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Showing papers in "The Mathematical Gazette in 2020"


Journal ArticleDOI
TL;DR: In this article, an alternative graphical interpretation of the diagram reveals that it represents (1) in this form: 6[(k − 1)T12 + (k − 2)(T18 − T6) + (t24 − T12) +... +(T6k − T 6(k−2)))], where Tm = 2m (m + 1) from the standard formula for the sum of the first consecutive squares.
Abstract: We leave it to the reader to show that an alternative graphical interpretation of the diagram reveals that it represents (1) in this form: 6[(k − 1)T12 + (k − 2)(T18 − T6) + (k − 3)(T24 − T12) + ... +(T6k − T6(k − 2)))] , where . Tm = 2m (m + 1) From the standard formula for the sum of the first consecutive squares we find that (1) is given by . n 18k [k (4k + 1) − 5] An examination of parity confirms the result obtained graphically. 10.1017/mag.2020.21 PAUL STEPHENSON Böhmerstraße 66, 45144 Essen, Germany e-mail: pstephenson1@me.com

6 citations


Journal ArticleDOI
TL;DR: Josefsson and Markaryd as mentioned in this paper proposed the Mathematical Association 2020 Västergatan 25d, 285 37 markaryd, Sweden e-mail: martin.markaryd@hotmail.com
Abstract: References 1. T. Andreescu and D. Andrica, Complex numbers from A to ... Z, Birkhäuser (2nd edn.) 2014. 2. W. H. Echols, Some properties of a skewsquare, Amer. Math. Monthly 30 (March-April 1923) pp. 120-127. 3. M. Josefsson, Properties of equidiagonal quadrilaterals, Forum Geom. 14 (2014) pp. 129-144. 10.1017/mag.2020.62 MARTIN JOSEFSSON © The Mathematical Association 2020 Västergatan 25d, 285 37 Markaryd, Sweden e-mail: martin.markaryd@hotmail.com

5 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that triangles whose sides are in arithmetic progression are equivalent to triangles in which one of the sides is the arithmetic mean of the other two, and they also gave three geometric contexts in which such triangles appear.
Abstract: This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.

4 citations


Journal ArticleDOI
TL;DR: Weierstrass as discussed by the authors showed that any continuous function can be uniformly approximated by polynomials on a bounded, closed real interval, and this theorem can be generalized to continuous functions.
Abstract: A famous theorem of Weierstrass, dating from 1885, states that any continuous function can be uniformly approximated by polynomials on a bounded, closed real interval.

3 citations


Journal ArticleDOI

3 citations



Journal ArticleDOI
TL;DR: Bar Catalan's constant as mentioned in this paper is one of the most inscrutable constants in mathematics, and the question concerning its irrationality is not settled even though it has been studied extensively in the literature.
Abstract: There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by Scott in [1] quipped that this constant seemed almost as useful as the more widely known Euler–Mascheroni constant γ, particularly in the evaluation of definite integrals. And like γ, Catalan's constant continues to remain one of the most inscrutable constants in mathematics where the question concerning its irrationality is not settled.

2 citations


Journal ArticleDOI
TL;DR: The program goals include improving student motivation, attention, behaviour, attendance and focus, as well as introducing students to an outdoor skill with the hope that this may increase attention to wildlife conservation efforts in the future.
Abstract: The National Archery in the Schools Program (NASP) began in Kentucky, USA in 2002 and has rapidly expanded to thousands of students around the United States. The program teaches archery in physical education classes and organises tournaments for student archers in elementary school and high school. The program goals include improving student motivation, attention, behaviour, attendance and focus, as well as introducing students to an outdoor skill with the hope that this may increase attention to wildlife conservation efforts in the future.

2 citations


Journal ArticleDOI
TL;DR: Folded paper road maps are found next to sextants in the pile of obsolete navigation tools GPS navigation apps like Waze and Google Maps are available on all smartphones and most new cars.
Abstract: Folded paper road maps are found next to sextants in the pile of obsolete navigation tools GPS navigation apps like Waze and Google Maps, accurate to within a few metres, are available on all smartphones and most new cars These apps provide drivers with real time traffic conditions and suggest minimum drive time routes, giving drivers the ability to avoid congestion and delays caused by heavy traffic, accidents, road construction and other hindrances

2 citations


Journal ArticleDOI
TL;DR: Koner et al. as discussed by the authors presented a combinatorial proof of Fermat's Little Theorem and showed that it can be proved by a single theorem prover with respect to the number of vertices.
Abstract: 4. https://www.jstor.org/stable/pdf/30037444.pdf 5 Solomon W. Golomb, Combinatorial proof of Fermat's Little Theorem, American Mathematical Monthly, 63 (10) (1956) p. 718. 6. Giedrius Alkauskas, A curious proof of Fermat's Little Theorem, American Mathematical Monthly 116 (4) (2009) pp. 362-364. 10.1017/mag.2020.17 SOURAV KONER Tezpur University, Tezpur, Assam, India, 784028 e-mail: harakrishnaranusourav@gmail.com

2 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider discrete spirals, i.e., spiral structures that are different from the traditional Archimedean spiral (r = a + bθ) and the logarithmic spiral (R = aebθ).
Abstract: Most spirals are continuous spirals, certainly those famous in history such as the traditional Archimedean spiral (r = a + bθ) and the logarithmic spirals (r = aebθ Here we consider ‘discrete spirals’

Journal ArticleDOI
TL;DR: In fact, the Gauss circle problem is to be regarded in the same spirit as the notoriously difficult Gauss triangle problem as discussed by the authors. And the number of integer points on the hyperboloid with can be estimated as follows.
Abstract: More generally, any matrix that preserves the quadratic form carries a point on the hyperboloid to another one. 3 × 3 x2 + y2 − z2 Although one can obtain parametrisations such as above, one cannot expect a description of all integer points akin to Pythagorean triples for the circle problem. In fact, the problem is to be regarded in the same spirit as the notoriously difficult Gauss circle problem. The number of integer points on the hyperboloid with can be estimated as follows. (x, y, z) |x| , |y| , |z| ≤ t If denotes the number of ordered tuples of nonnegative integers with , then consider the solutions for a given . The number of solutions rd (n) (x1, ... , xd) n = ∑ d

Journal ArticleDOI
TL;DR: The material of elementary algebra is full of interesting results and challenging problems, but elementary algebra seemed different. as mentioned in this paper continue to be astonished at how the material of school mathematics provides scope for investigations of considerable depth.
Abstract: I continue to be astonished at how the material of school mathematics provides scope for investigations of considerable depth. Elementary geometry and number theory are full of interesting results and challenging problems, but elementary algebra seemed different. I thought that I knew a bit about the real quadratic function f (x) = ax − bx + c. (My choice of a negative sign for the second term is deliberate.) Certainly I knew that when there was something special about . I knew too that and give rise to different situations. However, it never occurred to me that , and had anything special about them, let alone and a host of other particular values. b = 4ac f (x) b > 4ac b < 4ac b = ac b = 2ac b = 3ac b = 2 (3 + 5) ac

Journal ArticleDOI
TL;DR: The Thebault configuration as discussed by the authors is a variant of the first problem, which states that the centres of the newly constructed squares also form a square, and hence the center of a parallelogram can be represented as a square.
Abstract: Given a parallelogram, construct squares outwardly on its sides; hereafter we will call this the ‘Thebault configuration’. Our name derives from Thebault’s celebrated first problem, which states that the centres of these newly constructed squares also form a square.

Journal ArticleDOI
TL;DR: Expectation values, which are often associated with waiting times for success, may, at times, speak more clearly and poignantly about the uncertainty of an event than a theoretical probability.
Abstract: Probability and expectation are two distinct measures, both of which can be used to indicate the likelihood of certain events. However, expectation values, which are often associated with waiting times for success, may, at times, speak more clearly and poignantly about the uncertainty of an event than a theoretical probability. To illustrate the point, suppose the probability of choosing a winning lottery ticket is 2.5 × 10−8. This information may not communicate the unlikely odds of winning as clearly as a statement like, “If five lottery tickets are purchased per day, the expected waiting time for a first win is about 22000 years.”

Journal ArticleDOI
TL;DR: In this paper, the authors elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way, and prove a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem.
Abstract: We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we prove a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem. Subsequently, we briefly discuss two fundamental MCMC methods, the Gibbs and Metropolis-Hastings sampler. Only very basic knowledge about matrices, convergence of real sequences and probability theory is required.

Journal ArticleDOI
A. J. E. Ryba1
TL;DR: Conway as discussed by the authors was a magnet for all mathematicians, and he welcomed all who came to him, and his insights changed our understanding of several very serious branches of mathematics, such as game theory and game theory.
Abstract: John Horton Conway lived to discover the Mathematics behind problems, always working to isolate a pure, essential kernel of truth. He loved to communicate these simple truths to others, often changing the way they thought. Everything John touched turned to Mathematics, and to very beautiful Mathematics. John was generous with his mathematical riches; he gave them to everyone that showed interest — whether at the coffee house, at the sushi restaurant, or in Mathematics departments. He was a magnet for all mathematicians, and he welcomed all who came to him. John would find a way to start with some simple calculation, a game or puzzle and turn it into whatever he wanted to explain. John often chose problems about games and recreational topics, but the insights he derived changed our understanding of several very serious branches of Mathematics.

Journal ArticleDOI
TL;DR: The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers as mentioned in this paper and open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics.
Abstract: The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analogue to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley-Hamilton theorem.

Journal ArticleDOI
TL;DR: The authors argue that teaching for reasoning is a powerful complement to teaching that is more focused on skills and procedures, and argue that we need to make sure students of all ages engage in a range of mathematical reasoning.
Abstract: In this paper I look at different aspects of mathematical reasoning, and argue that we need to make sure students of all ages engage in a range of mathematical reasoning, particularly given the evidence that teaching for reasoning is a powerful complement to teaching that is more focused on skills and procedures.

Journal ArticleDOI
TL;DR: The following theorems are famous landmarks in the history of number theory: as discussed by the authors Theorem 1 (Fermat-Euler): A number is representable as a sum of two squares if, and only if, it has the form PQ2, where P is free of prime divisors.
Abstract: The following theorems are famous landmarks in the history of number theoryTheorem 1 (Fermat-Euler): A number is representable as a sum of two squares if, and only if, it has the form PQ2, where P is free of prime divisors q ≡ 3 (mod 4)Theorem 2 (Lagrange): Every number is representable as a sum of four squaresTheorem 3 (Gauss-Legendre): A number is representable as a sum of three squares if, and only if, it is not of the form 4a (8n + 7)

Journal ArticleDOI
TL;DR: In this paper, the Feuerbach line is shown to be related to the points on the following relative coordinates: F (λ (μ − ν), μ (ν − λ), ν (λ − μ)), I (1 − μν, 1− νλ, 1 − ǫμ), N (1 + βγ, 1+ γα, 1 + αβ).Then, to see how these points are related, we need only work with one coordinate (, say).
Abstract: (x + y + z = 1) α = cot A β = cot B γ = cot C (βγ + γα + αβ = 1) λ = tan 2A μ = tan 2B ν = tan 2C (μν + νλ + λμ = 1, Σ = λ + μ + ν, Π = λμν) 2α = 1 λ − λ 2β = 1 μ − μ 2γ = 1 ν − ν [] () H [βγ + γα + αβ] G (1, 1, 1) Now further relative coordinates are F (λ (μ − ν) , μ (ν − λ) , ν (λ − μ)) , I (1 − μν, 1 − νλ, 1 − λμ) , N (1 + βγ, 1 + γα, 1 + αβ) . Then, to see how these points on the Feuerbach line are related, we need only work with one coordinate ( , say). x Thus for : N

Journal ArticleDOI
TL;DR: The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700-1782), Leonhard Euler (1707-1783), and Christian Goldbach (1690-1764) as mentioned in this paper.
Abstract: The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.

Journal ArticleDOI
TL;DR: The question arises about the long term affordability of end of life care to those having to fund it, a question that ever more people both nationally and globally are having to confront.
Abstract: To live to a ripe old age, untroubled by health problems, physical or mental, is an almost universal aspiration. But most people are not so lucky and will likely be in care homes for their final years, with varying levels of disease, disability and dementia. Kinley et al [1] maintain that over a fifth of the population of developed countries die in care homes. Moreover, the financial cost of this end of life care, which is the focus of this paper, can be daunting and require much planning [2]. It was reported in 2017 that, in the UK, care home costs are rising up to twice as fast as inflation [3]. Consequently the question arises about the long term affordability of such care to those having to fund it, a question that ever more people both nationally and globally are having to confront.

Journal ArticleDOI
TL;DR: Lukarevski et al. as discussed by the authors presented an alternate proof of Gerretsen's inequalities, Elem. Math. Gaz. 101 (March 2017) p.123.
Abstract: References 1. F. Leuenberger, Problem E 1573, Amer. Math. Monthly 71 (1963) p. 331; solution by L. Carlitz, ibid. 72 (1964) pp. 93-94. 2. M. Lukarevski, An inequality arising from the inarc centres of a triangle, Math. Gaz. 103 (November 2019) pp. 538-541. 3. O. Bottema, R. Z. Djordjevic, R. R. Janic, D. S. Mitrinovic, P. M. Vasic, Geometric inequalities., Wolters-Noordhoff, Groningen (1969). 4. M. Lukarevski, Exradii of the triangle and Euler's inequality, Math. Gaz. 101 (March 2017) p.123. 5. N. Altshiller-Court, College Geometry, Barnes & Noble (1952). 6. G. Leversha, The geometry of the triangle, UKMT (2013). 7. R. Johnson, Advanced Euclidean geometry, Dover (1960). 8. M. Lukarevski, An alternate proof of Gerretsen's inequalities, Elem. Math. 72 (1) (2017) pp. 2-8.


Journal ArticleDOI
TL;DR: Conway's book as discussed by the authors is a good introduction to the first year of a calculus course with a focus on real analysis, which is often a major difficulty for mathematics undergraduates.
Abstract: Reviews A first course in analysis by John B. Conway, £39.99, ISBN 978-1-10717-314-9, Cambridge University Press (2017). It's hard to give a pithy one-sentence summary of what ‘real analysis’ entails. Perhaps it's easier to list some of the unmissable early highlights in this area for any student of mathematics. One must develop a formalism for proving seemingly intuitive results about real numbers, sequences of real numbers, and sequences of functions of real numbers; one must test one's intuition for when intuitively-obvious results do and don't hold in general; and one must develop the notation and machinery to give formal definitions of the basic, hugely-applicable notions of calculus. The title of Conway's book follows in the tradition of Bannan, Burkill, Pedrick and Yau, among others. This title may not be appropriate to all potential readers. Indeed, the author explains that he is targeting senior undergraduates, or graduate students, perhaps making the transition into rigorous mathematics from a more applied or general perspective. Not unlike ‘baby Rudin’ [1], the style of the early chapters may be a bit more dense than ideal for a student covering the material as part of the first year or two of an undergraduate mathematics degree. Despite Conway's engaging and personable prose (‘Confused? Frankly, the distinction will affect very little that is said’ and so forth are unlikely to remind any reader of Rudin's treatment...), one notices that there are no diagrams in the roughly two hundred pages of Chapters 2 through 7. Fluency with ‘epsilons and deltas’, as required to formalise the early results in any analysis course, is regularly found hard, and a major hurdle to mathematics undergraduates. Real analysis generally provides a valuable testing-ground for these skills which are then applied throughout the pure mathematical journey. This account, although it includes some exercises and examples, is probably not the ideal starting point for a reader still getting to grips with ordering multiple quantifiers, equivalent definitions, and so on. That said, the second half of Conway's text is highly recommended. As it explores more technical theory, going towards higher dimensions and the famous theorems of Gauss and Stokes, the level of detail seems extremely appropriate. This would be valuable both as an introduction to these topics (in a much more formal way than one might require for applications), or as a repository of the main results. An unusual touch is the addition of quarter-page biographies of all the mathematicians whose work is referenced, which goes well beyond the basic details, and in many cases discusses how the work fitted into the mathematicians’ personal and broader context. It also provides a valuable reminder of the interconnectedness of scientific progress – it was certainly news to me that Hermite (of the polynomials, and the transcendality of ) was the doctoral advisor of Poincare! e

Journal ArticleDOI
TL;DR: In this article, Lyness challenged readers of the Gazette to find a recurrence relation of order 2 which would generate a cycle of period 7 for almost all initial values, and the reader was challenged to find the shortest possible cycle.
Abstract: In 1942 R. C. Lyness challenged readers of the Gazette to find a recurrence relation of order 2 which would generate a cycle of period 7 for almost all initial values [1].

Journal ArticleDOI
TL;DR: Activities should provide students with a variety of challenging experiences through which they can actively construct mathematical meanings for themselves in a process of meaning-making.
Abstract: Nowadays there is considerable agreement among educators that learning mathematics fundamentally involves making mathematics [1]. Students learn mathematics while working on tasks that they consider meaningful and worthwhile, and their interest is aroused when they can see the point of what they are being asked to do. Given that learning mathematics involves a process of meaning-making - the use of mathematical language, symbols and representations as learners negotiate ideas – activities should provide students with a variety of challenging experiences through which they can actively construct mathematical meanings for themselves.

Journal ArticleDOI
TL;DR: The importance of the if condition in L'Hôpital's Rule has been discussed in this article, where the use of shared asymptotic behaviours such as and, when is suggested as a complementary technique for solving limits.
Abstract: Students are fascinated by L'Hôpital's Rule when they see it for the first time. However, they must understand the importance of the ‘if’ condition in L'Hôpital's Rule [1, 2]. Also they should learn other complementary techniques for solving limits, including the use of shared asymptotic behaviours such as and , when . Some examples of the limitations on the use of L'Hôpital's Rule without a second thought are ln (1 + x) x x → 0

Journal ArticleDOI
TL;DR: This article shows that, for any valid set, the authors can always construct the net of a corresponding tetrahedron, and shows that there are always several distinct infinite subsets that are constructible and whose edge lengths can be determined exactly.
Abstract: We tackle an unusual problem that, as far as I know, is not in the standard literature. To state it concisely I use what I call ‘valid sets’. We know that in any three or more positive quantities, only the largest can be half or more of the total value; and is then obviously greater than the sum of all the others. But if the largest is less than half the total of the set, it must be less than the sum of the others; and this is true for every element in this set. I call such a set ‘valid’. For example, the sides of a triangle are valid, as are the face areas of a tetrahedron. Our problem relates to the converse for a tetrahedron: given any four valid quantities, is there always a tetrahedron with those face areas? In this article I answer this by showing that, for any valid set, we can always construct the net of a corresponding tetrahedron. In fact, for any given valid set there is always an infinity of non-congruent tetrahedra with the given face areas. Although in general there are no formulae that give the exact edge lengths of these solids, I show that there are always several distinct infinite subsets that are constructible and whose edge lengths can be determined exactly.