bluuebunny/crossref_metadata_2025_split huggingface dataset:
Dissecting the Decagon
Authors: Lindgren H. 
Published: January 2007

The regular decagon will be dissected into other regular polygons and into Greek and Latin crosses by the methods outlined in my previous article (Math. Gaz. 45, 1961, p. 94).The first step is to find tessellations. In one of those that are used in this note, shown in 1 and 2, the decagon is cut into three pieces, and in the other only two. A further tessellation element, not used here but worth noting, is shown in 5.
Scheduling a Two-Way Bottleneck
Authors: Anderson D. V. 
Published: January 2007

There is grave difficulty in showing elementary school students how mathematics is applied in practical situations (at least in Canadian schools). Since current emphasis is on pure mathematics, applied mathematics (and applying mathematics)—procedures of model building, approximation, analytical manipulation and interpretation—may remain obscure or unknown to students leaving school despite essential simplicity of principles.Thus it is important to keep a look-out for problems which have an obvious practical interest or economic moment, but which can be tackled with the most elementary mathematics. Emphasis on the strategies of model building in the schools seems all the more sensible (and feasible) now that automatic numeration is permitting easy command of even complicated analyses.
The Multinomial Theorem
Authors: Bolton D. W. 
Published: January 2007

As the name suggests, the Multinomial Theorem is an extension of the Binomial theorem, and it was when I first met the latter that I began to consider the trinomial and the possibility of a corresponding Pascal's triangle.My mathematics master suggested that I construct the triangle myself. It became apparent that such a triangle may not be drawn, as the result is a pyramid or a tetrahedron where the apexes of the tetrahedron are occupied by the coefficients of xn, yn, zn in the expansion of (x + y + z)n.This may be shown by forming expansion coefficient triangles for a trinomial of power 0, 1, 2, 3.
Cliques and Groups
Authors: Vaughan H. E. 
Published: January 2007

Mr. Parrack's paper on cliques [in the Gazette of February 1966, pp. 43-46] links up with some results which I obtained in 1962 on “ double-identities ”. [Mimeographed copies of the paper are available for any who wish them.]Briefly, the simplest kind of double-identity is an equation which, like:is an identity in any additively-written abelian group when “ o ” is replaced by “ + ” and, also, when “ o ” is replaced by “ - ”. More precisely, an (e, o)-double identity is an equation between terms constructed from variables, “ o ”, and a constant, “ e ”, which is an identity when “ e ” is replaced by “ 0 ” and “ o ” is replaced by “ + ” or by “ - ”.
Cooperation in Mathematics Teaching between Primary and Secondary Schools
Authors: Tammadge A. R. 
Published: January 2007

Several lines of thought led up to the present experiment in cooperation which has just started in Abingdon. I would like to start by trying to summarize them.We are apt, in the grammar schools, to criticise the universities. We may say that their influence on our syllabus is bad. We may think that the teaching they give to our prize pupils is poor, the supervision unsuitable, the courses wrongly devised. These things we say and think and I am not concerned here with the rights and wrongs; but do we realize that these are precisely the things that primary teachers are saying and thinking about us?In the concluding chapter of “On Teaching Mathematics” (1), we said (a) Mathematics is a subject which must be taught well at an early stage of a child’s schooling, and (b) the youngest children should not be taught by those insensitive to the subject. How much do we in the grammar schools know of the primary teacher and of what is being taught today in the primary schools?
On Inequalities Connecting Arithmetic Means and Geometric Means of Two Sets of Three Positive Numbers
Authors: Oppenheim Alexander 
Published: January 2007

Suppose that a1, …, an and c1, …, cn are two sets of n positive numbers. If the arithmetic mean of the ci is at least equal to that of the ai, when can we say non-trivially that the geometric mean of the ci is at least equal to that of the ai and that equality will require equality (in some order) of the ai and ci?A well-known answer of course occurs when each c is equal to the arithmetic mean of the ai. When the two sets each consist of three numbers I can give an answer including this important case. To obtain this result it is sufficient to confine the ci to the interval from the least to the greatest of the ai.
A Geometric Duality for Two Metrics for the Co-Ordinate Plane
Authors: Rhodes Frank , Cox David , Dixon Patricia , Fever John 
Published: January 2007

who arrived at the result through discussions in an undergraduate class at the University of Southampton.Of the various metrics which can be placed on the co-ordinate plane R×R, the Euclidean metric is universally known, and two others are becoming more widely known in schools. With the co-ordinate notation for points x = (x1, x2), these metrics are
Natural philosophy
Authors: McCrea W. H. 
Published: January 2007

In this lecture we shall see how natural philosophy—or what we traditionally call applied mathematics—is continually changing. I hope to show that the subject is far livelier and more adventurous than we sometimes suppose, and that each advance opens ever wider vistas of further progress.We are concerned with the relation between mind and experience of the physical world. Our minds are such as to be most strongly persuaded that there is a significant relationship by their being able to predict experience that no one has ever had before.
The mathematics of population growth
Authors: Andrew A. L. 
Published: January 2007

The word “lesson” came back to Pooh as one he had heard before somewhere. “There’s a thing called Twy-stymes”, he said. “Christopher Robin tried to teach it to me once, but it didn’t.” A. A. Milne.In assessing the value of the mathematics taught in schools, considerable emphasis is normally given to the use which a minority will later make of it in their jobs. It is the purpose of this article to point out that even those who never use mathematics professionally could better understand important social issues if they had a good basic training in mathematics. This is illustrated by a set of mathematical models related to one such problem— the population explosion.
Dangerous loads and lattices
Authors: Lloyd E. Keith 
Published: January 2007

There are probably few readers of the Gazette who also read the weekly periodical Commercial Motor, but it was a table appearing in an article in the latter which caught the author’s eye and provided the inspiration for the present work.In recent years legislation has been introduced which requires that lorries carrying dangerous substances must also bear panels on which appear, inter alia, codes identifying the substances and giving an indication to the fire brigade of what action should be taken in the event that the vehicle is involved in a collision or other emergency. Examples of the codes are shown in Figs. 1 (a) and (b). The former is probably the most commonly seen code as it is that displayed on petrol tankers; the latter is a code seen on a vehicle carrying liquid nitrogen.
Random triangles
Authors: Easingwood Trevor 
Published: January 2007

“What is the probability that a randomly drawn triangle is acute-angled?” The problem sounds simple enough. However, what is interesting about it is the interpretation given to “random”. We investigate three possibilities(a) that the angles of the triangle are chosen;(b) that the lengths of the sides are chosen; and(c) that the coordinates of the vertices of the triangle are chosen.
What the eye doesn't see, …
Authors: Quadling Douglas 
Published: January 2007

We are just contemplating the purchase of a new car, and have been testdriving a number of competing models. A mirror fixed to the driver's door is nowadays standard equipment, and on most of the models this is attached near the front of the door, more or less in line with the hinges. But in the VW Golf it is set further back; a good feature, claimed the salesman, since it reduces the driver's “blind spot”.
I confess that it was not obvious to me that this is so. If true, then perhaps it would compensate for the inconvenience of having to turn one's head further round to look in the mirror. But mathematical investigation seemed to be called for.
Who gets good degrees?
Authors: Anderson Johnston 
Published: January 2007

In a previous article in the March 1979 Gazette I looked at a population of 589 students entering a particular university department between 1971 and 1975 to read for a degree in mathematics, or in mathematics-with-another-subject. These students graduated between 1974 and 1978.The present article provides updated information for students entering between 1976 and 1983 (graduating 1979 to 1986 respectively). Several factors have changed since the earlier survey. A significant change is the increase in the number of entrants offering single-subject mathematics at A-level. A second important factor has been the consequences flowing from the UGC cuts in universities in 1981; admission targets are now much more closely controlled and the level of offer to intending entrants has often tended to rise (certainly the case at this institution). There has been, as a result, a decrease in the number of students entering with poorish grades.
An interception problem
Authors: Lord Nick 
Published: January 2007

The problem A balloon which ascends with constant vertical velocity—is released from ground level into a steady wind which blows it towards an archer who is situated at ground level and able to fire an arrow at any elevation. What is the minimum speed of release of the arrow needed to achieve a hit?(He is only allowed to fire after the balloon is released (!) and we shall assume the flight of the arrow to be unaffected by the wind.)
An optimisation problem for triangles
Authors: Crilly Tony , Millward Stanley 
Published: January 2007

Here is a problem which can be tackled at various levels. It has the merit of being straightforward and requiring little abstract mathematics, though it is far from trivial.Our investigation began with the well known problem of fencing a field one of whose borders runs along a straight river. Given 100m of fencing, describe the rectangular field which has maximum area, where no fence is required along the riverbank.
Piecewise linear recurrence relations
Authors: Crampin Mike 
Published: January 2007

The properties of linear, homogeneous, constant coefficient, second order recurrence relations such asare well known. Consider however the recurrence relationSuch things are rarely studied; it may not be obvious that equation (2) generates a sequence which is periodic with period 9, which is to say that for any initial values x0, x1, the sequence x0, x1, x2, x3, … satisfies xn+9 = xn for every n. This article records some discoveries which resulted from an attempt to understand this interesting fact. These discoveries open up a field for exciting, elementary, but by no means trivial, investigations.
A problem in squash strategy
Authors: Brooks W. J. , Hughston L. P. 
Published: January 2007

Two players A and B have a game of squash. Player A serves at the start of the game. They play a sequence of rallies. If the player serving the rally wins the rally, he scores a point, and serves the next rally. If he loses the rally, then the other player serves the next rally, with the score unaltered.Generally the first player to reach nine points wins the game. However, if the score becomes eight-all then the first player to reach eight has a choice: he specifies whether the game is to be played to nine points or to ten ponts.What is the correct choice? We propose to analyze the problem by use of an elementary probabilistic model.
Quasi-primes and the Goldbach conjecture
Authors: MacKinnon Nick 
Published: January 2007

In 1742 Goldbach sent a letter to Euler which contained the conjecture that:Every even number greater than 2 is the sum of two prime numbers.It is easy to verify Goldbach's conjecture numerically up to some quite high limits but no proof has yet been found. I have long had the uneasy feeling that the conjecture is not about the prime numbers at all but would equally apply to any ‘sufficiently dense’ sequence of odd numbers. However a recent attempt to generalise the sieve of Eratosthenes has led me to regard Goldbach's conjecture with a good deal more respect.
An exploration of hyperpower equations nx = ny
Authors: Voles Roger 
Published: January 2007

My interest in this fascinating area of mathematics was aroused by a recent article [1] in which it was shown thathas an infinity of non-trivial solutions (i.e. u ≠ v) in positive integers, rationals and algebraic irrationals. Replacing u and v by their reciprocals, x and y, permits the equation to be written as
How to lose at Tetris
Authors: Burgiel Heidi 
Published: January 2007

Tetris is a computer game which has obsessed many computer users and attracted much attention, despite the simplicity of its rules. This paper addresses the question: ‘can you “win” the game Tetris?’Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989. Versions of Tetris are sold for most personal computers. There are Tetris arcade games, Tetris Nintendo cartridges, and hand-held Tetris games; Tetris has been played on machines ranging from mainframes to calculators. The game's success has prompted the invention of several similar games, including Hextris, Welltris, and Wordtris.
A geometrical creation myth
Authors: Cable John 
Published: January 2007

It has struck me that axiomatic treatments of mathematics resemble the creation myths of primitive religion. In both cases the attempt is being made to provide an ordered account of something that already exists. In the one case the ordering is by deduction; in the other by the temporal sequencing of narrative. By way of illustration I offer a paraphrase of the opening pages of Hubert's Grundlagen in the style of a creation myth.Of course, I do not use ‘myth’ with any derogatory intention. Nor do I wish to disparage historical attempts from Euclid onwards to present mathematics as a deductive system; indeed I will join the current chorus of regret that this aspect of the subject is now absent from school mathematics. The trouble is to know how best to present axiomatics. It is no longer possible, as it was for Euclid and even for Frege, to suppose that deduction leads back to ‘primitive truths’. One cannot pretend that the mathematics lies wholly in the deduction itself; as Wittgenstein observes, this is like saying that cabinet-making consists in gluing. To say that an axiomatic system is a ‘free creation of the human mind’ does not begin to address the question of purpose and motivation. Perhaps the geometrical deductive systems of Euclid and Hilbert should be seen as conceptual models, designed to make sense of geometrical phenomena, rather as a creation myth is designed to make sense of biophysical phenomena.
Steiner-Lehmus and the automedian
triangle
Authors: Parry C. F. 
Published: January 2007

The celebrated Steiner-Lehmus theorem
states that if two internal bisectors of a
triangle are equal then the triangle is isosceles.
It beautifully illustrates the point that the
converse of a theorem may be far more difficult to
prove than the direct statement. The literature of
the Steiner-Lehmus theorem is considerable with
many contributions in the
Mathematical
Gazette.
An interesting variation on this theme may
be found by examining the triangle formed by the
second intersections of the medians with the
circumcircle. Let the median point of a triangle
ABC be
G and let the medians
meet the opposite sides at
DEF and the
circumcircle again at
LMN.
‘Problem-solving’? Or problem solving?
Authors: Gardiner A. 
Published: January 2007

David Hilbert (1862–1943) was one of the most outstanding mathematicians of the modern era. At the International Congress of Mathematicians in Paris in 1900 he presented twenty-three major research problems which he felt would be important for the development of mathematics in the twentieth century. These problems all seemed very hard, but in bringing them to the attention of other mathematicians Hilbert felt the need to stress that this should not be used as an excuse to put off trying to solve them.‘ However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes. This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason. In Mathematics there is no ignorabimus.’ [1]
Rigorous thinking and the use of instruments
Authors: Gardiner Tony 
Published: January 2007

The title is a section heading from the lovely biography [1] by Florian Cajori of the English mathematician William Oughtred (15647-1660). Though Cajori was writing history he chose his subject and style to give modern readers (of 1916), and especially teachers of mathematics, something to think about.Oughtred was a fine amateur mathematician and a thoughtful teacher. John Aubrey in his brief lives (quoted in [1p12]) wrote:He was a little man, had black haire, and black eies (with a great deal of spirit). His head was always working. He would drawe lines and diagrams on the dust… did use to lye a bed till eleaven or twelve a clock… Studyed late at night; went not to bed till 11 a clock;… and on top of his bed-staffe, he had his ink-horne fix't. He slept but little. Sometimes he went not to bed in two or three nights.
The remarkable Ibn al-Haytham
Authors: Smith John D. 
Published: January 2007

‘I saw that I can reach the truth only through concepts whose matter are sensible things and whose form is rational.’The achievements in experimental and theoretical science of the Arab scholar al-Haytham (also known as Alhazen, from his latinized first name al-Hasan) make him as much a figure of the sixteenth and seventeenth centuries as of his own tenth and eleventh centuries.When his writings become known in the West the importance of his contribution to optics was widely recognized and he was studied by Galileo, Kepler, Fermat, Snell and Descartes. Mathematicians remember al-Haytham chiefly for Alhazen's Problem on the reflection of light from a circular mirror, which he solved by the method of conic sections; Huygens, Gregory, l'Hospital, Barrow (and many others) later took up the problem with the new analytical methods of geometry. Al-Haytham also wrote a commentary on the postulates of Euclid, and his attempted proof of the parallel postulate has similarities to Lambert's quadrilateral and Playfair's axiom in the eighteenth century. His theory of cognition may produce yet further interest in his work.
Diophantine boxes
Authors: MacHale Des 
Published: January 2007

I have a soft spot for geometric problems which lead to diophantine equations, that is, equations for which only positive integer solutions are sought. I have not seen the following naturally occurring problem discussed anywhere.

Problem A

A closed rectangular box has its volume numerically equal to its total surface area. If the length, breadth and height of the box are all positive integers, find all possible sets of dimensions of the box.
The solution of this pretty problem is completely elementary and can be fully understood by those with a basic knowledge of algebra.
We remark that to solve any three-dimensional problem it is often useful to step down a dimension and to consider the corresponding problem for plane figures.
Excavations at Polatli: A New Investigation of Second and Third Millennium Stratigraphy in Anatolia
Authors: Lloyd Seton , Gökçe Nuri 
Published: January 2007

In our 1949 budget, the Directing Committee of the Institute put at my disposal a sum of one hundred pounds, for the purpose of an excavation in collaboration with the Turkish Department of Antiquities. On discussing the matter with the Director, Dr. Hamit Koşay, I found that the mound in which he was at the moment most interested was near the small modern town of Polatli, about forty miles from Ankara, on the main railway line to Istanbul. I understood from Dr. Koşay that a large part of this mound had been quarried away by peasants in need of earth for making bricks, and that, as a result, a number of whole pots and metal objects of the Copper Age had found their way to Ankara. I was subsequently shown these objects by Bay Nuri Gökçe, Director of the Hittite Museum, and a few days later visited the site in his company.The hüyük, which takes its name from Polatli (Plate II), is situated on the outskirts of the modern town, near a spring of water. Its height is impressive, though the area which it covers suggests that the ancient settlement which it represents would have been hardly more than a village.
Harran
Authors: Lloyd Seton , Brice William , Gadd C. J. 
Published: January 2007

The great interest and importance attaching to the ruins of Harran, particularly in relation to the city's age-old association with the Mesopotamian moon-cult, has never been in doubt. The remarkable degree to which, during the past generation, they have escaped the attention of archaeologists has been due entirely to their geographical inaccessibility. So many references to Harran, either under its own name or in the classical guise of Carrhae, occur throughout the length of Mesopotamian, Roman and mediaeval Arab literature, that it has acquired a strong historical personality, almost without reference to its material remains.In fact, the few and brief investigations of the site so far accomplished by archaeological explorers, have failed to throw any light whatsoever on the outstandingly important matter of local topography in relation to the several religious shrines for which the city was famous during nearly three millennia. In the middle of the last century, it was successively visited by members of Chesney's Euphrates Expedition and the English missionary, G. P. Badger.
Le dieu Baal et le dieu Moloch dans la tradition Biblique
Authors: Dhorme Par Edouard 
Published: January 2007

A mon vieil ami, le Professeur J. Garstang, en souvenir de nos visites archéologiques aux sites bibliques de jadis.—E. Dhorme.Si J'associe ces deux dieux dans une même étude, c'est non seulement parce qu'ils représentent l'un et l'autre une conception similaire de la divinité dans les anciennes religions sémitiques, mais aussi parce qu'ils ont subi, l'un et l'autre, un même traitement de faveur, ou plutôt de défaveur, dans la transmission du texte massorétique appuyée par le grec des Septante. En revanche, si la personnalité du Baal n'a fait que s'agrandir en se précisant, grâce surtout aux découvertes de Ras Shamra, l'ancienne Ugarit, celle du Moloch a été l'objet de contestations énergiques depuis une vingtaine d'années, au point qu'une brochure, parue en 1935, annonçait dans son titre “la fin du dieu Moloch”. Nous allons donc examiner les faits qui, selon nous, exigent la présence de Moloch à côté de Baal dans le plus ancien panthéon sémitique attesté par la Bible.