The Mathematical Gazette
Number of Followers: 1 Subscription journal ISSN (Print) 00255572  ISSN (Online) 20566328 Published by Cambridge University Press [353 journals] 
 MAG volume 106 issue 566 Cover and Front matter

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Pages: 1  2
PubDate: 20220622
DOI: 10.1017/mag.2022.109

 MAG volume 106 issue 566 Cover and Back matter

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Pages: 1  3
PubDate: 20220622
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: 20220622
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: 20220622
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 wellknown 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 base2 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: 20220622
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 rightangle 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 handheld 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: 20220622
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: 20220622
DOI: 10.1017/mag.2022.63

 FibonacciLucas hyperbolas

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Authors: Sporn; Howard
Pages: 242  246
Abstract: Let us define a FibonacciLucas 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 FibonacciLucas hyperbolas.
PubDate: 20220622
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 pointmass centroid, the perimetercentroid, and the laminacentroid. 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 pointmass centroid and then a lingering view of the laminacentroid.
PubDate: 20220622
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 fourpiece dissection as follows:
PubDate: 20220622
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. 179188], 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: 20220622
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: 20220622
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: 20220622
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 nonnegative integrands to represent the gaps between π and fractions.
PubDate: 20220622
DOI: 10.1017/mag.2022.70

 106.17 An interesting spinoff

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Authors: De; Prithwijit, Bhattacharya, Sutanay
Pages: 310  312
PubDate: 20220622
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: 20220622
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: 20220622
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: 20220622
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: 20220622
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: 20220622
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: 20220622
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: 20220622
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: 20220622
DOI: 10.1017/mag.2022.79

 106.26 The nested polygons problem revisited

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Authors: Lord; Nick
Pages: 335  338
PubDate: 20220622
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: 20220622
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: 20220622
DOI: 10.1017/mag.2022.82

 106.29 An improvement on the GarfunkelBankoff inequality

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Authors: Jiang; WeiDong
Pages: 342  344
PubDate: 20220622
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: 20220622
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: 20220622
DOI: 10.1017/mag.2022.86

 On ‘A pretty series revisited’

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Authors: Jameson; Graham
Pages: 350  350
PubDate: 20220622
DOI: 10.1017/mag.2022.87

 On ‘Correct answer – dodgy method’

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Authors: LundHansen; Lars
Pages: 350  351
PubDate: 20220622
DOI: 10.1017/mag.2022.88

 On 105.28

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Authors: Beardon; Alan
Pages: 351  351
PubDate: 20220622
DOI: 10.1017/mag.2022.89

 Problem Corner

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Authors: L; N.J.
Pages: 352  357
PubDate: 20220622
DOI: 10.1017/mag.2022.90

 Student Problems

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Authors: Woollacott; Beth
Pages: 358  360
PubDate: 20220622
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 9781911616085,
The Mathematical Association (2020)
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Authors: Hall; Peter
Pages: 361  361
PubDate: 20220622
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: 9780691182643, Princeton University
Press (2020)
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Authors: Crilly; Tony
Pages: 362  363
PubDate: 20220622
DOI: 10.1017/mag.2022.93

 Frank Ramsey: a sheer excess of powers by Cheryl Misak, pp. 537, £25
(hard), ISBN 9780198755357, Oxford University Press (2020)
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Authors: Crilly; Tony
Pages: 363  366
PubDate: 20220622
DOI: 10.1017/mag.2022.94

 The secret formula by Fabio Toscano, pp. 161, £22 (hard), ISBN
9780691183671, Princeton University Press (2020)
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Authors: Toller; Owen
Pages: 366  367
PubDate: 20220622
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: 20220622
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 9783030040369
Springer Verlag (2019).
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Authors: Mala; Firdous Ahmad
Pages: 370  371
PubDate: 20220622
DOI: 10.1017/mag.2022.97

 Mathematics is the poetry of science by Cédric Villani, translated by
Malcolm DeBevoise, pp 69, £9.99, ISBN 9780198846437, Oxford
University Press (2020)
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Authors: Leversha; Gerry
Pages: 371  372
PubDate: 20220622
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 9781108722629,
Cambridge University Press (2020)
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Authors: Hunacek; Mark
Pages: 372  373
PubDate: 20220622
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: 20220622
DOI: 10.1017/mag.2022.100

 An introduction to functional analysis by James C. Robinson, pp 248,
£29.99 (paper), ISBN 9780521728393, Cambridge University Press (2020)

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Authors: Hunacek; Mark
Pages: 375  376
PubDate: 20220622
DOI: 10.1017/mag.2022.101

 A modern introduction to differential equations (3rd edn.) by Henry J.
Ricardo, pp. 539, £115 (hard), ISBN 9780128234174, Academic
Press/Elsevir (2020)
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Authors: Toller; Owen
Pages: 376  377
PubDate: 20220622
DOI: 10.1017/mag.2022.102

 The wonder book of geometry by David Acheson, pp. 280, £12.99, ISBN
9780198846383, Oxford University Press (2020)
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Authors: Leversha; Gerry
Pages: 377  378
PubDate: 20220622
DOI: 10.1017/mag.2022.103

 Thinking probabilistically by Ariel Amir, pp. 242, £39.99 (paper), ISBN
9781108789981, Cambridge University Press (2021) (ecopy reviewed)
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Authors: Toller; Owen
Pages: 378  379
PubDate: 20220622
DOI: 10.1017/mag.2022.104

 Fundamentals of graph theory by Allan Bickle, pp. 336, $85, ISBN
9781470453428, American Mathematical Society (2020)
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Authors: Hunacek; Mark
Pages: 379  380
PubDate: 20220622
DOI: 10.1017/mag.2022.105

 Algorithms by Panos Louridas, pp. 312, $15.95 (paper), ISBN
9780262539029, MIT Press (2020)
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Authors: Aaronson; Hugo
Pages: 380  381
PubDate: 20220622
DOI: 10.1017/mag.2022.106

 The best writing on mathematics 2019 by Mircea Pitici (ed.), pp. 272,
£20.00, ISBN 9780691198354, Princeton University Press (2019)
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Authors: Leversha; Gerry
Pages: 381  382
PubDate: 20220622
DOI: 10.1017/mag.2022.107

 The mathematics lover’s companion by Edward Scheinerman, pp. 274,
£12.99 (paper), ISBN 9780300255393, Yale University Press (2021)
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Authors: Toller; Owen
Pages: 383  383
PubDate: 20220622
DOI: 10.1017/mag.2022.108
