bluuebunny/crossref_metadata_2025 huggingface dataset:
On Langford’s Problem (I)
Authors: Priday C. J
Published: January 2007

For numbers a ≥ b ≥ l we shall denote by (a, b) the set of numbers b, b + 1, …, a. We shall say that a set S of numbers is perfect if there exists a sequence containing just one pair of each of the numbers in S, satisfying the condition: for every number r in the set, the two r’s are separated by exactly r places, and having no gaps (a perfect sequence).
Geometry and Girard Desargues
Authors: Swinden B. A.
Published: January 2007

It is a far cry from the Conics of Apollonius to the theory of linear spaces and matrix algebra and many men have helped to build the road between. This article is about one of the builders whose work for a time was discarded and whose name was almost forgotten. But his name finds frequent mention now, when the latter aspect of geometry bears so little resemblance to the former, and metrical geometry has given place to projective geometry with its non-metrical homogeneous coordinates, which make it possible to publish a book on coordinate geometry which does not contain a single figure.
Some New Theorems on Divisibility
Authors: Goodstein R. L. , Rumney M.
Published: January 2007

Introduction. If we call the number pa + qb the Tp,q transform of the number n = 10a + b, then in general the Tp,q transform of n will differ from n; it easily seen, however, that given any number n, we can always find a pair p, such that the Tp.q transform of n is n itself. In fact, if n = 10a + b and p = 10 − b q = a + 1 then the Tp,q transform of n is (10 − b) a + (a + 1)b = 10a + b = n.
On the Multiplication of Complex Numbers
Authors: Newing R.A.
Published: January 2007

In the development of the theory of complex numbers, it is important to give a definition of them dependent only upon real numbers. In the usual algebraic treatments the product is postulated in the form

or obtained from a matrix representation. The object of the present note is to show by elementary methods that the assumption that the multiplication of complex numbers is distributive, associative, and such that the vanishing of a product implies the vanishing of at least one factor, is sufficient to define the product. This is indeed merely a special case of results well known in the theory of linear algebras.
Industrial Mathematics
Authors: Citninghame-Green Raymond
Published: January 2007

With his unaided faculties, a man is comparatively limited in performance. His vision, hearing and legs operate only over small distances, he cannot lift very heavy weights nor weave much cloth in an hour. But by making use of devices such as the aeroplane, the telephone, and the powered loom, he can exploit natural properties and forces, and thereby attain what is literally a superhuman standard of performance in all kinds of activities.
Free Variable Axioms for Groups
Authors: Goodstein R. L.
Published: January 2007

The growing importance of the elements of group theory in sixth form teaching has been recognised by a number of recent publications, but a very simple presentation of the theory of groups by means of free variable axioms which was introduced by Lorenzen more than twenty-five years ago, and which is particularly well suited for elementary teaching, does not appear to be as well known as it deserves to be.

Lorenzen defines a group in terms of divisionx/y, instead of multiplication, a device which obviates the need to postulate the existence of a neutral element and an inverse.
Junior and School Mathematical Societies
Authors: Sherlock Alan
Published: January 2007

In the Gazette (176) for May, 1925, Miss J. M. Brown wrote a delightful article on “ Mathematical Clubs ” in Schools. It is interesting today to recall some of the ideas mentioned.

“ The obvious function of a Mathematical Club is to stimulate the interest which may have been aroused in the classroom, or if possible to awaken an interest hitherto dormant.”

The History of Mathematics was considered a starting point but was not found to be a very suitable topic for arousing enthusiasm. Mention is made of graph work (there was no Cundy and Rollett, Mathematical Models, in those days)—and of interest shown in Flatland (Dr. Abbott).
Vectors: A Special Case?
Authors: Mansfield D.E. , Bruckheimer M.
Published: January 2007

J. Cable [1] suggests that if the word “vector” is going to be introduced into secondary school mathematics then it should at least be done properly Probably nobody would explicitly disagree with this, although there is a fair amount of evidence that implicitly many have and still do. But why must “vector” be introduced properly? Why the special fuss about vectors? The basic mathematics necessary to introduce vectors properly is common to many other mathematical ideas. (We wish to make it clear at the outset that our remarks in this article only refer to the printed text of the paper mentioned above. We do not wish to imply, for instance, that Mr. Cable does make a special fuss about vectors, but that the implication of his article is that vectors deserve special attention.)
The Cubic Curve and an Associated Structure
Authors: Macnab D.S.
Published: January 2007

The purpose of this article is to show how elementary properties of the cubic curve can give rise to an abstract structure with interesting properties and to investigate some of the simpler of these.
Lanchester’s “Potted Logs”
Authors: Wooldridge R.
Published: January 2007

The collection of papers and books of the late Dr. F. W. Lanchester, F.R.S., held in the library of the Lanchester College of Technology, Coventry, contains a paper cover booklet entitled Lanchester’s “Potted Logs”, Parts I and II, printed and published by Taylor and Francis Ltd., Red Lion Court, Fleet Street, E.C.4, in 1939, priced 2s. The booklet measures 4ξ × 7¼ and consists of 27 pages devoted to “a concise tabulation (slide rule auxiliary)” of common and natural logarithms for engineers. In Part I (14 pages) “a new system is disclosed by which the Logarithms of Numbers running to seven or eight figures may be evaluated to seven places of decimals and vice versa” Of these 14 pages, 5 only are required for the tables, the remainder being devoted to preface, introduction and worked examples. In Part II (8 pages, 4 of which are tables) the system is extended to cover natural logarithms. The remaining pages contain a note on the musical scale (“of interest to the musician, it is a table in antilog form to the base 1-059463 . . and is the foundation of the system of equal temperament”) and lists of papers, lectures, reprints, reports, memoranda and books by Lanchester.
Non-Convex Pentahedra
Authors: Norgate M.
Published: January 2007

Three straight lines are needed to enclose a finite region of a plane, a two-dimensional space. The polygon formed is a triangle. Different types of triangle are described by adjectives; scalene, isosceles, equilateral and acute angled, right angled, obtuse angled. All the triangles have a property in common: they are all convex.

Four lines form a quadrilateral. The convex examples are well known. There are two further types: those which are “re-entrant” having an interior angle greater than 180 degrees, a reflex angle; those in which a pair of opposite sides cross within the quadrilateral, a “crossed” quadrilateral.
An Old Puzzle
Authors: Lockwood E.H.
Published: January 2007

This puzzle is described in Wallis’s Algebra (Cap. CXI), with detailed instructions both for making it and solving it. The apparatus consists of two parts, which are easily made from simple materials, as follows:
Economical Number Bases
Authors: Phythian J.E.
Published: January 2007

Throughout this article, all numerals are assumed to be in base ten notation, unless otherwise stated.

Under many of the new mathematics programmes currently in use in secondary schools, the topic “Numbers to different bases” is introduced. Indeed, provided such an elementary textbook series has chapters on Sets, Counting to different bases, Symmetry and/or Transformation Geometry, and, possibly, Vectors and Matrices, it is then allowed to write “Modern Mathematics” on its title page almost regardless of the methods employed in the book. However, my topic does not concern the vagaries of modern mathematics textbooks, their content and their methodology, but rather the question of why ten is used as a number base. Undoubtedly many of you already feel that you know, and that the reason is concerned with numbers of fingers and toes. However, one might try to decide more rationally whether ten is a sufficient base to represent the wide range of numbers with which we are normally concerned.
A Mathematician’s Experiences as a Member of an Industrial Research Group
Authors: Metzer K.
Published: January 2007

It has been suggested in The Listener lately that industrial mathematicians may have something to contribute to mathematical education by describing some of their work in articles to be published in The Mathematical Gazette. As the personal approach is often more graphic and interesting, I have decided to set down an outline of some of my experiences as a member of an industrial research group.
The Rate of Growth of Mathematics During Four Centuries
Authors: Wallis P. J.
Published: January 2007

The exponential law of increase has been described by Professor D. J. de Solla Price as
“the fundamental law of any analysis of science”, and considered to hold over long periods of
time [1]. The same writer suggested that low-grade work doubled in about ten years, but that
the period taken was about twice as long for very high quality activity. It is useful to
remember that a doubling every fifteen years corresponds to a multiplication by about a
million in three centuries, or an annual increase of nearly five per cent.
A Problem in Projective Geometry
Authors: Rigby J. F.
Published: January 2007

In his article “A Problem in Projective Geometry” (Gazette, December 1960) T. G. Room says that he has not been able to find a projective geometric proof of his theorems. May I give the following proof?
Some Problems in Combinatorics
Authors: Forder H. G.
Published: January 2007

The following problems are merely different forms of one

1. In how many ways can a convex polygon of n + 1 sides be split into triangles by non-intersecting diagonals? We represent such a splitting by a symbol. Take one side of the polygon as base and letter the other n sides in order by a, b, c,…. If a triangle with sides lettered x, y is cut off, denote the diagonal by (xy). If this diagonal and side z form the sides of another triangle of the dissection, denote the third side of that triangle by ((xy)z). Similarly for two diagonals, as the figure illustrates. Finally the symbol for the unlettered base is the symbol for the dissection.
Robbins—and all that
Authors: Busbridge I. W.
Published: January 2007

Ladies and gentlemen—It has become customary for the President of the Association, when a university teacher, to speak of some aspect of mathematics in which he or she is an expert. When the Council of the Association did me the very great honour of inviting me to preside over this conference, the first, I believe, ever to be held in Oxford or Cambridge, I decided to break away from tradition. This decision was strengthened when I knew that I was to have the privilege and responsibility of addressing you in the Shel-donian, which is furnished with no apparent visual aids.
Envelopes of Zylbertrest-Lines
Authors: Steller E. T.
Published: January 2007

S. Zylbertrest in his interesting generalisation of the Wallace-line has opened the geometric gate to a new field of research. D. G. Tahta then followed with an algebraic approach and derived a few theorems from his equation of the Zylbertrest-line.
The Representation of Lines by Dual Vectors
Authors: Eastham M. S. P.
Published: January 2007

In Note 2968 a definition of dual numbers a1 + ∊a2, where a1 and a2 are real numbers and ∊2 is put equal to zero whenever it occurs, is given. Similarly, dual vectors are expressions q1 + ∊q2, where q1 and q2 are ordinary three-component vectors (see [1]). The dual vector q1 + ∊q2 is said to be proper if |q1| 0, and normal if |q1| = 1, q1 . q2 ≠ 0.
The Newsom Report—“Half Our Future” A Commentary on Chapter 18: Science and Mathematics
Authors: Eperson Canon D. B.
Published: January 2007

Few could cavil at the general views expressed about the place of mathematics in schools by the writers of the Newsom Report, as most of the ideas have been advocated in many Training Colleges for at least ten years. It is strange, however, to see that the section devoted to mathematics is the second part of Chapter 18 on “Science and Mathematics.” Does not this lend colour to the heresy promulgated by Hogben in “Mathematics for the Million” that the subject is to be valued principally, if not entirely, as the “Handmaid of Science”? Everyone will agree that “Mathematics has much to offer to boys and girls of average ability at the Secondary stage. It has a self-evident usefulness which can be a great advantage in arousing the interest of pupils.” Is it true that the only appeal to such pupils is through experience of the practical utility of mathematics in the world around them—the modern world of technology and science and commerce? Doubtless we need more and better mathematicians in all such activities, but they will not be drawn in any large numbers from these pupils. Numbers, shapes and patterns have an intrinsic fascination of their own that is independent of their practical value. For instance, it might be argued that the fact that “all numbers divisible exactly by 9 have a digit sum that is a multiple of nine” may be of some practical value—it was the basis of the process of “casting out the nines” that was formerly advocated as a quick check on the result of a calculation. Yet the pattern of the “nine-times table” is of aesthetic value as well as an aid to memory. Is the test to discover whether a number is a multiple of 11 of much practical value? Lewis Carroll used it to devise a method of dividing by 11 that depends solely on addition and subtraction, but its real value is the insight into number relations that is gained from demonstrating its validity: it certainly has an appeal to most people as a kind of “mathematical magic.” Do you know the test for discovering whether an amount of money expressed in shillings and pence is divisible exactly by 11?—just add the number of shillings to the number of pence, and if their sum is a multiple of 11, the amount of money is divisible by 11. There is an equally simple test for an amount of money to be exactly divisible by 13—can you find it? Experience shows that facts such as these stimulate the interest of those who dislike arithmetical calculations, whether young people or adults: the film-strip “Alice in Numberland” was designed to undermine the prejudice that many women students have against mathematics, by showing Lewis Carroll’s delight in puzzles and paradoxes, and his humorous approach to Symbolic Logic. Even in Secondary Modern Schools at all levels it should not be forgotten that mathematics can have an “entertainment value” comparable to that of music or any other form of art, if presented in the right way.
A Generalized Permutation Problem
Authors: Goodstein R. L.
Published: January 2007

In Note 2590 (Gazette 1956) I gave a very simple method for constructing a ring of 2n 0’s and 1’s in such a way that each set of consecutive n digits formed a unique number in the scale of 2. For instance with n = 4, the cycle 0111101100101000 provides in turn the numbers 0111 = 7, 1111 = 15, 1110 = 14, 1101 = 13, 1011 = 11, 0110 = 6, 1100 = 12, 1001 = 9, 0010 = 2, 0101 = 6, 1010 = 10, 0100 = 4, 1000 = 8, 0000 = 0, 0001 = 1, 0011 = 3.
A Circular Tour of Some “A” Level Mathematics
Authors: Simpson E. J.
Published: January 2007

In a previous article on the Spirograph, published in The Mathematical Gazette of December, 1966, I suggested that it could be used at sixth form level to illustrate some advanced work on cartesian coordinates. Following this up, I have found a number of links with calculus, algebra and trigonometry.
What Can Be Taught in Primary Schools?
Authors: Williams E.M.
Published: January 2007

Recent changes in the content of some Secondary School courses in mathematics have been brought about chiefly to bridge a discontinuity which is often found between the mathematics presented to undergraduates and the examination requirements for sixth forms. It is mathematicians who have been the leading advocates of change at this stage. There has been no such pressure from Secondary Schools to Primary, though the discussions on new Secondary programmes have roused a considerable interest among Primary teachers and have already begun to influence the changes which are taking place in so many Primary Schools and at such great speed.
Higher Purchase
Authors: Love E. R.
Published: January 2007

What is the effective rate of interest when a loan of £L is paid for by n equal payments of £P at equal intervals of time, as in hire purchase and other forms of time payment?
Estimating Bounds on Athletic Performance
Authors: Deakin Michael
Published: January 2007

A sequence of athletic records forms by definition a monotonic sequence. Since it is reasonable to assume this to be bounded, it follows that a limit must exist to future performance. It has been proposed [1, 2] that this can be estimated by use of a curve-fitting procedure on existing data, such as may be found in many compendia of athletic records (e.g. [3]).
A Note on Derangements
Authors: Bizley M. T. L.
Published: January 2007

One of the best known of all classical elementary combinatorial problems is the enumeration of the derangements of n letters A1 A2,…, An, i.e., the permutations of these letters in a row so that, for every i, At is not the ¿th letter. The question is frequently presented in the form of an office boy who puts n differently addressed letters into n envelopes; we then seek the number of ways (w„) in which he can send every letter to a wrong address. Textbooks almost invariably solve the problem by deducing from general reasoning the recurrence relation :
A Simple Method of Constructing Certain Magic Rectangles of Even Order
Authors: Phillips J. P. N.
Published: January 2007

Magic squares might reasonably be supposed to be amongst the most useless of mathematical diversions. Yet, like other apparently fruitless amusements such as Eulerian squares and Kirkman’s school-girls problem, they have found applications in the field of the design of experiments. In an investigation of the effects of two variables upon a phenomenon, where the observations are obtained one after the other, magic squares make it possible to take account of linear trend in the results. For example (the writer is only competent to offer illustrations in the field of psychology : others in bis own field will doubtless occur to the reader) Miller and Selfridge investigated how much subjects could remember of series of words which varied both in their length and the degree to which they resembled everyday English. Again Fitts and Annett, Golby and Kay investigated how fast a subject could alternately tap with a metal stylus two separate metal plates, and how his speed was affected by the width of the plates and the distance between them. In both cases the subjects performed the task a number of times under different combinations of levels of the variables (length and resemblance to English, and width and distance respectively), so that any analysis of the data is liable to be vitiated by the fact that each subject will probably improve with practice throughout the experiment, unless special account is taken of this trend.
Going one Better in Geometric Dissections
Authors: Lindgren H.
Published: January 2007

Methods have been described that facilitate the discovery of economical dissections. They are summarized in §§ 1 and 2 that follow.

1. Each of the figures to be dissected is made an element of a strip with parallel sides; 1 and 2 show strips formed from a Latin or tau cross and a pentagon. The first is a P-strip (prototype parallelogram), in which all elements are the same way up. The second is a T-strip (prototype trapezium), in which alternate ones are inverted. Points such as A and B in 1 are called congruent, a term copied from Whittaker and Watson's Modern Analysis, p. 430.
Maestro Puzzles
Authors: Reeve J. E. , Tyrrell J A.
Published: January 2007

A few years ago there appeared on the commercial market a certain puzzle called “Maestro”. This consists of packing into a 6 x 10 rectangle the twelve shapes depicted in Fig. 1, (where they are shown packed into an annular region bounded by two concentric squares). It is also possible to pack these shapes into other simple geometrical figures. The aesthetic attraction of this problem lies in the fact that the twelve shapes concerned form a complete set in the sense that any other shape consisting of five abutting squares is congruent to one of them.
A Souvenir from Paris
Authors: Cundy H. Martyn
Published: January 2007

Two years ago I was in Paris for the New Year. The shops were gay with decorations, and polyhedra were obviously in fashion. I was particularly impressed with a lampshade which was a feature of many of the decorations; it was in the form of an icosidodecahedron, the pentagonal faces being dimpled, and with “petals” added, so that the whole resembled the flower of a Golden Chalice (Solandra grandifloral). Eventually I found a little shop by Notre Dame which would sell me one; I found myself the purchaser of a small box containing twenty strips of card, about 35 cm by 7 cm, which I was assured would make up into the lampshade I required.
Transcendentals galore
Authors: Wilker J. B.
Published: January 2007

The natural numbers are the elements of the set N = {1,2,3,…}, the integers are the elements of the set Z = {0,± 1,±2,…} and the rational numbers are the elements of the set Q = {p/q:pεZ, qε6 N}. A real number is called algebraic if it is the root of a polynomial equation

P(x) = anxn + an−1xn−1 +…+ a1x + a0 = 0
Integer-sided triangles with an angle of 60°
Authors: Gilder John
Published: January 2007

We are looking for all triangles with sides of integer lengths a, b, and c say, with an angle of 60° opposite the side of length a.
Primary mathematics: towards 2000: 1986 Presidential address
Authors: Shuard Hilary
Published: January 2007

It is a great pleasure to welcome the Mathematical Association to Cambridge, which since Newton has always been one of the most important homes of mathematics in this country. Cambridge is a traditional home of advanced and research mathematics, and it therefore gives me particular pleasure to have the honour of giving my Presidential Address to the Association in Cambridge on the topic of primary mathematics the foundation and growing point of the ‘mathematics for all’ on which our increasingly scientific and technological culture makes greater and greater demands.
The mathematics of scout’s pace
Authors: Drummond J. E.
Published: January 2007

Scout‘s pace, according to my old boy scout manual, consisted of alternately walking and running 22 yards. I believe it is also used by military scouts and messengers who wish to travel at a moderate speed without too much effort. As I have also discovered, it is useful for elderly, overweight athletes with a low pain tolerance.
How many times does n go into n ǃ
Authors: MacKinnon Nick
Published: January 2007

Sometimes nǃ is very divisible by n. For example, 12ǃ is divisible by 125 while 720ǃ is divisible by 720178. On the other hand, if p is prime then pǃ is divisible by p but not by p2.

In general it is quite difficult to calculate the maximum exponent m(n) such that nm(n) divides nǃ

In this short article I will locate some values of n, for which nǃ is very divisible by n, find a tight upper bound for m(n), and point the reader in the direction of related problems.
Some triangle identities
Authors: Clarke Robert
Published: January 2007

Consider any point P inside ΔABC. The trilinear coordinates of this point are p1, p2 and p3, which are the perpendicular distances from P to the sides BC, CA and AB respectively.
Runs in Bernoulli trials
Authors: Colwell D. J. , Gillett J. R.
Published: January 2007

In a sequence of Bernoulli trials suppose the outcomes are either successes (S) or failures (F). A list of these outcomes will consist of runs of S’s and F’s and it is the purpose of this article to investigate such runs. Initially the investigation will be pursued theoretically. Then the theoretical model will be applied to data provided by the Test Matches which have been played between England and Australia for many years, to give a comparison between the expected results and those that have actually been achieved.
It doesn’t work
Authors: Barton Alan
Published: January 2007

“The total work done by the external forces acting on a rigid body is equal to the change in the kinetic energy of the body.” O.K. ? Well, it all depends on definitions.

The work done by a constant force is usually (in nearly all textbooks ?) defined as the product of the magnitude of the force and the displacement of its point of application, this displacement being measured in the direction of the force. Consider then a sheet of paper on which a student has drawn in pencil a straight line which he is now about to erase. He causes the eraser as it moves along to exert a constant force on it, and he exerts another constant force with his other hand to keep the paper still. The desk (supposed smooth) exerts constant normal forces. Of all these forces on the paper the first force does some work, according to the definition, while the others do none. The total work done by them is positive, yet there is no change in kinetic energy of the paper.
Investigational work at A-level
Authors: Fielker David S.
Published: January 2007

I read the first page of John Hersee’s article in the June 1984 Gazette and sat down immediately, “stimulated to contribute comments” on his question. “What do we mean by ‘investigationary work’… in the context of A-level?”
Applications of Euler’s formula to partition problems
Authors: Detemple Duane
Published: January 2007

In this article several combinatorial problems are posed for the partitioning of certain plane figures. The solutions are obtained by means of Euler’s formula for plane graphs or networks, V - E + R = 1, where V is the number of vertices, E is the number of edges, and R is the number of bounded regions of the figure. Being unified by the common thread of Euler’s formula, the sequence of problems forms an interesting unit on solving related problems.
Planetary orbits and approximations
Authors: Edgar S. B.
Published: January 2007

How refreshing to read C. D. Collinson’s [1] article in the March 1986 Gazette on the importance of stressing approximations in teaching mechanics. Once students get over their initial scepticism that mechanics has anything to do with the real world, I too find an enthusiastic response from classes invited to contribute their ideas on the sort of approximations and assumptions which have to be made to model a physical situation. Approximations and assumptions are fundamental to the art of mathematical modelling in mechanics and cause more difficulties to students than is often realised; class discussions reveal that many of the problems which students have with mechanics are related to approximations, and such discussions can also bring out some unsuspected fundamental misconceptions. One such misconception, related to approximations, is discussed here and the proposed method of clarification includes developing a little more Collinson’s use of “the linear approximation”.
Mr Peter Nicholson, The Practical Builder and Mathematician
Authors: Lloyd E. Keith
Published: January 2007

I am unable to answer Ian Cook’s query (Gazette, note 65.25) concerning the original discoverer of the construction which he quotes from Nicholson’s Carpenter’s new guide, but readers may be interested in learning more about Peter Nicholson, who is variously described in his own books and in biographies as: architect, carpenter, surveyor, practical builder and private teacher of the mathematics.
Last words on adventitious angles
Authors: G. A. Q.
Published: January 2007

The June 1975 Gazette (59, No. 408, 98-106) carried an article by Colin Tripp about the so-called “adventitious angles” problem: in the isosceles triangle in Fig. 1, for certain given values of the angles marked a, b, c, to find θ. Correspondence to which this gave rise was summarised in a “progress report” in March 1977 (61, No. 415, 55-58). This cleared up certain points raised by Tripp, using trigonometric methods; and its publication also had the desired effect of stimulating some further response, which has settled most of the outstanding questions. It seems now to be an appropriate time for the editor to produce a “final summary” for those readers who have taken an interest in the problem.
Two group isomorphisms and a little projective geometry
Authors: McDonough T.P.
Published: January 2007

Among the first examples of groups encountered in group theory are the symmetric and alternating groups and the general linear groups. The purpose of this article is to describe these groups and to describe two unusual functions between groups of this type, in a language which can be understood by those with only an elementary knowledge of group theory and no knowledge of linear algebra or projective geometry.
Reduction of a square matrix to triangular form
Authors: Gerrish F.
Published: January 2007

Given a square matrix A with real or complex entries, there may not exist an invertible matrix S such that S-lAS is diagonal. For instance,

cannot be diagonalised in this way (the contrary assumption leads by easy direct calculation, to a contradition); however, this matrix is already upper-triangular.
More means worse: A comment on university expansion
Authors: Ord Keith
Published: January 2007

The slogan which forms the title of this note was coined in the 1960s by the opponents of large scale university expansion. It was maintained that an increase in student numbers could only lower academic standards. The purpose of this note is to show that more can mean better or worse according to the criterion used. The theorems proved in the main part of the paper give formal statements of intuitively obvious results, but the amount of misunderstanding generated on this topic suggests that such an exposition is worthwhile. The conclusion of the note must be that comparisons of academic achievement, based on varying proportions admitted to higher education, should be interpreted with great care.
Reasonable elections don’t exist!
Authors: Baylis John
Published: January 2007

In 1972 Kenneth Arrow was awarded the Nobel prize for economics. Much of his work was in the theory and philosophy of decision making, in particular the theory of voting, and in this field “Arrow’s theorem” is a truly remarkable result. It is a genuine application to the social sciences of some elementary mathematics and as such I feel that it is a pity that it is not in the repertoire of more teachers. Hence this article.
A statistical paradox—(2)
Authors: Downton Frank
Published: January 2007

The apparent paradox posed by Stephen Ainley (Gazette, December 1982, note 66.43) arises from two interconnected causes.
Self-descriptive strings
Authors: Mckay Michael D. , Waterman Michael S.
Published: January 2007

Take a look at the ‘string’ (or finite sequence)

6,2,1,0,0,0,1,0,0,0.

Count the occurrences of the numbers:

Notice that the bottom row is the string we started with! In fact the above string is the unique answer to the popular question (posed, for example, in a recent edition of the Trans World Airlines magazine, The Ambassador, August 1978) to find a string
Einstein and the art of coarse mathematics
Authors: Glenn J. A.
Published: January 2007

It is now widely accepted that the first or “kinematic” part of Einstein’s 1905 paper on Relativity [1] can be made accessible to interested non-specialist students. This part is mathematically straightforward, although its first readers found it conceptually puzzling. “From this”, remarks Einstein, as he follows one of the simpler results of applying the Lorentz transformation to our measures of space and time, “there ensues the following peculiar consequence…” It was these “peculiar consequences” that repelled many physicists of his own day. Later, the challenge to intuition caught public imagination and led to the many popular and semi-popular introductions to the Special Theory. Today there are also several accounts available at a mathematical level appropriate to sixth form or college students.
The shapes and sizes of Pythagorean triangles
Authors: Shiu P.
Published: January 2007

We shall try to show that, in some well-defined sense, every right-angled triangle is ‘approximately similar’ to a Pythagorean triangle: and then we’ll ask how large the Pythagorean triangle must be.
Jack-in-the-box
Authors: Helfgott A. , Potts R. B.
Published: January 2007

A few years ago, a well known laundry detergent manufacturer ran a sales-promotion competition in Australia called “Jack-in-the-box”, with the following rules:

1.

Draw a continuous line through 13 of the numbered squares (see Fig. 1). The numbers in the squares through which the line has been drawn are then added to give a total score. Entries achieving the highest total scores will qualify for one of the prizes.

2.

The line must not go through any square more than once. It also cannot be drawn diagonally through the corner of any squares_only up and down or across (see the example in Fig. 1, for which the score is 723).

3.

The line must begin at the square J (for Jack-in-the-box) and may finish in any square. The J square does not count as one of the 13 squares.
An application of catastrophe theory
Authors: Hughes Anthony
Published: January 2007

Popular expositions of the new subject of catastrophe theory have tended to concentrate on ‘soft’ applications, but, as I hope to show, it is perfectly possible to make a serious, important, and mathematically more demanding application intelligible to sixth formers with a minimum of differential calculus. The application is in that part of engineering science which deals with the stability of structures. Catastrophe theory produces nothing really novel in this field, but, by assuming Thorn’s theorem, which will be explained below, it is possible to make the whole subject accessible to those for whom Taylor’s theorem for a number of variables, and the associated concepts, are too difficult. Even for engineers already familiar with the results, catastrophe theory offers an attractive unity within the subject and can point to analogies in related fields.
On mountain tracks and motorways, men and mandarins
Authors: Flestrin Quinbus
Published: January 2007

. . . In one of the lands I visited I met a young explorer called Noisrot, who worked in the mountainous and undeveloped central regions of Spuorgetinif. He was one of many government employees who were charting these unexplored regions, looking for minerals and for passes in the rugged central mountain range towards the north, for it was hoped one day to open up the interior. Whenever possible Noisrot lived in the wild. He spent his days walking in the hills, listing the flora and fauna, collecting specimens, and surveying; his evenings were spent writing reports and reading those prepared by his colleagues.
Groups in modular arithmetic
Authors: McLean K. Robin
Published: January 2007

It has often been noted that modular arithmetic provides a rich source of supply of groups. Indeed, a remarkable theorem asserts that any finite commutative group can be found by a sufficiently diligent search through the multiplicative groups and subgroups of modular arithmetic! During the last few years two articles in this area have appeared in Mathematics Teaching. In the first [1], Tim Brand drew attention to the fact that such a multiplicative group can have an identity element other than 1. (For example, 8 is the identity of the group {2,4,8} under multiplication mod 14.) More recently [2] Geoff Saltmarsh described an ingenious way of finding the identity and put forward an interesting conjecture about these groups.
Constructing a ring without unique factorisation
Authors: Fletcher Colin R. , Lidster M.
Published: January 2007

The study of unique factorisation is almost as old as mathematics itself. The so-called Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be factorised into a product of primes in only one way, was probably the first major theorem proved. It is interesting to note that Euclid (c. 300 B.C.) did not give the result in this form. He proved (IX 14) that “if a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it” (see [1]).
Napoleon revisited
Authors: Nelson R. D.
Published: January 2007

If on the sides of a triangle ABC equilateral triangles are constructed, inwardly and outwardly, and their circumcircles drawn, the 6 circles will in general have 11 points of intersection: A, B, C; 2 points on each of the 3 sides; a point F common to the circumcircles of the outer triangles, and a point N common to those of the inner triangles (Fig. 1).

This article uses complex numbers to investigate the figure, and has as its main object the establishment of an extremal property of N.
Further Logic Diagrams in Various Dimensions
Authors: Collings S. N.
Published: January 2007

Given n dimensions, the obvious way of representing n classes is by a hypercube of side 2. Such a hypercube contains 2n unit hypercubes, one for each combination of the n classes. With only one dimension available, it is still possible to represent each of the 2n combinations. Take a line OX with any point P1 marked on it. The interval OP1will represent all combinations excluding the first class A1 and the interval P1X all combinations containing it.
A computer verification of a pretty mathematical result
Authors: Leonard Bill , Shultz Harris S.
Published: January 2007

A very appropriate use of a computer in a mathematics class is to provide investigation vehicles for formulating conjectures to be verified theoretically. In other words, first we notice something and then we try to prove it. In this paper we shall reverse the process, first arguing the plausibility of a very fascinating result and then obtaining confirmation by looking at some computer generated data. The argument is based on the work of the eighteenth century mathematical giant, Leonhard Euler (1707-1783).
Equilateral triangles and the golden ratio
Authors: Rigby J. F.
Published: January 2007

This article could be subtitled ‘Thoughts on contemplating a model of five tetrahedra inscribed in a dodecahedron’; it is an attempt to communicate the pleasure that ensues when logical reasoning is combined with a visual delight in geometrical figures.
Fibonacci power series
Authors: Glaister Paul
Published: January 2007

A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.
Triangles and tetrahedra, quadrilaterals and cubes: A classroom investigation
Authors: Gardiner Tony
Published: January 2007

One way to make a tetrahedron is to use four equilateral triangles. Let’s see why this works. When we fit two equilateral triangles of the same size together along an edge, the edges match up.
Modular arithmetic and cryptography
Authors: Reade J. B.
Published: January 2007

Even the purest of pure mathematics can have a crucial influence on practical problems. In this article we show how a topic in pure mathematics (modular arithmetic) originally pursued for its own interest only, turns out to have unexpected application to an area of communication theory (cryptography). The fact that at the present time it is easy to construct large prime numbers but very difficult to factorise large composite numbers has made it possible to devise simple codes which are uncrackable by known methods.
Sums of powers of integers—a general method
Authors: Crilly Tony , Millward Stanley
Published: January 2007

This perennial problem has led to many articles in the Gazette (listed by Sullivan in the June 1987 edition) many of them geometric or combinatorial in nature. If we look at some older algebra texts, published before the 1960s say, we see that the method of differences is one of the standard methods for dealing with the problem. With the current interest in discrete mathematics this method merits re-examination. A feature of the method is its effectiveness in finding the solution to the problem for all powers and it is not limited to the sums of squares and cubes. In this article we hope to show that it can be treated more transparently using the calculus of finite differences.
0 = x−x
Authors: MacHale Des
Published: January 2007

Everyone knows that x−x = 0, but how many people realise the power of reversing this equality and writing 0 = x−x? What I mean is that we can always add and subtract the same quantity to and from any algebraic expression without altering its value. In general, we may choose infinitely many values of x, and the skill lies in choosing the most suitable x so that progress can be made in a specific problem. Beginners in mathematics are often mystified when a particular x is chosen and almost invariably respond with something like “I agree that what you are doing is correct, but why on earth did you pick that particular x?” There is no answer to this except perhaps “because it works!”.
A new Ceva-type theorem
Authors: Grünbaum Branko , Shephard G. C.
Published: January 2007

The classical theorems of Ceva and Menelaus make assertions about the value of certain products of ratios of lengths in configurations in the affine plane. We shall use the term Ceva-type to describe any result of this general kind: one that specifies a configuration in affine space of n dimensions, defined only by incidences, about which one can make an assertion about a product of ratios of lengths, areas, etc. Several results of this kind are known. Apart from the classical results there are, for example, Ceva's and Menelaus' Theorems for n-gons, Hoehn's Theorem for pentagrams [1], and the Selftransitivity Theorem of [2].
Minimal Steiner trees for three dimensional networks
Authors: Bridges Richard
Published: January 2007

I was intrigued by Brian Bolt’s note “The Home Stretch”, as it had not occurred to me to look at Steiner trees spanning networks in three dimensions before. Nor did I find any references to them in the (admittedly limited) literature I consulted, though plenty has been written about the two dimensional case (see, eg, Gardner, Wells, MacKinnon). I was quickly able to improve on Bolt's tree, and decided to investigate networks for other shapes.
Integer geometry: some examples and constructions
Authors: Peterson Blake E. , Jordan James H.
Published: January 2007

Integer Geometry is the geometry of points that are pairwise integer distances apart. An integer polygon is a convex set of n points in the plane such that no three points are collinear and the distance between any two points is an integer. Similarly an integer polyhedron is a convex set of n points in space such that no three are collinear, each face is an integer polygon, no two faces are coplaner, and the points are all integer distances apart. The measure used to order such polygons and polyhedra is perimeter-plus which is the sum of all of the edges added to the sum of all of the diagonals [1]. The purpose of this paper is to provide pictures of some fundamental examples as well as a brief explanation of some of the constructions. We begin by looking at polygons and then use those to build polyhedra and conclude by examining two closely related examples of sixfaced polyhedra.
“ … Assume the string is inextensible and elastic…”
Authors: Roper Tom , Hartley Ron
Published: January 2007

If the above phrase appeared in an A-level paper the Board responsible would receive many protests: “a contradiction in terms”, “impossible”. But is it? Our attention was drawn to this via this A-level problem from the J.M.B.

Three smooth spheres A, B and C, of equal radii and masses m, 2m and am, respectively, lie at rest in a straight line on a smooth horizontal table, with B between A and C. The sphere is connected to A and C by light inelastic strings, which are initially slack. The sphere A is then projected directly away from B with speed u. Find, in terms of a and u, the common velocity of the three spheres when C has been jerked into motion.

Find also in terms of a, m and u, the impulse in each string at the instant when C starts to move. Verify that the ratio of these impulses is independent of a.

(Further Maths, Paper II, 1988)
Triangles, surds and pell’s equation
Authors: Clarke Robert J.
Published: January 2007

This article shows a link between Pell’s equation and the right-angled triangles for 60° and 45° via the trigonometric ratios for 15° and the extraction of square roots.

It is well known that the trigonometric ratios of 60°, 30° and 45° are easily obtained from the triangles ABC and RST which are shown in Figure 1.
George Green, mathematician and physicist 1793 – 1841
Authors: Cannell D.M. , Lord N.J.
Published: January 2007

These past two years have seen the bicentenaries of Michael Faraday, Charles Babbage and John Frederick Herschel. A fourth contemporary, who deserves to rank with these, is George Green, the bicentenary of whose birth will be marked by the dedication of a plaque in Westminster Abbey. His memorial will be in proximity to those commemorating Newton, Kelvin, Faraday and Clerk Maxwell. Those to the Herschels (William and John) and Stokes are close by. Green’s memorial designates him “Mathematician and Physicist”. Most mathematicians will know of Green’s theorem and Green’s functions; physicists find his papers seminal to the study of, for example, solid state physics and elasticity and, since the mid-twentieth century, Green’s functions have become an indispensable technique for those working in nuclear physics.
Mathematica: A system for doing mathematics by computer
Authors: Hayes Allan
Published: January 2007

Mathematica is a computer software system for symbolic manipulation and numerical computation; it has excellent graphics, a high-level, mathematically-oriented programming language and, very importantly, it takes seriously the need for work to be interactive, properly presented and communicated. This article is in fact prepared as a Mathematica “notebook”. An extensive library of supporting software “packages” has been developed, and several books about using it are now available.
The bascule bridge—an unexpected cardioid
Authors: Cundy H. Martyn
Published: January 2007

In an article in a recent edition of Mathematics in School (May 1989), Ann Kitchen considers the mathematics of lifting bridges, among which is the bascule bridge shown below, where the counterpoise weight travels down the arc of a circle. This was chosen as a “simplification”, though even so an explicit analytical solution cannot be found. In her example the potential energy of the system was chosen to be the same in the closed and open positions, and there is then a position of unstable equilibrium in between.
Partitions and the dancing bear
Authors: Mackinnon Nick
Published: January 2007

A partition of n is a decomposition of n into a sum of positive integers. So the 5 partitions of 4 are

The number of partitions of n is written p(n) and (with the convection that p(0) = 1) the sequence p(0), p(1), p(2), ... begins

1, 1, 2, 3, 5, 7, 11, 15,....