bluuebunny/crossref_metadata_2025 huggingface dataset:
Aboriginal Applied Mathematics
Authors: Montgomery R. J.
Published: January 2007

Last week I felt that one of my sixth form applied mathematics sets was finding its subject too straightforward. Imaginations had been stuffed with smooth spheres, perfectly elastic and moving as in a vacuum. Methods relied almost exclusively on the less complicated aspects of Newton’s work, with even solid rigid bodies seeming to behave like the ubiquitous point masses. Advanced level text books contain a good store of examples and, needless to say, the class had worked solidly through its own book. “Solidly” was the right word! I wondered how to fan a flagging interest, whether to bring out my own constructional toys for another lesson and to set up a pulley system or project particles from one side of the room to the other, whether to take over the physics laboratory and let the pupils use some proper apparatus. Alas, my pupils are more sophisticated than I and therefore eschew the use of toys, while regrettably the physics staff arrange their physics experiments so much more ably than I could. Besides, it would be entertaining to see the sixth formers investigating a situation which could not be explained with the help of Advanced level mathematics. Let them flounder, as perhaps our forefathers floundered whilst discovering such trivia as Hooke’s Law or Newton’s Experimental Law!
An Early Best Seller: Francis Walkingame’s ‘The Tutor’s Assistant’
Authors: Wallis P. J.
Published: January 2007

Some seven years ago an attempt was made to tell something of the story of Francis Walkingame, the author of The Tutor’s Assistant, which was first issued in 1751 and became the most popular arithmetic text book for over a century afterwards. Recent contributions to the Gazette shows that this author and his famous book are still unknown to many readers. Interest was aroused by Dr. T. M. Flett’s description of Edward Winder’s Exercise Book, which was clearly based on an edition of The Tutor’s Assistant of about 1830. In his description of John Stewart’s Exercise Book, written at Dundee some thirty years earlier, Mr. W More remarked that the rules were the same as Winder used and quoted additionally the Rule of False in very similar words to those used by Walkingame. This example is interesting as showing that Walkingame’s book was used in Scotland at the turn of the century. Richard Hartley’s 1846 letter gives a list of the later sections in Part I (Integers) of Walkingame and shows that he had just started Part II on Vulgar Fractions.
The Laplace Transform
Authors: Hancock S.T R.
Published: January 2007

In technical colleges one is often called upon to introduce advanced mathematical techniques to students whose background is not very extensive. Such a method is the use of the Laplace Transform for the solution of linear differential equations with constant coefficients. Textbooks which deal with this topic, even those specifically written for engineers, derive the transform from the Fourier Integral, or from Heaviside’s Operational Calculus, or just brusquely define the process. Then certain standard forms are evaluated, the rules for application to differential equations are laid down and a selection of worked examples follows. But such a treatment can mystify a student who wants to know why, in following these rules, he should, in effect, firstly, multiply each term of the equation by e-px, secondly, integrate the resultant products from 0 to ∞, and, thirdly, obtain in this way the complete solution.
Patterns of Household Expenditure
Authors: Conway Freda
Published: January 2007

The way in which people spend their money is a matter of considerable interest. It is a topic which can be studied from many points of view and with varying degrees of sophistication, but it is well worth considering with the aid of a few graphs or Engel curves. Ernst Engels published his first account of Belgian working class family budgets in 1857 and his name is still associated with this study, though many of his conclusions concerning expenditure patterns have been falsified by subsequent changes in economic and social conditions.

These curves are used to illustrate the way in which expenditure on particular goods and services changes as household income increases. Data for their construction is obtained from household budget inquiries and is usually given in the form of average expenditure per household on a wide range of commodities for a limited number of income groups.
Groups from one Axiom
Authors: Forder H. G.
Published: January 2007

§1. Our first theory concerns a set of elements, denoted by letters a, b, c, … and words derived from them. The simplest word, apart from a single letter, has the form ab; other words are formed by adjoining a letter on the right to a previous word. Thus ab, abc, abcd are words. We shall say e.g. d is “hooked on” to abc.

The only statements about words in the theory are of the form X = Y, where X, Y are words, and the sign = has the usual properties. The statements are general in the sense that if any word be substituted for a letter in X or Y, and X', Y' be the new words so got, then X' = Y'. Furthermore if X = Y, then Xa = Ya. Note that if e be hooked on to abcd, we get (abcd)e, which is written abcde.
A Paradox
Authors: Thurston Hugh
Published: January 2007

The following definition of differential is not uncommon. It is to be found in Goursat’s and Hardy’s classical texts, as well as in a few more recent ones.
Vector Geometry in School
Authors: Giles G.
Published: January 2007

Now that formal Euclidean geometry is disappearing from the school syllabus the time is ripe for a reappraisal of the background to analytical geometry

Consider the following question. Find the coordinates of the fourth vertex of a parallelogram A BCD if the other vertices are A(2, 4), J5(l, 1) and (7(7, 2).
Henry Briggs. The Binomial Theorem Anticipated
Authors: Whiteside D. T.
Published: January 2007

One of the pleasant aspects of research into mathematical history is the way in which existing material, passed over in the conventional account, may allow us not only to establish the bare, if unexpected, fact of a priority but, more importantly, to assess the significance of major currents in mathematical thought with greater precision. Briggs’ partial discovery, in anticipation of Newton, of the general binomial expansion,

is an excellent case in point.
The Density of Prime Numbers
Authors: De Visme G. Hoffman
Published: January 2007

Proofs of the prime number theorem are extremely hard to follow, and leave the impression, at least among amateurs, that the essential property of prime numbers, namely their primeness, plays very little part in the argument.

Simple reasoning, based on the rules of probability, can however give a very fair indication of the way in which the density of primes in the region x varies with x.
An Investigation
Authors: Branfield J. R.
Published: January 2007

Situation: To find the number of unit squares the diagonal of a rectangle passes through when the rectangle has integer sides and is divided into unit squares.
Summing square roots
Authors: Gates L. W.
Published: January 2007

Formulae for Σl=1n ik when k is a positive integer are well known, and many methods for finding them have been described. This article gives an account of an approximate, but remarkably accurate, formula which I have found for the case k =1/2.
Halving a Triangle
Authors: Dunn J. A. , Pretty J. E.
Published: January 2007

It is well known that any line through the “mid-point” of a parallelogram bisects the area. What can be discovered about lines which bisect the area of a triangle?

Obviously any median bisects the area. It is also bisected by each
line which is parallel to one side and divides the others in the ratio
Problems: comments, corrections and further solutions
Authors: Cobb R. H.
Published: January 2007

The article Problems, solved and unsolved in the Gazette for last March (60 (411), 1–11) drew reactions from a number of readers. There is not sufficient space to show in full all the proofs sent in, but I will try to indicate the methods used. For statements of the problems themselves I must ask readers to refer to the earlier article.
Problem corner
Authors: V. W. B.
Published: January 2007

If you have a suitable problem (solved or unsolved) please send it to the editor, enclosing a solution if you have one. Solutions and comments are invited for the following two problems: I cannot reply to you all personally but I will publish the best.
The Edinburgh Walk-in Numeracy Centre
Authors: Jordinson Ralph
Published: January 2007

During the recent Association Conference at Dundee some interest was expressed in the formation of mathematics workshops. Experience indicated that the main idea behind such a workshop was a place to which anyone could go with a problem in mathematics and, hopefully, receive tuition or guidance on how best to resolve it. The workshop may be attached to an existing FE college and run by the staff mainly for the benefit of students at the college but also with the possibility of helping someone from outside.
Problem corner
Authors: H. G.T.Q.
Published: January 2007

Solutions are invited to the following problems. They should be addressed to Graham Hoare at Dr Challoner’s Grammar School, Chesham Road, Amersham, Bucks HP6 5HA, and arrive not later than July 30th please.
Linear approximation and school mechanics
Authors: Collinson C.D.
Published: January 2007

Oxbridge Tripos examination questions originating from the turn of the century have had a marked effect on the teaching of mechanics both at School and at University. These questions, reproduced in the standard textbooks of the day, were the stable diet of the undergraduate population over so many years and set the pattern for a style of question in which the student applies mechanical principles by rote without ever questioning the validity or relevance of those principles. An unfortunate consequence of this state of affairs is that mechanics has been largely overlooked as a vehicle for the teaching of mathematical modelling. There have been moves recently to make questions more “relevant” and it is not unknown to see illustrations of “real” boats, shotputters etc. However, the student is still expected to answer the question “mechanically”, if the reader will excuse the pun, without any discussion of the modelling process. How refreshing it is, even with first year undergraduates, to look at the problem of a tin can being knocked off a wall by a bullet and, instead of asking the students to apply two conservation laws in order to determine the speed with which the can hits the ground, to ask under what assumptions the two conservation laws would appropriately model the physical situation? It is quite staggering how the lecture theatre erupts with ideas, suggestions and interest.
Drug levels in the body
Authors: Bruce Ian
Published: January 2007

A recent unfortunate incident in the medical world leads to some interesting mathematics [1]. The incident centred around the half-life of a drug used in treating arthritic people; elderly patients having longer drug half-lives than young people, e.g. due to impaired liver or kidney function. Doses were being repeated before the half-life period of the patient, with a dangerous build-up of the drug in the system, leading to death in some cases. A mathematical model can be developed to describe the normal drug concentration variation with time, and the cumulative effect of repeated doses.
The birthday problem for boys and girls
Authors: Crilly Tony , Nandy Shekhar
Published: January 2007

The famous birthday problem is well known to regular Gazette readers. For those readers not familiar with it, the problem is to find the size of the smallest group of people for which there is a probability greater than one-half of two members of the group sharing a birthday. The answer (23 people) is a result which the eminent probability specialist William Feller described as “astounding”.
61.10 Another solution of a factorial problem
Authors: Ayres Frank
Published: January 2007

In the article Problems, solved and unsolved in the Gazette for March 1976, and in the postscript to its sequel in December 1976 (60, 1–11 and 284–291), proofs are given that

is integral when a and b are positive integers. The search for a combinatorial proof was unsuccessful.
Prime number formulae
Authors: Mackinnon Nick
Published: January 2007

The impossibility of a prime number formula seems to have entered the mathematical mythology. As I am continually challenged nonetheless to produce such formulae I thought I would invent some elementary ones. To put them in context I’ll first discuss some classic, and some quite recent, results.
Sharing a Cake
Authors: Austin A.K.
Published: January 2007

It is well known that if two people wish to share a cake fairly then one cuts it into two pieces and the other person chooses which of the pieces to have. This article considers the problem of sharing the cake fairly when there are more than two people. We are looking for procedures which ensure justice for an honest person even when there is dishonest collusion by the other people.
Collapsoids
Authors: Pedersen Jean J.
Published: January 2007

After constructing Platonic solids, Archimedean solids and some of the Kepler-Poinsot polyhedra, I became interested in zonohedra, like the rhombic dodecahedron and triacontahedron shown in Fig. 1. These

models were new to me and most of what I knew about them at that time is summed up by H. S. M. Coxeter in his book Regular polytopes as follows:
A construction for a regular heptagon
Authors: Johnson Crockett
Published: January 2007

Call the ends of a ruler A and Z, and (towards A) place on it a mark X. Draw a line BC the length of AX, and a square BCDE. Erect a perpendicular bisector of BC. With centre C and radius CE draw a circle. Now place the ruler (Fig. 1) so that AZ passes through B, with A on the perpendicular bisector of BC, and with X on the circle. Then ∠BAC = π/7.
Do you get the message?
Authors: Burn R. P.
Published: January 2007

The aim of coding theory is not to deceive, but to make information impervious to errors in transmission. When messages are being transmitted along wires, the likelihood of mistakes in transmission is small. When messages are being transmitted by radio, lightning can distort reception. When messages such as television pictures are being transmitted across interstellar space, there is a substantial possibility, because of meteorites, that the message received will differ from the message transmitted.
Data permutation tests
Authors: Backhouse John K.
Published: January 2007

This article is about a powerful method of carrying out tests of statistical significance. Its rationale depends on simple ideas of probability and is suitable for pupils at school. Practical applications require a considerable amount of computation but now that schools are equipped with computers this should not be a problem.
A-level mathematics in an information age
Authors: Ruthven Kenneth
Published: January 2007

Since the time of Gutenberg the volume of printed information has been doubling roughly every fifteen years. Only relatively recently, however, has the growth of knowledge reached a point at which it challenges human intellectual capacity. Our society has responded by a progressive specialisation of intellectual roles. Very broadly, the nineteenth century saw the development of specialism in distinct fields; the twentieth, the growth of specialism within those fields. To take just mathematics, it has been estimated that, after allowing for redundancy, our present knowledge fills the equivalent of something of the order of 60 thousand average-sized volumes. While at the start of this century, according to Ostrowski, it was expected that a mathematics graduate would be prepared to deal with any question in any branch of mathematics, by mid century Von Neumann was estimating that a skilled mathematician might be familiar with some 10% of the whole field.
A curious factorisation
Authors: Adewoye T. O.
Published: January 2007

Given any positive real number r there will exist some positive real numbers r1, r2,…,rn all less than 4 with

That surprising ‘simultaneous factorisation’ is established in this article. It is based on some fuller work which I did a couple of years ago, but the interest in it was revived when Nick Mackinnon’s and Nick Lord’s work on ‘Sums equal to products’ was published in the March 1986 Gazette.
A cardboard approach to factor groups
Authors: Austin Keith
Published: January 2007

Example 1

Take a cardboard equilateral triangle. It is well known that this has six symmetries. To display these symmetries we colour the vertices of the triangle on both sides of the card with three different colours.
Unbiased dartboards and biased calculators
Authors: Cook Ian
Published: January 2007

In his interesting article on Redesigning the dartboard, published in Gazette No. 413 (October 1976, pp. 171–178), Keith Selkirk refers to a dartboard he has designed by attempting to minimise the magnitude of the vector resultant obtained when vectors of magnitude equal to the score and in the appropriate direction are added.
Matrices, graphs and adjoints
Authors: Lloyd E. Keith
Published: January 2007

Several writers have drawn attention to the fact that there is a connection between determinants and sets of loops in directed graphs. This fact is sometimes useful in evaluating the determinant of a matrix M, as was explained by Greenman in his recent article [1]. Towards the end of his article he introduces Δij-subgraphs and relates them to the cofactors Cij of the matrix M. Now if M is invertible, then M−1 = adj M/det M, where the (j, i)-entry of the adjoint is just the cofactor Cij. Hence one would expect to be able to give a graphical explanation for the fact that M(adj M/det M) = I, and this will be done in the present article.
Is every absolutely convergent series convergent?
Authors: Cohen G. L.
Published: January 2007

There are various ways of introducing a discussion of the completeness of the system of real numbers. In this article we take the unusual approach suggested by the provocative question of the title.

In any elementary treatment of the convergence of sequences and series, a discussion of series of positive and negative terms will lead to the theorem that an absolutely convergent series is convergent.
A statistical paradox—(1)
Authors: O’Carroll M. J.
Published: January 2007

The interest of a problem, to me at least, depends on the simplicity of its statement and the subtlety of its solution. The four-colour problem is a good example. When the solution of the problem also brings surprise, the interest becomes fascination. Good paradoxes particularly fascinate me for they even start with a surprise, and in this case the resolution brings another.
Monge’s theorem in many dimensions
Authors: Walker Richard
Published: January 2007

Let three non-overlapping circles, no two with equal radius, be drawn in the plane. To each pair of circles belongs a pair of common external tangents. Monge’s theorem is that the three points where pairs of common tangents meet are collinear (see Fig. 1).
Stone circles: Megalithic Mathematics or Neolithic Nonsense?
Authors: Angell I. O.
Published: January 2007

To anyone remotely interested in the large number of megalithic stone ‘circles’ that are to be found in Britain, any discussion of their geometry, metrology or astronomy brings one name to mind—Professor Alexander Thom. The professor has spent a lifetime scouring these islands, and has produced surveys for many sites, from which he has evolved a most sophisticated picture of the society of the late Neolithic Age. His best known claim is the discovery of a universal unit of length, the megalithic yard (2·72 ft), which he says was used with extreme accuracy (±0·003 ft) during the whole period of circle construction, approximately 3000 to 1500 BC.
Simple projections
Authors: Smith Cedric A. B.
Published: January 2007

The simplest way of picturing a three-dimensional object in a plane is by orthogonal projection; that is, as it were, by taking a photograph from a distance. (Perspective, showing its appearance when viewed from nearby, is more accurate and lifelike but much more troublesome.) For technical drawing, it is customary to use a plan and elevation (and sometimes also side elevation). Quite often there are three perpendicular directions naturally associated with the object, such as those of the sides of a rectangular building. We call these the “left-right”, “front-back” and “down-up” axes, and the plan and elevations associated with these directions are usually easy to draw. Unfortunately, technical drawings of plans and elevations often appear rather artificial, and require training and practice to interpret properly.
Photomeasurement
Authors: Robin Anthony C.
Published: January 2007

A few years ago I tried making a model of an object from its photograph. Realising that, of course, one could not take direct measurements from the photograph, I had to try to guess the relative dimensions of the object. I ended up, unfortunately, with a model which did not seem to be in the correct proportions. Would it have been possible, I wondered, to find the precise ratios of the different lengths?
Bouncing Balls
Authors: Macmillan R. H.
Published: January 2007

In making a study recently of “whiplash injury” in vehicle impacts, I came upon a fact which surprised me, and further investigation led to some pleasing results involving only elementary mathematics, together with the design of a model which might have a future as an “executive toy” related to Newton’s cradle.
Random walks and concrete cracks
Authors: Middleton D.
Published: January 2007

The cracking or tearing of thin sheets of material is of considerable interest in modern-day engineering. A recent investigation into the behaviour of concrete indicated that it would be of interest to consider a probabilistic model in which a crack is regarded as some form of non-self-intersecting random walk in the plane. This paper considers some simple models and in the Appendix are included some results on paper tearing.
The two aspects of statistical estimation
Authors: Barnard G. A.
Published: January 2007

B. J. R. Bailey’s article in Gazette LVII, 169–174 (October 1973) worried me a little, because it seemed to suggest support for a widespread misapprehension, that mathematical statistics differs from other branches of applied mathematics in requiring a good deal of subjective argument. Every application of mathematics involves judgement, of course, but statistics does not really go beyond other applications in this respect.
Pascal (Mod p)
Authors: Broomhead W. Antony
Published: January 2007

This story is an object lesson in the art of picking other people’s brains.

After one of my fifth form had pointed out for me the result in Classroom Note 248 (ii) (June 1971), it occurred to me that some attractive patterns might result from re-writing Pascal's triangle with the numbers reduced modulo n. I tried it at first with n = 2, using a red dot for odd numbers and a green dot for even. The pattern was, as I had hoped, pleasing. Since we had an Open Day in the near future, I persuaded another fifth-former, P. W. Swinyard, to try the same thing for other small n. Attractive patterns resulted for n = 3, 4, and 5, a rather less attractive pattern for n = 6, but nice ones again for n = 7, 8, and 9. (I recommend readers to try this for themselves.)
What is centrifugal force?
Authors: Jagger Janet , Lord Kevi
Published: January 2007

The subject of centrifugal force comes up at all of our mechanics INSET meetings and it seems that it continues to be a problem when we teach circular motion. Consider the problem of the conical pendulum. The forces acting on the bob are its weight and the tension as shown in Figure 1:

Frequently students put a force outwards in addition to these forces and label it the centrifugal force (or mrω2). It is undeniably true that the lay person (and therefore perhaps all of us at an intuitive level?) feels that this force is there and this is presumably why it appears on force diagrams; after all, it seems quite natural, ‘how else does the bob remain out at an angle if not held by an outwards force?’
Mathematics in sport
Authors: Avery Peter
Published: January 2007

Illustrative examples which draw on everyday experience are particularly useful in the classroom. For this reason, sport can be a good source of examples. We present four illustrations of the ways that mathematics arises in different sports. The ideas should be understandable to those studying A-level mathematics.
That 3-4-5 triangle again
Authors: MacHale Desmond
Published: January 2007

A number of authors have considered the following (and related) problems:

Find all integer-sided triangles such that the perimeter equals the area.

In the March 1981 edition of Mathematics Teacher Lee Markowitz gives the following full set of solutions (though not the considerable calculations involved):

(5, 12, 13) (6, 8, 10) (6, 25, 29) (7, 15, 20) (9, 10, 17).

However, in many formulae involving the triangle, it is the semi-perimeter rather than the perimeter that arises naturally.
A singular impact?
Authors: Linton Matthew
Published: January 2007

While I worked at Teeside Polytechnic I had some difficulty in arriving in the morning by 9.00. Not being an early bird my desire was to pull into the carpark at 8.58, but this I did not seem able to achieve. If I left home at 8.15 the journey was smooth and trouble-free, and I was at my parking place by 8.50. Whereas if I left a moment later there were a couple more cars at the estate exit, each succeeding roundabout and set of traffic lights was busier, and eventually I drove into the car-park at 9.05—not only five minutes late, but also quite likely finding the spaces all full and having to resort to street parking. It seemed that no matter how I varied my start time the arrival interval of 8.50 to 9.05 was somehow inaccessible to me. Of course some people did arrive in this interval—if I did the sensible thing and left early I enviously watched them arrive at the time I would have wished. But they did not start from where I started. So I was drawn to the belief that the relationship between start and arrival time was a genuine discontinuous function, and not simply a continuous function with a rather steep critical section. Maybe the catastrophe theorists could analyse this further (but to call my misfortune a catastrophe would be a bit strong).
A Victorian mathematician: Arthur Cayley (1821-1895)
Authors: Crilly Tony
Published: January 2007

That Spring morning, the bent old man sat down at his large oak desk with his pen momentarily poised over the familiar blue foolscap sheet. He furtively glanced around the drawing room with its old china and flowers and books in abundance and wrote in one easy movement: The principles of book-keeping by double entry. He looked up once more but this time allowed a minute to gaze out through the window to the Cam gently meandering through the meadows towards the Cambridge colleges.
Two families of antiprisms
Authors: Aravind P. K.
Published: January 2007

The purpose of this note is to draw attention to two infinite families of antiprisms that differ slightly from the usual Keplerian kind. These figures have surely been noticed earlier by geometers and philomorphs, but I was unable to track down a reference to them. My attention was drawn to these figures by a Christmas gift that my family received recently. The gift was a cake in a box of a rather unusual shape (see Figure 1). On examining the box, I saw that it could be characterized variously as a deformed antiprism, an unfolded pyramid or a truncated prism. The box had two square faces parallel to each other, with the line joining the centres of the squares being perpendicular to the plane of either square. The squares were rotated 45° relative to each other, exactly as in a square antiprism, but one square had only half the area of the other. The separation of the squares was such that, of the eight triangles linking them, four were identical equilateral triangles while the rest were identical isosceles triangles.
A mathematician's view of bowling
Authors: Brearley M. N.
Published: January 2007

The motion of a bowl on a grass green may be investigated mathematically, using the principles of dynamics established by Sir Isaac Newton (1642 – 1727). The analysis gives a clear picture of how the bias operates, and reveals some surprising and little-known facts which are important from the bowler's viewpoint.
Searching for digital invariants
Authors: Cowell Don
Published: January 2007

As an elementary programming exercise I often ask students to find all three digit numbers equal to the sum of the cubes of their digits, for example 153 = 13 + 53 + 33. This is the usual solution:
From Pappus to today: The history of a proof
Authors: Deakin Michael A. B.
Published: January 2007

“Perhaps we do even worse, suppressing the creative instincts of our children at the same time as researchers try to imitate them with machines. A computer is said to have applied a problem solving programme to the proposition that the base angles of an isosceles triangle are equal. Instead of the Euclidean proof which proves two right-angled-triangles [on the left below] to be congruent, the computer produced a more elegant proof from the simpler construction (sic) [on the right below]. The conventional working must have been taught to millions of children: we may reasonably ask why none of them, apparently, have discovered this simple, new proof for themselves.”
Some groups of numbers
Authors: Weston J. D.
Published: January 2007

The succession of enlargements from ℤ (the system of the integers) to ℚ (that of the rational numbers) and thence to ℝ (the real numbers) and then ℂ (the complex numbers) is generally seen as one of increasing richness of structure. But gains are often accompanied by losses: for instance the step from ℝ to ℂ leads to the glories of complex analysis but away from ordinal properties once thought essential to any concept of number.

As far as group structure is concerned there are gains and losses at each stage of the progression. The following exposition is intended to demonstrate this without assuming more than a minimal knowledge of groups and of the theoretical foundations of analysis. Beyond some elementary arithmetical algebra, and an acquaintance with standard notation and vocabulary for maps in general, essentials taken for granted include the idea of a homomorphism (of groups), the existence and fundamental properties of the exponential function (of a real variable), and the Fundamental Theorem of Algebra (which is actually a theorem in complex analysis).
How long is a game of snakes and ladders?
Authors: Althoen S.C. , King L. , Schilling K.
Published: January 2007

“There is really only one Game, the Game in which each of us is a player acting out his role. The Game is Leela, the universal play of cosmic energy.” Thus begins Harish Johari in Leela: the game of selfknowledge, a serious commentary and religious interpretation of the Hindu board game Leela. Leela is essentially the game Snakes and Ladders, which in the U.S. is the popular children’s game of snakes and ladders, England’s famous indoor sport,” for 50ø in 1943 by the Milton Bradley Company of Springfield, Massachusetts. This game is really a 101 state absorbing Markov chain, which is amenable to mathematical as well as moral analysis. In what follows we investigate only the mathematical side of this diversion, specifically the expected playing time.
Evangelista Torricelli
Authors: Robinson Philip J.
Published: January 2007

This article provides a brief introduction to Evangelista Torricelli and to one aspect his work. It is divided into three parts. Part 1 gives some biographical details and a short survey of his work and achievements. Part 2 is concerned with a geometrical approach towards the study of projectiles as seen in Torricelli's early work De motu. Part 3 shows how Torricelli applied his projectile studies to practical ballistic problems.
On solving a quadratic
Authors: Milner W. W. , Watson F. R.
Published: January 2007

“Find the modulus and the argument of each root of the equation:

(University of London A level, part question.)

We were investigating iterative methods of solving the equation, using a micro-package (in BASIC) which performs complex arithmetic and displays the results in an Argand diagram. The facility for “doodling” which this provides led to some interesting conjectures and results, of which this note is
an account.
Napoleon triangles and adventitious angles
Authors: Fox Michael
Published: January 2007

In this article I investigate Napoleon triangles, generalisations of the mysterious equilateral triangle in Napoleon’s theorem. I start with that theorem, develop some analogous results, find configurations with unexpected integer angles, and return to an extension of Napoleon’s theorem. Many of the geometrical proofs depend upon spiral similarities, and the numerical work uses some unfamiliar trigonometrical identities.
A brief history of Statistics in the last 100 years
Authors: Lindley D. V.
Published: January 2007

When the Mathematical Gazette was founded, statisticians were mainly concerned with the collection and presentation of data. The mathematical side consisted mainly of developments from the work of Gauss and Laplace in the early part of the 19th century, though some surprisingly modern work was done by Edgeworth towards the close of the century. 1889 saw the publication of Galton’s Natural Inheritance, introducing correlation and regression. These two workers influenced Karl Pearson who, in 1900, began the modern era of mathematical statistics with the concept of the significance test usually known by the symbol he used, chi-squared.
What use are abstract spaces?
Authors: Sawyer W. W.
Published: January 2007

In school we learn classical mathematics, the mathematics known before the year 1900. It has a direct relation to the actual world. Arithmetic is related to counting and measurement, algebra to general truths about calculations, geometry and trigonometry to the sizes and shapes of objects, calculus to the velocities of moving objects and a host of other applications. Students have no difficulty in knowing what it is about and what it is for. Students who go on to modern mathematics, the mathematics of the present century, either in university or by private study, find it considerably different. There is no immediate contact with reality, and they may find it hard to visualise what is being discussed.
On the geometry of the sriyantra
Authors: Tahta Dick
Published: January 2007

Practitioners of Tantric yoga are said to use certain diagrams, called yantras, as aids to meditation [1]. The most interesting of these, from a geometrical point of view, is the sriyantra (the great yantra). A version of diis, shown below, has been reproduced in various recent books and articles as an example of the mathematical achievements of early non-European cultures. In this article I want to make some observations about the geometric structure of the sriyantra, and to comment on some of the claims being made about it. I also hope, in passing, to be offering an example of the sort of geometric situation that can be explored with the powerful Cabri geometry program [2].
An extension of Holditch’s theorem on the area within a closed curve
Authors: Cooker Mark J.
Published: January 2007

While reading a book by David Wells on curious and interesting geometry, I came across the following remarkable theorem named after Holditch.

In Figure 1 the point R divides a straight stick ST into lengths p and q, where p, q 0. We restrict the end points of the stick, S and T, to lie on a plane, simple, closed, convex contour, C1, and ST slides around C1. Assuming C1 is such that ST can pass around C1 once, the locus of R is another plane closed contour, C, inside C1.
Improving the efficiency of racing shell oars
Authors: Brearley Maurice N. , de Mestre Neville J.
Published: January 2007

By analysing the forces acting on an oar during the power stroke it is possible to estimate the mean velocity given to a racing shell. Use is made of oarlock force data, obtained experimentally by the Australian Institute of Sport, to obtain accurate predictions in the case of a racing pair. Similar conclusions would hold for boats with any number of oars. The effect of altering the angle between oar shaft and blade is investigated, leading to a suggestion for a design change to improve oar efficiency. Predictions are made of the resulting improvement in boat position in a 2000-metre race.
What shape is an ellipse?
Authors: Hirst Ann E.
Published: January 2007

An ellipse is ellipse-shaped, of course! Every mathematician knows what an ellipse looks like – at least, this is generally true in the Cartesian plane, with the standard distance function or metric. However, there are other metrics which can be applied to the plane ℝ2 and it might be useful to know what an ellipse looks like when distances are defined by these metrics. The set of points under consideration will be ℝ2, the real plane, and the metrics under consideration will be the standard metric, the Manhattan (or taxi-cab) metric and the maximum modulus metric.
Fractal tilings derived from complex bases
Authors: Hagey Sara , Palagallo Judith
Published: January 2007

Tilings have appeared in human activities since prehistoric times. The mathematical theory of tilings contains a rich supply of interesting and sometimes surprising facts as well as many challenging problems. A vast literature exists on the subject of tiling, and almost every imaginable variant of the question ‘How can a space be tiled by replicas of a set?’ has been discussed. Figure 1 shows a tiling of a portion of the plane where each tile is of the same size and shape and has a fractal boundary.
Critical sets in latin squares: an intriguing problem
Authors: Keedwell Donald
Published: January 2007

A latin square of order n is an n × n array (or matrix) containing n distinct symbols (which are often taken to be the symbols 0, 1, …, n - 1, though any n symbols will do) arranged in such a way that all n of these symbols occur in the n cells of each row and also in the n cells of each column. Such squares occur in many guises: for example, as appropriate field layouts in the statistical design of experiments, as the addition or multiplication tables of groups, as a coding theory device and as a means of representing a finite geometry algebraically.
The high jump – the neglected Straddle style
Authors: Brearley Maurice N. , de Mestre Neville J.
Published: January 2007

The world high jump record for men is now an amazing 2.45 m (8 ft 0½ in). This requires an athlete to raise his or her the centre of gravity through a vertical distance of about 1.3 metres, something which is quite impossible by leg thrust alone. Through experience, coaches and athletes have developed techniques which maximise performance, and this paper attempts to describe these in mathematical terms. In this article, the various factors contributing to the height of a jump are investigated with the aid of information gained from a film of a Straddle-style jump. The same principles are involved in a jump using the Fosbury Flop.
Beycesultan Excavations: First Preliminary Report
Authors: Lloyd Seton , Mellaart James
Published: January 2007

Beycesultan Hüyük is situated three-and-a-half miles south-west of the kaza town of Çivril in the vilayet of Denizli, and about one hundred and thirty miles east of Smyrna. The Çivril valley, of which Beycesultan must in ancient times have represented the provincial capital, is watered by the upper course of the Meander River and has an elevation of some three thousand feet above sea-level. Its length is traversed by a modern road, taking the line of a very ancient trade-route, which, at the south-west end of the valley, leaves the upland country and drops down suddenly, some two thousand feet, into the great rift of the lower Meander, whose olive groves slope towards the Aegean Sea. This must always have been one of the traditional approaches to the Anatolian Plateau from the west. The climate of Çivril is accordingly that of central Anatolia: very low winter temperatures, succeeded by perpetual summer sunshine and moderate heat.
The Sultantepe Tablets (continued) VI. A Letter of Gilgamesh
Authors: Gurney O. R.
Published: January 2007

The text here chosen for translation can hardly be compared as literature with that presented in the preceding volume of this journal; it is, however, in some ways even more remarkable. It is a fictitious letter, purporting to be addressed by none other than Gilgamesh, the famous hero of the epic and legendary king of Uruk (Erech), to the king of another city, the name of which is only partly legible and appears to be hitherto unknown. Gilgamesh, who introduces himself as king of Ur, requests his correspondent to go to a certain mountain (or country) and send him various animals, precious stones, metals and other commodities, in quantities which are completely fantastic and absurd. They are apparently to be floated down the river to Babylon. The text concludes with threats of dire retribution if he should fail to comply with this request.

No less than three copies of this strange document have been found at Sultantepe: no trace of it has been discovered at any other site. It contains a number of specifically Assyrian grammatical forms; and its inappropriate use of the phrase “from the horizon to the zenith” to indicate the extent of a king's dominions (l. 4) is paralleled only in the annals of the Assyrian king Tiglath-pileser III (745–727 B.C.).
Alahan Monastery, Fourth Preliminary Report
Authors: Gough Michael
Published: January 2007

This report should really have appeared in Anatolian Studies XVI (1966). Dis aliter visum. During the autumn of 1965 and the first half of 1966 the present writer was either ill or in hospital, and so unable to work to any purpose.

After a fallow season in 1964, excavations at Alahan were resumed in 1965 between the 26th June and the 21st August, a period of exactly eight weeks. The average number of men employed was twenty-five. The Director was again in charge, with Mr. R. P. Harper as his deputy. Mr. G. Bakker again acted as our architect, while Mrs. Bakker dealt with small finds and conservation. Field assistants were Miss Margaret MacDonald, Messrs. T. B. Mitford (Institute Fellow for 1964–1965) and Anthony Ray. Towards the end of the excavation Mr. J. W. Hayes visited the site to advise on pottery and coins, while Miss Margaret Ramsden assisted in the tedious but skilled task of cleaning up skeletons for photographing.
Early Cultures of the South Anatolian Plateau
Authors: Mellaart James
Published: January 2007

The purpose of this article is to describe here some of the material found during our survey of the Konya Plain in 1958. In two previous articles pottery of the 2nd millennium and the Iron Age found here have already been published. That of the Early Bronze Age, the most prosperous period in this area, will be described at a later date, and the present article will only describe the pottery of the Neolithic and Chalcolithic periods, under the following headings:

I.

Neolithic cultures of the South Anatolian Plateau.

II.

The Early Chalcolithic in the Konya Plain.

III.

The Late Chalcolithic in the Konya Plain.

The importance of the Konya Plain in Anatolian prehistory is obvious. It is the largest single plain on the whole of the Anatolian Plateau with alluvial soil, and as such it is the granary of Turkey. No other region on the plateau shows such numbers of ancient mounds, or so many mounds of great size. The survey of this region, geographically as well as archaeologically a distinct unit, has at last linked the western plateau with Cilicia, and the results have shown that there is now a cultural continuum from the borders of Syria to the Aegean Sea since the Neolithic period.
Roman Senators in Cappadocia
Authors: Harper R. P.
Published: January 2007

An inscription at Comana Cappadociae (Şar köyü, Mağara, Adana), hitherto unpublished, provides the evidence for this note (Pl. XLIIa, Fig. 1). The stone is now in the yard of the school at Şar. It is a white marble slab, 0.90 × 0.40 m., recut below and at the lower right corner. The building which now houses the school was a church when the village was inhabited by Armenians and seems to have been built by them entirely out of ancient materials. This stone is said to have formed the door-sill of the church and to have been taken out when the position of the door was changed recently. Lines 4 and 5 certainly appear to have been worn by the passage of feet.
Two Lycian Families
Authors: Jameson Shelagh
Published: January 2007

Perhaps the best known of all the aristocratic Lycian families is that of the Licinnii of Oenoanda, whose ramifications are set out in their genealogy (IGR. III, 500). This document, perhaps the most complete of its kind, not only reveals the complicated nexus of relationships forming the structure of such a family, but also provides much incidental information which demonstrates how its various members became prominent.

It seems probable that the earliest members of the family to bear Roman nomina, Licinnius Musaeus and Marcius Thoas, obtained the franchise under Nero and Vespasian respectively. Licinnius Musaeus will have received citizenship from the governor C. Licinnius Mucianus (a fact borne out by the name of Musaeus' grandson Mucianus), Marcius Thoas from Sextus Marcius Priscus, legatus Augusti pro praetore of Lycia and Pamphylia under Vespasian.

The first few generations of the enfranchised family produced a number of high-ranking officials of the Lycian Koinon, but were not otherwise especially significant. In the early stages of its history there was a good deal of intermarriage among members of this family—the product, perhaps, of a combination of insularity and social exclusiveness. Such marriage partners as were not chosen from within the family itself came from the higher strata of Lycian society, from families which were socially acceptable to the Licinnii and locally distinguished. For the majority of the Licinnii this remained the pattern for so long as we have any knowledge of them.
The Urartian Cemetery at Igdyr
Authors: Barnett R. D.
Published: January 2007

In 1943 the Academy of Sciences of the Georgian S.S.R. published at Tiflis, as Part XIII.B of the Bulletin of the Georgian Government Museums, a book of 171 pages by the late B. A. Kuftin (1892–1953), entitled Urartskii “kolymbarii” u podoshvy Ararata i Kuro-Arakskii Éneolit [An Urartian “Columbarium” on the slopes of Ararat and the Copper Age of the Kur-Araxes basin]. The terminus technicus “columbarium” (Latin = a pigeon-cot) is translated in German as “Urnenhalle”. More appropriate here would be to call it an “Urn-field” but “columbarium” is retained here as having no adequate English translation. In the original the Russian text is followed by a version in Georgian and a very brief English summary. This book in fact contains two essays, the first dealing with the “columbarium”, the second with the Copper Age remains under the lava to the south of the “columbarium”, and called the “Clay Citadel”. We are here only concerned with the first.
Early Cultures of the South Anatolian Plateau, II
Authors: Mellaart James
Published: January 2007

The end of the Çatal Hüyük West culture is shrouded in mystery. Both Çatal and Kara Hüyük South were apparently deserted and never reoccupied and it is only at Can Hasan Hüyük east of Karaman that later deposits have been recognised overlying remains of the early Chalcolithic culture. Elsewhere the evidence lies buried in the cores of the numerous city mounds of the Early Bronze Age period. Late Chalcolithic remains are fairly common in the Konya Plain, but they were in nearly every case found on sites where no earlier or later remains were encountered. This might suggest a shift in the settlement pattern of the plain after the end of the Early Chalcolithic period (see map, Fig. 1).