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The Mathematical Gazette
Number of Followers: 1

Full-text available via subscription

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ISSN (Print) 0025-5572 - ISSN (Online) 2056-6328
Published by Cambridge University Press

[353 journals]

MAG volume 106 issue 566 Cover and Front matter
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  Pages: 1 - 2
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.109
MAG volume 106 issue 566 Cover and Back matter
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  Pages: 1 - 3
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.110
Dropping plates
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  Authors: Hopkins; David
  Pages: 193 - 205
  Abstract: A friend of mine [1] mentioned a problem to me, which he was told had an interesting solution involving an unexpected square root. I have not seen this problem described elsewhere, so I have carried out my own analysis, which I will present here. In fact the solution involves not only square roots, but also higher roots … and a logarithm.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.59
The real solutions of x = ax
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  Authors: Beardon; A. F.
  Pages: 206 - 211
  Abstract: We denote the real logarithm of a positive number a by ln a, so that ax = exp (x ln a), and we shall discuss what is known about the real solutions x of the equation(1)First, as exp t> 0 for all real t, each real solution x of (1) is positive.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.60
Winning strategies: the emergence of base 2 in the game of nim
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  Authors: Friedman; Eric J., Landsberg, Adam S.
  Pages: 212 - 219
  Abstract: Many players know that the secret of winning the game of nim (and other “impartial” combinatorial games) is to write the sizes of the game’s piles in base 2 and then add them together without carry. The proof of this well-known procedure (described below) is both straightforward and convincing. Nonetheless, the procedure still appears magical, as though a rabbit has been pulled out of a hat. Astute students (and frustrated professors) often ask why the winning strategy for such games involves base 2, and not some other base. After all, nothing about the game of nim itself – the game rules, the configuration of the tokens, etc. – provides any hints about the origin of base 2 in this setting. Minimal insight is offered by most published proofs, which themselves tend to either appear almost wizardly in nature (i.e. assume the base-2 method and show that it miraculously solves the problem) or employ combinatorial arguments that supply little abstract intuition (at least to the authors of this article).
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.61
A mathematical approximation in the physical sciences
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  Authors: Mahony; John D.
  Pages: 220 - 232
  Abstract: The business of making mathematical approximations in the physical sciences has a long and noble history. For example, in the earliest days of pyramid construction in ancient Egypt it was necessary to approximate lengths required in construction, especially when they involved irrational numbers. Similarly, surveyors in early Greece seeking to lay out profiles of right-angle triangles or circles on the ground invariably ended up making approximations regarding measurements of required lengths, as indeed is the case today. Practitioners have always faced the problem of having to decide when parameters in theory have been met satisfactorily in the practice of measurement. Further, before the advent of hand-held calculators, students in schools in the UK would have been very familiar with the approximation 22/7 for the transcendental number π, obtained perhaps by comparing (as this author did) the measured circumferences of many laboriously drawn circles of different sizes with their diameters. Despite the advent of sophisticated calculating devices and facilities, such as computers and spreadsheets, the practice of making approximations is still much in evidence in theoretical work in fields associated with physical phenomena. Such approximations often result in formulae that are easy to use and remember, and moreover can produce theoretical results that support directly, or otherwise, results from measurements. In this respect, the practical mathematician does not have to seek results to many decimal places when measurement facilities allow for accuracy to only a few. The purpose of this Article is to illustrate this point by discussing an example drawn from the realms of antenna theory, relating to the performance of a dipole antenna. It is not the purpose here to delve into the derivation of dipole theory, but to extract the relevant information and show how useful mathematical approximations can be employed to simplify a relationship between parameters of interest to an antenna engineer. To this end, it will first be necessary to introduce some antenna concepts that might be new to the reader.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.62
A computer look at N!
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  Authors: Sullivan; Jerry
  Pages: 233 - 241
  Abstract: The product over the integers 1 to N is written as N!, is called N factorial, and is defined as: (1).
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.63
Fibonacci-Lucas hyperbolas
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  Authors: Sporn; Howard
  Pages: 242 - 246
  Abstract: Let us define a Fibonacci-Lucas hyperbola as a hyperbola passing through an infinite number of points of the form (Fm, Ln), where the Fm are distinct Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, 21,…, where F0 = 0), and the Ln are distinct Lucas numbers (2, 1, 3, 4, 7, 11, 18, 29,…,. where L0 = 2). The simplest examples are 5x2 - y2 = 4, which contains the points (Fk, Lk) with odd subscripts, e.g. (1, 1), (2, 4), (5, 11), and 5x2 - y2 = -4, which contains the points with even subscripts, e.g. (0, 2), (1, 3), (3, 7); (see [1, 2]). These follow immediately from the identity(1)Our goal is to find more of these Fibonacci-Lucas hyperbolas.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.64
Locus problems concerning centroids of a cyclic quadrilateral and two
classic cubic curves
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  Authors: Fried; Michael N.
  Pages: 247 - 257
  Abstract: On his website dedicated to questions and investigations arising out of dynamic geometry technology, Michael de Villiers has a series called Geometry Loci Doodling [1]. These are locus problems connected to the centroids of cyclic quadrilaterals – ‘centroids’ in the plural, for there are three different kinds of centroid depending whether one understands the quadrilateral in terms of its vertices, perimeter or area. The corresponding centroids are the point-mass centroid, the perimeter-centroid, and the lamina-centroid. In each case, de Villiers keeps three vertices of the quadrilateral fixed on the circumcircle, and then traces the locus of the different centroids as the fourth point moves round the circle. In this paper, I shall take a brief look at the point-mass centroid and then a lingering view of the lamina-centroid.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.65
Dissecting attached squares
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  Authors: MacHale; Des
  Pages: 258 - 268
  Abstract: Many papers and indeed books have been written about the problem of dissecting a number of squares of different integer side length and reassembling the pieces to form a single square (see [1], [2] and [3]). For example, in the case of 32 + 42 = 52, we can achieve a four-piece dissection as follows:
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.66
New dualities in convex quadrilaterals
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  Authors: Dalcín; Mario
  Pages: 269 - 280
  Abstract: In [1] de Villiers points out the duality between sides and angles of quadrilaterals. The first objective of this article is make explicit two new dualities in the quadrilaterals. For this we will consider the diagonal segments: if diagonals AC and BD of a quadrilateral ABCD intersect at O, we call the diagonal segments OA, OB, OC, OD. According to [2, pp. 179-188], hierarchical classifications of the convex quadrilaterals are made taking as classification criteria the quantity and position of sides, angles and diagonal segments. In hierarchical classifications the more particular concepts form subsets of the more general concepts. Through classification according to the number and position of equal sides it is possible to define six families: four sides equal, at least three equal, two opposite pairs equal, two consecutive pairs equal, at least one opposite pair equal, at least one consecutive pair equal. In the families two opposite pairs equal and two consecutive pairs equal, the pairs may be equal to each other and then the four sides are equal. So in these two families the possibility DA = AB and AB = BC is excluded. Families analogous to the previous ones can be defined taking as a criterion of hierarchical classification the quantity and position of equal angles or the quantity and position of equal diagonal segments.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.67
The intriguing mechanics of a tractrix of cards
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  Authors: De; Subhranil
  Pages: 281 - 290
  Abstract: Figure 1 shows the photograph of a tractrix made of cards. This is a simple yet captivating way to make a tractrix — an arrangement that has recently appeared in some engaging pedagogical resources and literature [1, 2]. When closely spaced cards lean like this, one after another, over a horizontal plane, the contour created is a tractrix — a curve of significance in mathematics, since its revolution around its asymptote produces a pseudosphere: a curved surface with constant negative Gaussian curvature [3, 4].
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.68
On the class of an integer triangle
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  Authors: Read; Emrys
  Pages: 291 - 299
  Abstract: Any mathematics student who has ever used the cosine rule to investigate simple properties of an integer triangle will immediately have realised that the cosine of each angle of the triangle must be a rational number. It is clear, however, that the same is not in general true for the sines. In [1], it is shown how to use a property of the sines of the angles of an integer triangle to categorise the triangle as being of a particular class. In this article, we develop some of the concepts and results of [1] to derive a method for generating integer triangles of a given class. Finally, we apply our results to find all primitive integer triangles in the particular case of Heronian triangles.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.69
Monotonic series for fractions near π and their convergents
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  Authors: Lucas; Stephen K., Nimbran, Amrik Singh
  Pages: 300 - 309
  Abstract: We describe various methods to derive monotonic infinite series for fractions near π and obtain a variety of series for the special case of its convergents. These series immediately show that π is clearly different from these fractions, replicating with series the results in Dalzell [1, 2] and Lucas that used integrals with non-negative integrands to represent the gaps between π and fractions.
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.70
106.17 An interesting spin-off
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  Authors: De; Prithwijit, Bhattacharya, Sutanay
  Pages: 310 - 312
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.71
106.18 Impossibility of solving the quintic using Cardano’s Solution
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  Authors: Ohyama; Hiroshi
  Pages: 312 - 315
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.72
106.19 Some observations on inequalities related to Huygens’
inequality
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  Authors: Lord; Nick
  Pages: 316 - 318
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.73
106.20 When do we have 1 + 1 = 11 and
2 + 2 = 5'
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  Authors: Padmanabhan; R., Shukla, Alok
  Pages: 319 - 323
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.74
106.21 A class of interesting integrals
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  Authors: Levrie; Paul, Nimbran, Amrik Singh
  Pages: 323 - 325
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.75
106.22 The golden ratio represented by a tangent
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  Authors: Yoshida; Norio
  Pages: 325 - 329
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.76
106.23 Proof without words: sin 3x = 3 sin x − 4 sin3x
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  Authors: Subramaniam; K. B., Thomas, Aji
  Pages: 330 - 330
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.77
106.24 Proof without words: a Riemann sum
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  Authors: Plaza; Ángel
  Pages: 331 - 331
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.78
106.25 Three discs for the incentre
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  Authors: Lukarevski; Martin, Scott, J. A.
  Pages: 332 - 335
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.79
106.26 The nested polygons problem revisited
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  Authors: Lord; Nick
  Pages: 335 - 338
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.80
106.27 An interesting application of Ptolemy's inequality
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  Authors: Tho; Nguyen Xuan
  Pages: 338 - 340
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.81
106.28 Inequalities involving the inradius and altitudes of a triangle
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  Authors: Tho; Nguyen Xuan
  Pages: 341 - 342
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.82
106.29 An improvement on the Garfunkel-Bankoff inequality
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  Authors: Jiang; Wei-Dong
  Pages: 342 - 344
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.83
106.30 Threshold functions and the birthday paradox
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  Authors: Bevan; David
  Pages: 344 - 348
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.84
On ‘What makes a good Proof without Words’
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  Authors: Lukarevski; Martin
  Pages: 349 - 349
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.86
On ‘A pretty series revisited’
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  Authors: Jameson; Graham
  Pages: 350 - 350
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.87
On ‘Correct answer – dodgy method’
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  Authors: Lund-Hansen; Lars
  Pages: 350 - 351
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.88
On 105.28
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  Authors: Beardon; Alan
  Pages: 351 - 351
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.89
Problem Corner
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  Authors: L; N.J.
  Pages: 352 - 357
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.90
Student Problems
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  Authors: Woollacott; Beth
  Pages: 358 - 360
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.91
Pack up a penguin, journeys into the mathematics of area by Chris
Pritchard, pp. 240, £19.00 (MA members £13.30), ISBN 978-1-91161-608-5,
The Mathematical Association (2020)
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  Authors: Hall; Peter
  Pages: 361 - 361
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.92
Arts & minds: how the Royal Society of Arts changed a nation by Anton
Howes, pp. 387, £30 (hard), ISBN: 978-0-69118-264-3, Princeton University
Press (2020)
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  Authors: Crilly; Tony
  Pages: 362 - 363
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.93
Frank Ramsey: a sheer excess of powers by Cheryl Misak, pp. 537, £25
(hard), ISBN 978-0-19875-535-7, Oxford University Press (2020)
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  Authors: Crilly; Tony
  Pages: 363 - 366
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.94
The secret formula by Fabio Toscano, pp. 161, £22 (hard), ISBN
978-0-69118-367-1, Princeton University Press (2020)
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  Authors: Toller; Owen
  Pages: 366 - 367
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.95
The flying mathematicians of World War I by Tony Royle, pp. 269, £22.50
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  Authors: Crilly; Tony
  Pages: 367 - 370
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.96
Africa and mathematics: from colonial findings back to the Ishango Rods by
Dirk Huylebrouck, pp. 229, £27.99 (hardback), ISBN 978-3-03004-036-9
Springer Verlag (2019).
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  Authors: Mala; Firdous Ahmad
  Pages: 370 - 371
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.97
Mathematics is the poetry of science by Cédric Villani, translated by
Malcolm DeBevoise, pp 69, £9.99, ISBN 978-0-19-884643-7, Oxford
University Press (2020)
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  Authors: Leversha; Gerry
  Pages: 371 - 372
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.98
Representations of finite groups of Lie type (2nd edn.) by Francois Digne
and Jean Michel, pp 172, £37.99 (paper), ISBN 978-1-10872-262-9,
Cambridge University Press (2020)
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  Authors: Hunacek; Mark
  Pages: 372 - 373
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.99
Quantitative reasoning by Eric Zaslow, pp. 227, £26.99 (paper), ISBN
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  Authors: Hall; Peter
  Pages: 374 - 375
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.100
An introduction to functional analysis by James C. Robinson, pp 248,
£29.99 (paper), ISBN 978-0-52172-839-3, Cambridge University Press (2020)
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  Authors: Hunacek; Mark
  Pages: 375 - 376
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.101
A modern introduction to differential equations (3rd edn.) by Henry J.
Ricardo, pp. 539, £115 (hard), ISBN 978-0-12823-417-4, Academic
Press/Elsevir (2020)
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  Authors: Toller; Owen
  Pages: 376 - 377
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.102
The wonder book of geometry by David Acheson, pp. 280, £12.99, ISBN
978-0-19-884638-3, Oxford University Press (2020)
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  Authors: Leversha; Gerry
  Pages: 377 - 378
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.103
Thinking probabilistically by Ariel Amir, pp. 242, £39.99 (paper), ISBN
978-1-10878-998-1, Cambridge University Press (2021) (e-copy reviewed)
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  Authors: Toller; Owen
  Pages: 378 - 379
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.104
Fundamentals of graph theory by Allan Bickle, pp. 336, $85, ISBN
978-1-4704-5342-8, American Mathematical Society (2020)
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  Authors: Hunacek; Mark
  Pages: 379 - 380
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.105
Algorithms by Panos Louridas, pp. 312, $15.95 (paper), ISBN
978-0-26253-902-9, MIT Press (2020)
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  Authors: Aaronson; Hugo
  Pages: 380 - 381
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.106
The best writing on mathematics 2019 by Mircea Pitici (ed.), pp. 272,
£20.00, ISBN 978-0-691-19835-4, Princeton University Press (2019)
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  Authors: Leversha; Gerry
  Pages: 381 - 382
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.107
The mathematics lover’s companion by Edward Scheinerman, pp. 274,
£12.99 (paper), ISBN 978-0-30025-539-3, Yale University Press (2021)
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  Authors: Toller; Owen
  Pages: 383 - 383
  PubDate: 2022-06-22
  DOI: 10.1017/mag.2022.108