathematics all they are
bound to be is personal.
Beauty is truth: geometric inequalities
Authors: Mavlo Dmitry
Published: January 2007

The publication of [1], which contains all, or almost all, relevant results from all over the world since Euclid’s time up to the middle 1960s, confirmed that this field of study has good traditions, is alive and in good shape. Two decades have passed and the appearance of [2] proved that this part of mathematics is flourishing more than ever. This field (which we will call GI) is a peculiar one: firstly because the subject matter doesn’t presuppose any great mathematical knowledge or technique; and secondly because it is so attractive that the best achievements in this field suit the refined taste of professional mathematicians, the unaffected taste of amateurs, and the yet unsophisticated taste of inexperienced intelligent students. Moreover, due to the specific features of this field of study, with some luck, all of them, as well as the enthusiastic mathematics teachers, have more or less equal chances to make remarkable contributions. As in any other branch of mathematics, a talent, good luck and/or hard work, can produce real gems, but in GI the achievements are more visible, more tangible for simple mathematical folk’s appreciation, and have undoubted educational value. They are well ready for easy usage by mathematical educators both at school and university level, due to their main advantage: many of them are simple, but by no means trivial.
The factorial function: convex functions, the Bohr-Mollerup-Artin theorem, and some formulae
Authors: Fowler David
Published: January 2007

How might we define x!? This is the next in my series of articles [1-3]; by now we know quite a bit about the factorial function, except that we have no real definition of it, nothing that would allow us to evaluate, say, ½!. So how do we move on to the next stage?

There is a range of strategies. We could find or manufacture a suitable function f that satisfies the basic relations

and adopt it as our definition.
Lewis Carroll – mathematician and teacher of children
Authors: Eperson Canon D. B.
Published: January 2007

It is well known that Lewis Carroll enjoyed the company of children and entertained them with fantastic stories, whilst on boating expeditions or on beaches, but his diaries also reveal that he enjoyed teaching children mathematics and other school subjects. On April 16th 1855, he recorded his concern with the education of his younger sister Louisa, who had an aptitude for mathematics: ‘Went into Darlington – bought Swale’s Chamber’s Euclid for Louisa. I had to scratch out a good deal he had interpolated, (e.g. definitions of words of his own) and put in some he had left out. An author has no right to mangle the original writer whom he employs: all additional matter should be carefully distinguished from the genuine text. N. B. Pott’s Euclid is the only edition worth getting – both Capell and Chamber’s are mangled editions.’ Three days later he recorded: ‘Advanced Louisa’s mathematics to simple Equations (third day of Algebra), and the first 12 propositions of Euclid.’ On the next day, he left his home at the rectory of Croft, and returned to Oxford for the Easter Term.
The Hale-Bopp comet explored with A level mathematics
Authors: Corbishley H. R.
Published: January 2007

On 15 March 1997, when boarding an aeroplane bound for Spain, I was handed a copy of the Independent newspaper. One page of the paper was devoted to the Hale-Bopp comet; from two pieces of information about the comet’s orbit I set out to find out as much as I could about the orbit using A level mathematics, and in some cases further mathematics. I was interested in finding the fastest and slowest speeds of the comet, and the size of its orbit in relation to the planetary orbits. I also found speeds and times at other points on the orbit, where distances from the sun could be calculated. These findings, along with an approximate calendar for the next orbit of the comet, are presented in a diagram at the end of this article. During the course of my working, I made two discoveries of a more general nature. The first of these is that the radius of curvature of a conic at the point nearest to a focus is equal to the semi-latus rectum. The second was one of Kepler’s laws, that for an elliptical orbit T2 ∝ a3, where T is the time of a complete orbit, and a is the semi-major axis.
Shooting for π: the bowstring lemma
Authors: Longuet-Higgins Michael S.
Published: January 2007

Some remarkable new series for π, in which the convergence is exponentially rapid, have been discovered by Bailey et al. [1, 2] using machine subroutines; see also Wagon [3]. An example is equation (17) below. Although the authors also provide algebraic proofs, these seem at first sight to be very special. The purpose of this note is to provide a more general demonstration and to deduce other instances of such series. It will be shown that all the results follow naturally from one or other of two simple lemmas.
Chains, froths and a ten-bead necklace: systems of circles and spheres
Authors: Fox Michael
Published: January 2007

Figures 1–4 set the scene, with configurations of tangent circles that may perhaps be new. The labels are curvatures: the curvature or bend being the reciprocal of the radius. By convention, a circle that surrounds another has negative bend and radius. In the first part of this paper I give some easy formulae for configurations like these (see Note a). I show how we can develop fractal structures—I call them froths—having an infinity of tangent circles in each ‘triangular’ region; and give methods for finding integral froths. The second part extends these ideas to systems of tangent spheres, which may also be integral. Throughout, an integral object is one whose bend is an integer, circles and spheres are generally referred to by their bends, and examples are marked with bullets (•).
The curious rectangles of Rollett and Rees
Authors: Ruane P. N.
Published: January 2007

Over sixty years ago, a schoolmaster by the name of A. P. Rollett set a problem for a lower fifth form class which was unwittingly based upon the famous Archimedean problem of the Arbelos. This caused his pupils substantial difficulties and eventually it prompted Rollett to send a Note to the Mathematical Gazette seeking the mathematical insights of the readership regarding this question. He received very helpful responses from over twenty readers whose numbers included many notable mathematicians of the day such as E. A. Maxwell, E. H. Neville and R. S. G. Rutherford etc. Solutions were also received from two members of the clergy, (Canon) D. B. Eperson and the Rt Revd the Bishop of Kootenay, which may suggest that this problem was of sufficient intractability as to be deserving of divine inspiration! But this was not the end of it because, during the next twentythree years, there followed a series of very interesting related notes and articles continuing up to 1960 all of which made reference to Rollett’s initial homework problem of 1937.
A proof of Lester’s Theorem
Authors: Shail Ron
Published: January 2007

The object of this note is to draw readers’ attention to a very new theorem in the Euclidean geometry of the triangle and to provide a straightforward Cartesian proof. This remarkable theorem, due to Lester, asserts that in any scalene triangle the two Fermat points, the nine-point centre and the circumcentre are concyclic. Lester’s original computer-assisted discovery and proof make use of her theory of ‘complex triangle coordinates’ and ‘complex triangle functions’ as expounded in, and . A proof has also been given by Trott using the advanced concept of Grobner bases in the reduction of systems of polynomial equations to ‘diagonal’ form. Trott’s work uses the computer algebra system Mathematica as an essential tool and he also provides an animation of the Lester circle as one vertex of the triangle is varied. Further information on the configuration is given in.