bluuebunny/crossref_metadata_2025 huggingface dataset:
Mathematical Education in Russian Schools|A Report on a Visit to Russia in August 1966
Authors: Thwaites Bryan
Published: January 2007

The National Council of Teachers of Mathematics in the USA took the occasion of the 1966 International Congress of Mathematicians as the opportunity for arranging with the Russian authorities a tour of Soviet mathematical centres by a small party of selected American mathematicians with special experience and interests in various aspects of education. The tour covered Kiev, Moscow and Leningrad and it was hoped that, in each city, a full range of mathematical education would be seen, from the department of mathematics in the University to the famous special boarding schools for pupils of outstanding mathematical ability and finally to the elementary level.

I was honoured to be invited to join the party as the only non-American and am very grateful to the Schools Council for enabling me to go by offering to meet the costs of travel and subsistence.
On Scoring in Games
Authors: Watson Donald
Published: January 2007

The purpose of this paper is to consider the question: if p is the probability that a player P will win a given rally against a player Q, what is the probability that P will win a given game? The answer, of course, depends upon the method of scoring; and the methods considered here will be those used in squash and, for comparison table tennis.

In table tennis P earns a point each time that he wins a rally. But in squash he must serve the rally that he wins in order to gain a point; and he relinquishes the service when he loses a rally that he serves.
On Sums of Progressions of Positive Integers
Authors: Goodstein R. L.
Published: January 2007

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.
On the Equations xn±yn = zm
Authors: Beerensson R. G.
Published: January 2007

1. We determine first an infinity of solutions of the equation xn-yn=zm when m, n are relatively prime. Let p be an odd prime; then q= pm has a primitive root g, say, and so, if ϕ is Euler's function, x= gϕ(q) satisfies xn= gnϕ(q)) = l (mod q).
The Cambridge Mathematical Tripos
Authors: Bushell W. F.
Published: January 2007

The last examination of the Cambridge Mathematical Tripos, when the candidates were placed in each class in Order of Merit and not in alphabetical order, was held in 1909. Last year was therefore the fiftieth anniversary of the end of a famous epoch. The University Calendar used annually to reproduce all the lists from 1747, the year when they were first printed, and they made fascinating reading, well known to mathematicians. Of course candidates were examined before that date, but full details have not been preserved. It was earlier called ‘The Senate House Examination’, and not before the start of the Classical Tripos in 1824 did the name gradually change.
Problems of Arithmetical Geometry
Authors: Mirsky L.
Published: January 2007

The subject matter of my talk is geometry in the sense that I shall deal with such entities as lines, circles, and squares. But the results I aim at bear little resemblance to the theorems of traditional geometry, which describe properties of given configurations. Our present object is quite different; we shall begin by specifying not configurations but processes (for example processes of selection), and then seek to determine how economically or how effectively these processes can be carried out. Questions of this nature are quantitative and the solution, if it is known, is generally expressed by a number. It is for this reason that I speak of 'arithmetical' geometry. The problems of the type I shall discuss are usually simple to state and require very little in the way of mathematical equipment for their understanding. However, their solution often presents formidable difficulties and is in many cases as yet far from complete. Let us now consider in some detail a few of the characteristic problems in this field.
Pantactic Squares
Authors: Astle B.
Published: January 2007

Consider a square array of 25 grid squares, the grid squares being of two types distinguished by the colours black and white. A basic square of four adjacent grid squares can be selected from such a square array in 16 possible ways. If these 16 basic squares are all distinguishably coloured, then the 5x5 square array is said to be pantactic.
The Mathematical Background of Teachers in Training
Authors: Buckland P.R.
Published: January 2007

There are many ideas, fundamental to the whole of mathematics, that are difficult to teach explicitly and are rarely examined. They include the concept of mathematical proof, the idea of a converse, necessary and sufficient conditions, and those elements of elegance and rigour without which mathematics can lose its beauty and even its claim to be called mathematics. It is these ideas I shall refer to as mathematical background; it must be emphasised that this background is common to both traditional and “modern” mathematics, and that its cultivation in their pupils would seem a desirable objective for all teachers, both conventional and progressive. Most of these ideas were clearly expressed in the Mathematical Association’s booklet, “Suggestions for Sixth-Form Work in Pure Mathematics”, which seems admirable in everything but its title, since most of the work should be begun much earlier than the sixth-form. (If it is left until then, there is justification for one reviewer’s remark that it is “an attempt to paper over the cracks of sixth-form mathematics”.)
Evaluating Determinants by Pivotal Condensation
Authors: Blyth T.S.
Published: January 2007

As far as the author is aware, there does not seem to be currently available in the literature a proof of the little-publicised pivotal condensation method of evaluating determinants. The purpose of this note is to supply a simple proof of this.
A Functional Equation Characterising the Sine
Authors: Rosenbaum R. A. , Segal S. L.
Published: January 2007

Recent articles on equations characterising the trigonometric functions ([1], [2]) prompt a consideration of still another equation, which might be classified under the heading of “Students’ Mathematical Mythology” In [3], Mr. Heafford multiplies the “obvious” sin (x + y) = sin x + sin y by the equally “clear” sin (x − y) = sin x − sin y to obtain the (correct!) result, sin (x+y) sin(x − y) = sin2x − sin2y.
Algebra Aids Electronics
Authors: Cuninghame-Green R. A.
Published: January 2007

This investigation grew out of the need to design a piece of electronic test-equipment. The purpose of the equipment was to study the effect of varying capacity upon the behaviour of a number of electric circuits, and it consisted basically of an automatically-stepping switch which presented at its output terminals a controlled sequence of values of capacity. The nature of the test demanded that the set of capacity values should as nearly as possible cover a given range in a logarithmic scale, and to economise in components the equipment was to generate the set of values from a basic set of condensers, by creating all possible parallel-connected combinations. The problem was, how to choose the basic set? The analysis has a number of points of combinatorial interest, and illustrates fairly well the ad hoc approach of the industrial mathematician.
Two Properties of Integers
Authors: Andrew A.L.
Published: January 2007

In a recent competition (see M.R. 23 (1962), A1482) it was required to show that the set {2, 3, 4} has the property, here called Pv that every number x in (0, 1] (i.e. 0 < x ≤ 1) can be expressed in the form , where m1 is an integer and belongs to the set for all i.
Impulsive Motion of a System
Authors: Snell K. S. , Parsonon S. L.
Published: January 2007

In an article entitled “Two Problems on Impulsive Motion” (M. G., XLIV, p. 95) the writer (J. O’Keeffe) complains of the lack of method in elementary books on problems such as the initial motion of two jointed bars under the action of an impulse. In our view, such problems are best treated, in an elementary course, by reducing the total momentum of the system to a single linear momentum and a momentum-couple. The necessary equations are then obtained by using the principle of equivalence of this system with that of the impulses. This treatment makes it clear why and in what form it is correct to take moments about the point at which the bars are hinged. Below we give the general treatment, and also the application to this particular problem.
July Thoughts
Authors: Chapman L. R. , Butler P.
Published: January 2007

It is dangerous to generalize from unique experience, but the season of examination results is round again. Apart from the annual surprises, the ranked order is much as our departmental meetings predicted. The range of marks is similar and the percentage qualifying to proceed to Part II of the B.Ed, degree is of the same order. No one failed, but the tail is as abysmal as in previous years. These “tail” students are perhaps the most important ones, for they will teach over a wide band of education: the spectrum may start at the reception class in an infants school and end in the higher forms of a secondary or comprehensive school. Ability as a teacher doesn’t necessarily correlate with skill as a mathematician, but the quality of their contribution is questionable. So in this summer season efforts are again directed towards finding reasons for comparative failure.
Playing with Probability; Return Match
Authors: Lawrance A. E.
Published: January 2007

It was encouraging to receive some letters following the article, in the 1969 December Gazette (Vol. LIII, No. 386), entitled “Playing with Probability”. Most of the letters made reference to the use of “a steam hammer to crack a walnut”. The purpose of the article was to review the teaching of probability at three levels in a secondary school, and since stochastic matrices are to be found in the further mathematics of the School Mathematics Project it seemed appropriate to give an example which used them. The fact that the particular example chosen could be answered in one line does not detract from the interest of the method used in the article—although, of course, as mathematicians we are always looking for elegance.
Mathematics, Logic and Undecidability
Authors: Meltzer B.
Published: January 2007

Among the greatest advances in knowledge this century one would include relativity, quantum mechanics and the explanation of hereditary replication in living things by the structure of the DNA molecule. But it would not be absurd to include in this category a discovery which is not at all well-known and goes back to 1930, namely, that logic and mathematics as we know them have undecidable propositions.
Formulae For Primes
Authors: Goodstein R. L. , Wormell C. P.
Published: January 2007

The criticism by T. B. M. Neil and M. Singer in The Gazette for October 1965 of C. P. Willans’ formula for the nth prime number is in my view quite unjust. Willans’ formula expresses two properties of the nth prime, namely that the number of numbers x such that 1 ≤ π(x) < n is two less than the nth prime, and that the nth prime is less than 2n. Willans gives a trigonometrical representation of π(x), the number of primes less than or equal to x. The analysis which Neil and Singer give is an exposure of the formula only in the sense that a clear proof is an exposure.
Numerical Methods and the Computer
Authors: Dean P. G.
Published: January 2007

The transfer from pen and paper methods to hand calculator methods which has been made by many teachers still uses visible denary digits. We are now beginning the transfer to a computer and I feel that the Mathematical Association’s admirable new report does not place enough emphasis on the possibilities of computer system errors which are invisible to the user. The teacher looking at numerical error must not put untested faith in its abilities, and must not expect it always to behave in a regular denary fashion, as will be shown in the tests below. He must also be warned that the computer reference manual rarely has the reliability of a text-book.
Whither Geometry?
Authors: Willmore T. J.
Published: January 2007

This Association came into existence nearly one hundred years ago with the prime object of improving the teaching of geometry and although its field of interest has widened considerably the teaching of geometry remains a very important issue. Of course, the question “Whither Geometry?” is rather meaningless unless one specifies whether one is talking about geometry as taught to primary, junior or secondary school levels, to Colleges of Education and the Polytechnics, to undergraduates and postgraduates at University, or geometry as understood by professional research mathematicians. Moreover the motivation for studying geometry may well be quite different at these different levels. I shall try to deal with the question as it affects all these various levels of mathematics.
What is Modern Mathematics?
Authors: Smithies F.
Published: January 2007

The first thing to be said about what is commonly called “modern” mathematics is that it is not really so modern as one might suppose. A great deal of it was in fact created more than fifty years ago. What gives it an air of novelty is that only in the last generation has it had a serious effect on the undergraduate teaching of mathematics in our universities, and only in the last few years has its impact on school mathematics begun to be felt. Some such time-lag is natural and inevitable ; new mathematics is usually created as a research tool, and so long as it is nothing more there is no compelling reason to seek the simplest way of expounding it or to re-examine more elementary mathematics in its light. It is when one starts to teach a new subject that simplification begins, and when this has been accomplished one begins to appreciate the possible advantages of the new point of view at a more elementary level.
On Fermat’s Last Theorem
Authors: Long Louis
Published: January 2007

In my note in the December 1960 Gazette I proved that if there is an odd prime p and numbers a, b, c prime to p such that

then p necessarily has one of the forms 120k+1, 120k+49.
Shanks, Ferguson and π
Authors: Broadbent T. A. A.
Published: January 2007

Fifty or sixty years ago, any forward schoolboy (to borrow a phrase from Macaulay) knew that a man called Shanks had calculated the value of π to 707 places of decimals. Who Shanks was, why he should have performed this Gargantuan feat, and why he should have stopped, if not from exhaustion, at the 707th place, were questions that may have been asked but were seldom answered. The bald statement is still to be found in a number of recent books, but it is not often brought up to date by noting the interesting sequel, that round about 1944 a member of this Association, D. F. Ferguson, re-calculated πby a formula similar to but different from that used by Shanks, and discovered that Shanks had made a mistake and that his value was incorrect after the 527th place.
Units and Symbols in Mechanics
Authors: Lowey H. V.
Published: January 2007

In many textbooks on applied mathematics, mechanics, or applied mechanics, the impression is still given that the only absolute systems of units are either the CGS system, using the gramme and dyne, or the British system using the pound and poundal and also that the only alternative system is a gravitational one in which either the unit of force or both the unit of force and the unit of mass vary with position; a few bodies do include the MKS system.
Sylvester’s Chemico-Algebraic Theory A Partial Anticipation of Modern Quantum Chemistry
Authors: Griffith J S.
Published: January 2007

In 1877, the English mathematician J J. Sylvester (1) announced the discovery of a rather complete formal analogy between the mathematical theory of the invariants and covariants of binary quantics and the chemical theory of valency as it then stood.
Some Geometrical Thoughts II
Authors: Croft H. T.
Published: January 2007

I consider here another 3 problems, whose solutions are “not easy to see, but easy when seen”. Each of them involves in some way the notion of the convex hull of n points in a plane.
A Vectorial Dedekind
Authors: Cable John
Published: January 2007

To continue the debate (Mansfield and Bruckheimer [1], Broomhead [2]) :

1. Scalar Quantities are things like length, area and mass. Vector Quantities are things like force, velocity and acceleration (in 2 or 3D). But let us not, for the present anyway, call vector quantities “vectors.”
Statistical Evaluation of π
Authors: Langner A. J.
Published: January 2007

A sheet of graph paper, marked in inches and tenths or (preferably) smaller subdivisions, is placed on a horizontal surface.

A halfpenny is placed flat near its edge, and flicked along the paper.
The Diffraction Rainbow
Authors: English D. R.
Published: January 2007

This is a very beautiful natural phenomenon, seen on occasions when the primary bow is a strong one. As will appear later, the occurrence necessitates raindrops of a homogeneous size, and of course bright sunshine at a low angle. The diffraction rainbow is therefore most commonly seen in tropical countries, in a thunderstorm that has already passed over and is moving away from the setting (or rising) sun.
Matrix Proof of a Theorem In Mechanics
Authors: Green D. J.
Published: January 2007

An interesting application of elementary matrix theory is to prove that at any point of a rigid body there exists at least one set of principal axes of inertia.
A Method of Solving Some Diophantine Equations By Means of the Equation
Authors: Blackburn W. T.
Published: January 2007

The following discussion indicates how the equation
A Problem in Projective Geometry
Authors: Room T G.
Published: January 2007

The theorem to be discussed in this note depends only on the plane incidence axioms (with Desargues), the Pappus axiom, and the property that a point and its harmonic conjugate are in general distinct. When algebra is applied to the geometry it may be taken over any commutative field of characteristic other than 2.
A Note on Fermat’s Theorem
Authors: Long Louis
Published: January 2007

It was conjectured by Fermat that if n > 2, the equation has no solution in positive integers. If n is composite, say n = lm, the equation implies Am + Bm = Cm, where A = al, etc. So the insolubility of the equation with exponent m ensures the insolubility of the equation with exponent n.
The Open University
Authors: Collings S. N. ,   the Mathematics Foundation, Course Team
Published: January 2007

In any consideration of the Open University it is as well to recall the words of Lord Crowther its Chancellor. He said “We are open first as to people…. We took it as axiomatic… that no formal academic qualifications would be required for registration as a student. Anyone could try his or her hand and only failure to progress adequately would be a bar to continuation of studies…. We are open as to methods…. Every new form of human communication will be examined to see how it can be used to raise and broaden the level of human understanding…. We are open finally to ideas.”
A new look at functions in modular arithmetic
Authors: Bird Marion H.
Published: January 2007

When a function in arithmetic modulo n is represented through a cartesian graph, the resulting set of points reveals little information about the form of the function (for example, see Fig. 1). It is not easy to discern connections between the points; a restriction to the values 0, 1, 2,…, n — 1 would have to be placed on x and y; and the fact that n — 1 is as ‘near’ to 0 as 1 is, is not apparent in the graph.
The Contribution of the Scots to Mathematics
Authors: Schlapp R.
Published: January 2007

The task you have set me, of surveying in the space of 50 minutes the contribution of the Scots to mathematics, is a well-nigh impossible one. Such a survey must begin with the revival of learning in western Europe, and may appropriately end shortly before the middle of the nineteenth century, when the concept of Scottish mathematics begins to lose definition; an exact date might be fixed, rather arbitrarily, by the first appointment of an Englishman to a Scottish mathematical chair in 1838, when Philip Kelland succeeded William Wallace at Edinburgh. These limits will exclude many distinguished mathematicians of Scottish birth active during the last 100 years, such as Sir Thomas Muir and Maclagan Wedderburn, not to speak of the applied mathematicians Kelvin, Tait and Clerk Maxwell.
Population in cities
Authors: Kilmister C. W.
Published: January 2007

I want to deal with one piece of applied mathematics which is a good example of the way in which—in nearly all applications of mathematics—the important steps are those of tentative model-building, and the testing of such models against reality. It is this feature that makes mechanics such an atypical case; there the model, Newton’s laws, is made for one beforehand. In the present instance the initial problem, whose partial solution leads, as usual, to other problems, can be stated very simply: in most cities the population density, σ, tends to fall off with distance, r, from the centre.
Looking at graphs through infinitesimal microscopes, windows and telescopes
Authors: Tall David
Published: January 2007

The differential triangle of Leibniz for a real function f is found by taking an increment dx in the variable x, finding the corresponding increment dy in y = f(x) and drawing the ‘triangle’ in Fig. 1. Here ds is the increment in the length of the graph, where

and the derivative of f is
Archimedes in n dimensions
Authors: Val Patrick du , Saban Giacomo
Published: January 2007

“And though he made many elegant discoveries, he is said to have besought his friends and kinsmen to place on his grave after his death a cylinder enclosing a sphere, with an inscription giving the proportion by which the including solid exceeds the included.” Plutarch, on Archimedes, in The lives of famous men, the Life of Marcellus.
Anamorphoscopes: a visual aid for circle inversion
Authors: Kuchel Philip W.
Published: January 2007

The use of reflecting surfaces to unscramble distorted pictures has been apparent over the past three centuries, at least, in an art form known as anamorphic (ana = back again, morphe = form) art. Of course, working in the opposite sense, the mirrors in fun parlours give distorted images from an undistorted (!) object. A famous portrait of “Bonnie Prince Charlie” in the West Highland Museum, Fort William, Scotland, appears as a series of incomplete concentric circular streaks of colour, but when a cylindrical reflecting surface (the anamorphoscope) is placed at the centre of the painting a recognisable image is seen on looking into it. Niceron and DuBreuil both wrote treatises in the seventeenth century on the techniques of producing anamorphograms, and Gardner has given a recent discussion of the general topic. DuBreuil described a method for generating a distorted object (drawing) that will be transformed into an undistorted image by the use of a conical reflecting surface whose base sits on the plane of the object. The drawings are very easily done by following simple rules of geometry.
Affine planes: An example of research on geometric structures
Authors: Pickert Günter
Published: January 2007

At the end of the nineteenth century David Hilbert made the decisive step in the foundations of geometry (for the historical development see [1]): a road was opened to a theory of geometric structures. In spite of the fact that one spoke (and speaks still today!) strangely enough of ‘implicitly’ defining the geometrical concepts through the axioms, it became clear in the sequel that the axiom system of Hilbert constitutes an ‘explicit’ definition: not of course for ‘point’, ‘line’, etc. but of the concept ‘euclidean geometry’ (or ‘euclidean space’).
Counting triangles by group theory
Authors: Stratton A. E.
Published: January 2007

The introduction of group theory in the sixth form often raises the question of its relationship with other areas of mathematics. The purpose of this note is to develop a group theoretical solution of a simply described counting problem; a problem which may, in fact, be solved in several other ways. The development of the solution yields useful examples of several algebraic notions; in particular the notion of equivalence relation is exploited in a variety of ways. The particular group theoretical/combinatorial result that finally yields the answer to our problem is known as Burnside’s Lemma. This result is not normally presented until a second or third year university course. I hope to convince the reader that the manner of presentation given here is perfectly accessible at a more elementary level.
The mathematical background of mathematics graduates and undergraduates
Authors: Cornelius Michael
Published: January 2007

After the publication of the Mathematical Association’s booklet on sixth-form work in pure mathematics in 1967, P. R. Buckland published the results of a short multiple-choice test, given to graduates training to teach mathematics, in the Gazette in 1969. He discovered what he called “alarming gaps” in students’ mathematical backgrounds. Is the situation any different 15 years later? In an attempt to obtain an answer, Buckland’s test (with a few changes and a new ‘historical’ section) was given to a group of graduates and undergraduates.
Some sequences by inverse iteration
Authors: Clegg David , Bryant Victor
Published: January 2007

This article was inspired by the following problem in a recent addition of the Bulletin of the IMA:

Problem. The sequence a0, a1, a2, a3,… is defined by a0 = 3 and, for n ⩾ 1, an is the real root of the equation

Prove that the sequence converges to 5.
An attack on the Erdos conjecture
Authors: Mackinnon Nick , Eastmond John
Published: January 2007

5, 11, 17, 23, 29 is an arithmetic progression of prime numbers. It is conjectured that there are such progressions with arbitrarily many terms (there cannot be infinite sequences because a, a + d, a + 2d,… sooner or later contains a + ad). The longest such known to the authors has first term 4943 and common difference 60 060 with 13 prime terms, but longer ones will have been found by now. In general, additive problems about the multiplicatively defined prime numbers are inevitably difficult (e.g. the Goldbach conjecture and twin primes) and may even be candidates for undecidability. We have been trying to prove a much more general conjecture of Erdos, spurred on by his offer of $3000 to the first proof or disproof of it.
61.9 A permutation algorithm
Authors: Owen G. I.
Published: January 2007

A computing colleague asked me recently if I could devise an algorithm for obtaining the ith permutation in an alphabetical list.

Suppose that the n! permutations of n distinct letters A, B, C,..., N are listed in alphabetical order. The problem is: Given i (where i = 1,2,...,n!), find the ith permutation.
An improbable game of darts
Authors: Crilly Tony , Suen Stephen
Published: January 2007

The general utility of the ‘pigeon hole principle’ (that if n + 1 pigeons are placed in n boxes one of the boxes contains more than one pigeon!) is neatly illustrated by the problem of showing that:

If seven darts land on a dartboard in any configuration whatsoever, there will always be two darts which are not more than a distance of one radius apart.
Great Circles
Authors: Cook Ian
Published: January 2007

Spherical trigonometry has fallen somewhat out of fashion and so it was a salutary experience to be consulted recently by a would-be circumnavigator of the world.

A great circle route can be analysed graphically by first using a gnomonic chart. Such a chart is obtained by projection from the centre of the earth on to any plane. The distance of the plane from the centre determines the scale of the chart. On a gnomonic chart every great circle appears as a straight line and every straight line on the chart represents a great circle. Of course chart can only represent one hemisphere and even then the whole hemisphere would need an infinite chart.
The importance of being 259
Authors: Goodstein R. L.
Published: January 2007

The number of the Mathematical Association’s new headquarters in London Road, Leicester is 259.

This number has some quite interesting features. It is for instance the sum of the three squares

32 + 52 + 32 × 52,

and the sum of four cubes

23 + 23 + 33 + 23 × 33 and 13 + 23 + 53 + 53.
Geometric programming
Authors: Bagnato Robert A.
Published: January 2007

During and after World War II, operational research teams needed new, computer-adaptable methods for optimization problems in industry, engineering, economics and the armed forces. The calculus methods developed in the sixteenth century to solve problems in astronomy, physics and geometry, even when they were applicable, were not always computer-adaptable. Among these new operational research methods is geometric programming, an optimization technique which in its beginning utilized the theorems in this article. The word “geometric” comes from its use of the geometric mean as well as geometrical concepts such as orthogonality and normalization.
Using computers in the teaching of statistics
Authors: Holmes Peter
Published: January 2007

Over the past few years more and more schools have been able to gain access to computers. Many have their own terminals, and can work on-line, others send their material to a centre for batch processing. This article looks at some examples of the way such computing facilities can be used in the teaching of statistics. Clearly much more work can be done with interactive programs carried out on-line, but a great deal can be done with off-line facilities. Two difficulties with statistics are that experiments can take a long time and that the resulting computations can become tedious. In both cases the main statistical ideas can get lost in the sheer drudgery involved. One undeniable advantage of a computer is that it does things quickly. It is putting these two facts together that can make computers a useful tool in the teaching of statistics.
The Transition from School to University Mathematics
Authors: Cornelius M. L.
Published: January 2007

There is a widely held view that the jump from school to university mathematics is a big one. Recently James and Fiala have dealt respectively with the interdependence of school and university courses and the teaching of mathematics at university level. It seemed pertinent to obtain the views and opinions of students at the start of their mathematical studies at university.
61.17 The diurnal solar energy falling on the earth’s surface
Authors: Pursey H.
Published: January 2007

In a recent Note [1] a nomogram was described for calculating the times of sunrise, sunset and twilight in terms of the latitude of the observer and the time of year.
61.19 Pascal + Fermat → Gauss
Authors: Häggmark Per
Published: January 2007

Some years ago an interesting article appeared in the Gazette [1] about a young student J. E. Cubbon who had found a theorem on the number of odd coefficients appearing in a row of Pascal’s triangle. Whilst trying to prove Cubbon’s conjecture in my own way, I found an unexpected relation between Pascal’s triangle and the construction of regular polygons.
Notes: 61.23 The skewness of the binomial distribution
Authors: Rainer C. W. O.
Published: January 2007

Many results in A level statistics textbooks are quoted with the observation that the proof is “outside the scope of this book”. The result discussed here, which many students find surprising when they meet it for the first time, is obtained by a method which requires only some knowledge of moments and of the use of a probability generating function in the calculation of the mean and variance. It may be found by the students themselves with a little prompting.
The pigeonhole principle: “Three into two won’t go”
Authors: Walker Richard
Published: January 2007

When I began teaching secondary mathematics after several years in industry, I hadn’t been trained as a teacher. Having been away from school mathematics for so long, I was sometimes nonplussed by pupils’ questions.

One day I was showing a mixed-ability first-year class how to obtain the decimal expansion of a vulgar fraction. After explaining the how and, I hoped, the why of the method I did several examples on the blackboard and concluded by telling them that the expansion would either terminate or recur (or words to that effect). At once a small boy demanded to be told how I knew this. This floored me: I hadn’t got a ready answer and I had to promise to think about it.
Re-designing the dartboard
Authors: Selkirk Keith
Published: January 2007

Have you ever looked at a dartboard and wondered why it is numbered in its distinctive order?

If we ignore the bull, the inner, and the double and treble rings, the standard dartboard takes the form of a circle divided into 20 equal sectors numbered as in Fig. 1. (There are boards to be found in the Manchester area—and perhaps elsewhere—with a different arrangement, but the one shown is by far the commonest.)
Why teach mathematics? 1985 Presidential Address
Authors: Coaker Peter B.
Published: January 2007

As an industrial mathematician, who has done some teaching and has had some connections with the educational establishment, I feel extremely honoured to have been asked to serve the Association, this year, as its President. I feel it is a culmination of the growing links and dialogue between education and employment. I am happy to have been a part of
that process.
Cayley graphs for some well-known groups
Authors: Budden Frank
Published: January 2007

First it may be necessary to remind some readers of what is meant by the representation of a group by a Cayley graph. Each element of the group is represented by a point, the points being connected by lines (or edges) each of which represents a generating operation of the group. Suppose, for example, that x and y are two elements of a group, while another element;? is taken as a generator, and that these are connected: xp = y. (Throughout this article we shall take the identity element of a group to be 1, inverses, to be indicated x-1, while xp means “first x, thenp”.) This would be graphed:
Silver foil wrapping on bars of chocolate
Authors: Dean Peter G. , Scott Nigel
Published: January 2007

This investigation began in the refectory, when we noticed that the unwrapped foil on a four-section Kit Kat did not have its edges parallel to the edges of the chocolate bar. We wondered whether or not the angle between the two had been selected to reduce the cost of the foil by minimising the area covered by more than one sheet of foil, for example by overlap at the corners.
Numerical integration: a teaching approach
Authors: Mills John
Published: January 2007

Many elementary accounts of numerical integration use the trapezium rule and only rarely is the mid-ordinate rule mentioned. This is a great pity for the mid-ordinate rule is often simpler to apply and more accurate.
Changes in the teaching of mathematics in France
Authors: Revuz A.
Published: January 2007

The movement for reform of mathematical education in France, of which the first signs appeared about 1955, had two main objectives:

(a)

to reduce the mis-match which had grown up between contemporary mathematics and mathematical education at all levels;

(b)

to equip all pupils, as far as possible, with a view of mathematics which would enable them to understand and to master the increasing involvement of mathematics across an ever-widening spectrum.
Three progressions in a foursome
Authors: Cundy H. Martyn
Published: January 2007

In their valuable book Board games around the world. Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia. One such investigation gives rise to the following problem:

Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression).

The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.
A circular tour of some circle theorems
Authors: Lord Nick , Wells David
Published: January 2007

The following theorem about circles (which can be traced back at least as far as Archimedes) does not seem to be widely known and may be used to “pep-up” a discussion of circle theorems.

THEOREM. Let two chords of a circle meet at P and make a fixed angle θ (radians, say) with each other. Then the sum of the (opposite) arc lengths, L + L′, is independent of the position of P inside the circle (and thus, taking P at the centre, this sum equals 2rθ, where r is the radius of the circle).
Polynomials with equally spaced zeros
Authors: McLean K. Robin
Published: January 2007

Polynomials like (x + l)x(x - 1) or 5 (x - 2)(x - 4)(x - 6)(x - 8) are particularly easy to sketch because their zeros are real, distinct and equally spaced along the x-axis. As a simple exercise, please picture in your mind the graph of

y=(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)
Lagrange’s inversion formula and function matrices
Authors: Smith Cedric A. B.
Published: January 2007

Lagrange's inversion formula is usually presented in the following form. Let f be a regular (= analytic or holomorphic) complex function with the properties

Then it is a standard theorem that it has a regular inverse function g, such that g (f (z)) = z, with similar properties. (I assume here standard results to be found in appropriate text books.
Seeing is believing
Authors: Smith Cathy
Published: January 2007

Much of the investigational work that pupils do in schools reduces to a routine of generating data, spotting number patterns and describing those patterns [1]. I will refer to this routine as the inductive approach. Many GCSE candidates only use algebraic notation as an alternative translation of what they have already articulated in words. Yet surely the power of algebra is in its potential for providing proof and suggesting further enquiry, based on the similarity of structure between the algebraic model and the original situation.
A modular approach to sixth form mathematics
Authors: Wilson Richard B.
Published: January 2007

Writing in the Mathematical Gazette, MacKinnon [3], pinpoints several factors which are bringing pressure to bear on the development of the sixth form mathematics curriculum for the 1990s and beyond. Some of these factors are not new and their development has been chronicled by Bibby [1] and Wilson [9]. However these recurrent issues are still important and still unresolved. The changes in education pre-sixteen are prompting a reassessment of sixth form mathematics and this provides an ideal opportunity to develop a mathematics curriculum which addresses and hopefully resolves these issues. To achieve this goal I propose a modular approach to sixth form mathematics.
Mathematical people: Roger Heath-Brown
Authors: Heath-Brown Roger
Published: January 2007

Roger Heath-Brown (37) is a Reader in Pure Mathematics at Oxford University, and a Tutorial Fellow at Magdalen College. He works in analytic number theory, with interests covering the distribution of prime numbers, Waring's problem and Fermat's Last Theorem

This term I'm giving a first year lecture course—Groups, Rings and Fields. I'm not at my best at 9 am and despite my efforts the undergraduates find me rather dull. The weaker ones have to have everything spelled out in full, with the result that the interesting things are too sparsely scattered for the stronger students. I'm back in college by 10.30 am having collected a pile of mail. There's a request from the Mathematical Gazette for a “day in the life of” style article, and I toy with the idea of describing an occasion on which a journal asks me to describe a typical day's work …. I also receive a referee's report on a paper submitted to me as an editorial advisor to the London Mathematical Society. The report is not encouraging and I reject the paper—a frequent enough duty, but not one I relish. The associated paperwork fills the next three-quarters of an hour. This particular manuscript is rather too similar to something published in 1963. While double-checking this in “Reviews in Number Theory” I notice an abstract of a quite unrelated paper, by a Finnish mathematician.
Recent calculations of π : the Gauss-Salamin algorithm
Authors: Lord Nick
Published: January 2007

Recent years have seen a marked increase in the tempo of the hunt for digits of the decimal expansion of π as shown in the table below.
A symmetrical pursuit problem on the sphere and the
hyperbolic plane
Authors: Aravind P.K.
Published: January 2007

The following is a well known pursuit problem: n bugs are at rest at the vertices of a regular polygon. At a certain instant each bug starts crawling towards its neighbour on the right with a constant speed, always altering its course so as to be headed directly towards its neighbour. How long does it take for the bugs to meet at the center of the polygon and what paths do they trace out in getting there? The earliest reference I could find to this problem was in a collection of mathematical puzzles edited by L.A.Graham. A more recent reference to it occurs in the amusing and informative collection of geometrical curiosities by David Wells. The “n-bug” problem was also discussed briefly in two of Martin Gardner’s columns in Scientific American. The cover of the July 65 issue of that magazine shows the equiangular spirals traced out by four bugs as they “square dance” their way towards their tryst at the centre.
The probability of finding an adjacent pair in a deck
Authors: Singmaster David
Published: January 2007

Robert Harbin, in Waddingtons Family Card Games, gives an item called Prediction. You ask a friend to name two values of cards and to shuffle a deck. Then you bet that his two values will come up together as the deck is turned over. Harbin says he has worked out the odds as 32 to 19 in favour of success— “but I could be wrong. See what you make it.”
Trial by jury
Authors: Tahta Dick
Published: January 2007

In French law, the 12 members of a jury are not required to return a unanimous verdict. It was decreed in 1831 that a vote of 8 or more would suffice. Four years later, it was proposed to reduce the requirement to 7 or more. There was a heated discussion in the Chamber of Deputies of this proposal which was strongly challenged by Arago, a mathematical physicist. Despite his ardent use of probability theory, the law was in fact changed.
How to ask a question
Authors: Baker A. C.
Published: January 2007

The aim of this note is to discuss the relative values of different ways of posing questions on the same material. I then try to decide which methods are most effective as assessment questions. The environment of examination room, home, availability of books and notes are, of course, crucial to the selection of a method. The time available for marking must also form a realistic constraint on the choice of question.
Getting the right perspective
Authors: MacMillan Robert
Published: January 2007

In his letter in the March 1995 issue of the Gazette (79, p. 123), John Sharp points out that the correct viewing position for a picture is with the eye opposite the central vanishing point, that is, on the normal to the picture plane at the point of meeting of all lines in the scene represented that are perpendicular to the picture plane. In order to obtain a true view of the scene depicted, it is also necessary that the eye be placed at the correct distance from the picture.
The mathematics of bowls
Authors: Roper Tom
Published: January 2007

A little altering of the one side maketh the bowl to run biasse waies

Robert Recorde, The Castle of Knowledge, 1556

John Branfield’s request [1] for the mathematics of bowls was too tempting to resist. What follows comprises a simple model for the curve upon which the bowls move, an explanation of how and why they move as they do and an attempt to refine the model of the curve. The search for a model proved to be addictive and effectively displaced many other activities. One of the reasons for this was the way in which the investigation involved so many people with interesting contributions to make. In order to give a flavour of the investigation, I have taken the liberty of giving a somewhat personal rather than strictly mathematical account, the mathematics being no less important.
The enhancement of Machin’s formula by Todd’s process
Authors: Wetherfield Michael
Published: January 2007

1. Background

From the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series:

which converges for values of tanθ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind is

Writing x = cotθ, the series becomes

Expressions for π/4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’:
Which computer language?
Authors: Macmillan R.H.
Published: January 2007

Any discussion of languages suitable to be taught in school mathematics classes must first attempt to answer the question as to whether computer programming ought to form any part of the mathematics curriculum. The answer is by no means obvious, especially when the view has been expressed by the editor of this journal that “all programming languages (though particularly BASIC) are inimical to mathematical thinking at a higher level ”.
Difference-divisible sequences
Authors: Mason John
Published: January 2007

The sequence 1, 3, 7, 13, 21, … arises in many ways, for example from the integers placed in a spiral on a square grid, the terms arising where the spiral intersects a diagonal on the grid, and as the central axis of a triangular array of numbers:
Approximations to the normal distribution function
Authors: Morland Tim
Published: January 2007

The tide of technological advancement nearly swept away this article. I have had to rewrite this first section in the light of new developments, which I suspected might beat me to publication. For several years I have wondered why ‘pocket’ calculators, however well specified in other respects, have not had the pre-programmed ability to give values of the cumulative distribution function of a standard normal variable, commonly written as Φ (x), and its inverse. The A level mathematics student must be a prime target in the marketing of calculators and there can’t be many who don’t have to thumb through statistical tables a few hundred times in the course of their studies. The very latest generation of machines do include these functions, but it will, I suspect, be quite a few years before everyone is so well equipped.
Looking towards the GCSE applied mathematician
Authors: Whitworth Roger
Published: January 2007

I am looking forward with great enthusiasm to the arrival at advanced level of the GCSE student who has experienced a problem solving approach to mathematics through investigations. I hope examination boards are equally enthusiastic. Here for them is the opportunity to broaden their approach to syllabus content and methods of assessment. Can we hope that they are already working with this in mind? Advanced level mathematics courses, particularly in the options of applied mathematics, will need to nurture the good work of the GCSE courses and allow sixth-forms to produce a student capable of modelling real problems, in terms of sound and necessary mathematical techniques.
Cayley's theorem and a method for approximating e
Authors: MacKinnon Nick
Published: January 2007

A tree is a network of vertices connected by edges such that the removal of any edge disconnects the network. In the language of graph theory a tree is a connected simple graph in which every edge is a bridge. Here are all the trees with 4 vertices.