Showing papers in "The Mathematical Gazette in 2018"

102.53 The excentral triangle and a curious application to inequalities

Abstract: 102.53 The excentral triangle and a curious application to inequalities Introduction The celebrated Weitzenböck's inequality from 1919 is [1] a + b + c ≥ 4 3 , (1) where are sides of a triangle and is its area. Equality holds if, and only if, the triangle is equilateral. The inequality was given as Problem 2 at the third International Mathematical Olympiad held in 1961 in Hungary. One sharpening is the well-known Finsler-Hadwiger inequality from 1937 [2] a, b, c ABC

Fibonacci periods and multiples

Abstract: The well-known Fibonacci numbers Fn are defined by the recurrence relation Fn = F n – 1 + F n – 2. (1) together with the starting values F 0 = 0, F 1 = 1, or equivalently F 1 = F 2 = 1. We record the first few: The recurrence relation can also be applied backwards in the form Fn = F n + 2 – F n + 1 to define Fn for n 0.

Discrete version of the Pythagorean theorem

Abstract: The following observations are motivated by the facts that the area of a planar figure displayed on a screen can be expressed by a certain number of pixels; and if the figure is drawn by a plotter, then its area can be characterised by the total length of a line which fills it in. The generalisations of the Pythagorean theorem are of three kinds. Firstly, the squares on the sides of the right triangle are substituted by other geometrically similar planar figures (Euclid's Elements Book VI, Proposition 31 [1]). Secondly, the assumption of the right angle is omitted (the law of cosines), or both of these generalizations occur simultaneously (Pappus’ area theorem [2], see also H. W. Eves [3]). Thirdly, mathematical spaces other than the plane are considered (for example, de Gua-Faulhaber theorem about trirectangular tetrahedra [3], further generalised by Tinseau [4], Euclidean n-spaces, Banach spaces [5], see also [6]).

102.23 A visual proof that is irrational

Abstract: We now attempt to tile the diagonal of the large rectangle with copies of the small rectangle. One possibility is that, as in Figure 2(a), we exactly reach the bottom right-hand corner in which case (looking at the shorter sides), divides so that is an integer. Alternatively, as in Figure 2(b), we stop short of the bottom right-hand corner when the shaded rectangle left is strictly smaller than the rectangle, but is similar to it and with integer side-lengths. This is a contradiction – either because the argument may be repeated to give an infinite decreasing sequence of whole numbers or, by starting with , the side-lengths of the smallest rectangle with . b a N = a b

A vintage Listener crossword

Abstract: This Article deals with the following problem. Problem: Find positive integers x, y and z with x < y < z such that x + y, x + z, y + z and x + y + z are squares. (1) The interest in this problem arose from a conversation between the first two authors at The Times Listener Crossword Setters’ Dinner in March 2017. The Listener was a weekly magazine published by the British Broadcasting Corporation (BBC) from 1929 until 1991. Its main purpose was to reproduce the text of broadcast talks. However, from 1930 onwards it also contained a crossword, which was acknowledged to be the hardest cryptic crossword in a national weekly. When the magazine ceased publication, the crossword was ‘rescued’ by The Times newspaper and has appeared therein on Saturdays for over 25 years.

The seven circles theorem revisited

Abstract: The circles C1, & , Cn form a chain of length n if Ci touches Ci + 1, for i = 1, & , n − 1, and the chain is closed if also Cn touches C1. A cyclic chain is a chain for which all the circles touch another circle S, the base circle of the chain. If Ci touches S at Pi, then P1, & , Pn are the base points of the chain. Sometimes there may be coincidences among the base points; in particular, if Pi = Pj, then the line PiPj should be interpreted as the tangent S to at Pi.The seven circles theorem first appeared in [1, §3.1], and some historical details of its genesis can be found in John Tyrrell's obituary [2]. The theorem concerns a closed cyclic chain of length 6, and says that, if a certain extra condition is satisfied, then the lines P1P4, P2P5, P3P6 joining opposite base points are concurrent. Here and throughout, ‘concurrent’ should be read as ‘concurrent or all parallel’, that is, the point of concurrency might be at infinity.

102.44 Another proof of the irrationality of square roots

Abstract: References 1. Wikipedia, Convex function, https://en.wikipedia.org/wiki/Convex_function. 2. Jim Farmer, Using Taylor's formula to compare compound and simple interest, Math. Gaz. 97 (March 2013) pp. 150-153. 3. Wikipedia, Taylor's Theorem, https://en.wikipedia.org/wiki/Taylor's_theorem 4. J. Siegel and D. Swanson, The methods and materials of demography (2nd Edn.), Elsevier Academic Press (2004) Chapter 11. 10.1017/mag.2018.123 MING FANG, SHAHROOZ MOOSAVIZADEH, MUSHTAQ A. KHAN Dept. of Mathematics, Norfolk State University, Norfolk, VA 23504 USA e-mails: mfang@nsu.edu, smoosavizadeh@nsu.edu, makhan@nsu.edu

The existence of triangles, tetrahedra, and higher-dimensional simplices with prescribed exradii

Abstract: This Article is inspired by a problem that appeared recently in the American Mathematical Monthly, namely 11972 in [1, p. 369]. The problem asks the readers to prove that if r is the inradius of a tetrahedron, and if r1, r2, r3, r4 are its exradii, then By taking r = r1 = r2 = r3 = r4 = 1, one sees that (1) is not true for all positive numbers r, r1, r2, r3, r4. This is not surprising, since r is dependent on r1, r2, r3, r4 by the elegant relation which we shall prove in Theorem 2 below; see also, for example, [2, (5), §266, p. 92], [3. §41, 2°, (1), p. 76] and [4, Problem 6′(i), p. 39]. Using this relation, one can rewrite (1) as Intuitively, the four numbers r1, r2, r3, r4 are independent, and one may thus ask whether the inequality (3) holds for all positive numbers r1, r2, r3, r4 regardless of being the exradii of some tetrahedron or not.

102.20 The sum of the first n cubes is a square – two Proofs without Words

Abstract: 102.21 An addendum to Estermann's proof of the irrationality of 2 Estermann [1] gave an elegant proof of the irrationality of the square root of 2. As an afterthought, he remarks “If in this proof we replace 2 by any natural number whose square root is not an integer, and replace 1 by the greatest integer less than . we obtain a proof that the square root of an integer is either an integer or an irrational.”. Flanders [2] carried out the extension that Estermann remarked upon, and Siu [3] gave a geometric interpretation of Estermann's argument, making a plausible historical case that it could well have occurred to the Pythagoreans. Apostol [4] gave a pretty geometric proof of the irrationality of the square root of 2, which aligns perfectly with Estermann's proof. Unfortunately his geometric method only extends to prove that is an irrational number if the positive integer is not a perfect square. The point of this Note is to prove that Estermann's elegant argument, using only the observation that the h h

102.08 A method of evaluating ζ (2) and

Abstract: Introduction I have recently been involved in setting up a new DfE-funded resource aimed at students preparing for STEP* [1]. I came across an interesting question while trawling through the STEP data base [2] for suitable material. It is from 1998 and I am surprised that I failed to notice at the time that only a few more calculations are needed to provide a rather simple evaluation of , expressed as an infinite sum: ζ (2)

Central force fields and Kepler's laws

Abstract: In this survey we give very simple proofs of Kepler's Laws and other facts about central force fields using only Newton's second law, Newton's law of universal gravitation, basic notions of vector calculus, and an elementary double integral.Hopefully, this article will help undergraduate students of mathematics and engineering who wish to understand these fundamental scientific discoveries.In many textbooks (see, for instance, [1, 2, 3, 4, 5]), Kepler's Laws are obtained using conservation of energy and angular momentum, differential equations, mobile reference systems, or notions not so well-defined such as differentials or ‘infinitesimal elements’. Some of the arguments appear to be rather involved if one is not accustomed to them, whereas the proof of Kepler's Laws may actually be obtained from quite simple facts.

102.14 A Note on the Feuerbach triangle

Abstract: References 1. S. Grozdev and D. Dekov, Barycentric coordinates: formula sheet, International Journal of Computer Discovered Mathematics 1 (2016) no 2, pp.75-82. http://www.journal-1.eu/2016-2/Grozdev-Dekov-BarycentricCoordinates-pp.75-82.pdf 2. P. Yiu, Introduction to the geometry of the triangle, 2013, http:// math.fau.edu/Yiu/YIUIntroductionToTriangleGeometry130411.pdf 3. E. W. Weisstein, Tangential triangle, MathWorld A Wolfram Web Resource, http://mathworld.wolfram.com/ 4. G. Leversha, The geometry of the triangle, UKMT (2013). 5. C. Kimberling, Encyclopedia of Triangle Centers ETC, http:// faculty.evansville.edu/ck6/encyclopedia/ETC.html 10.1017/mag.2018.22 SAVA GROZDEV VUZF University of Finance, Business and Entrepreneurship, Gusla Street 1, 1618 Sofia, Bulgaria e-mail: sava.grozdev@gmail.com HIROSHI OKUMURA Maebashi Gunma, 371-0123, Japan e-mail: hokmr@protonmail.com DEKO DEKOV Zahari Knjazheski 81, 6000 Stara Zagora, Bulgaria e-mail: ddekov@ddekov.eu

Blocks of decimal digits

Abstract: Recently a friend kindly made me a birthday card whose background consisted of rows and rows of digits, some 3000 in all. There appeared to be no discernible pattern in the digits. Perhaps they had been taken from a table of random numbers. They were certainly not the opening digits of the decimal parts of π or , although they might, so far as I knew, have been consecutive digits of either number in some section remote from the decimal point.On thinking about this, I realised that they must be the opening digits of the decimal part of the square root of some whole number. Indeed, they must be the opening digits of the decimal parts of the square roots of infinitely many positive integers. It is remarkably easy to prove this and the argument is simple enough to be appreciated and understood in school classrooms at GCSE level.

Rational arc length

Abstract: Finding an expression for the length of a curve is one of the simpler geometric applications of the integral. If f is a function with a continuous derivative, then the expression gives the length of the curve y = f (x) on an interval [a, b]. However, after writing out the integrand for familiar functions such as y = x2 and y = sin x, it quickly becomes apparent that, in general, finding an antiderivative is a challenge. Of course, a computer can give accurate approximations for the value of the integral for the length of a curve, but it would be nice to find the exact length rather than a decimal approximation. In his work on geometry (from 1637), Descartes stated that he believed it was not possible to determine the exact lengths of curves. However, just twenty years later, William Neile was able to find the length of arcs of semicubical parabolas (see Katz [1]). These curves have the form y = kx3/2 and are usually the first examples or exercises given to students since the resulting integral is very easy to compute. In this paper, we are going to examine this curve and other related curves and consider problems such as the following: find rational numbers a and b so that the length of the curve over the interval is an [a, b] integer. As we shall see, problems such as this provide a variety of opportunities for undergraduate students to explore some interesting mathematics arising from a few simple and accessible questions.

Fisher’s F-ratio illustrated graphically

Abstract: This paper aims to make Fisher’s F-ratio more easily understood by representing it graphically, using model comparison. In the graph, the null and experimental models are plotted with the number of parameters on the horizontal axis and goodness of fit to data on the vertical axis. The value of F appears as the ratio between the slopes of two lines in the diagram. The diagram gives rise in an obvious way to the “Occam line”, which is shown to embody the principle of parsimony. There is an organic relationship between the F-ratio and the Occam line, such that the plot point representing the experimental hypothesis lies below the Occam line if, and only if, the F-ratio is greater than one. The paper also shows using a simple geometrical argument that the natural measure of effect size when using the F-ratio is adjusted R-squared, and argues that its ANOVA counterpart, epsilon-squared, is superior to the commonly used omega-squared measure.

102.38 A very elementary short proof of Conway's little theorem

Abstract: Theorem (Conway's Little Theorem): A triangle is equilateral if, and only if, the ratio between the length of any two sides is rational and the ratio between any two angles is also rational. This theorem first appeared in [1, last comment], and recently with some added witty remarks (e.g., why should only Fermat have a little theorem? .....) has re-appeared in [2], and Conway this time calls it his little theorem. We should admit that the re-publication of the theorem in [2], with those added witty remarks, made us read the theorem and its proof in [2] more carefully. Since the theorem is a nice characterisation of equilateral triangles (perhaps, an unusual characterisation in the classical sense of Euclidean geometry), one is tempted to present a geometric proof which applies tools as elementary as possible. Before proceeding with our promised proof, let us recall that as a consequence of two rather sophisticated theorems in [3, Theorems 3.9 and 3.11], it is shown in [3, Corollary 3.12] that the only rational values of the trigonometric functions of are given by: , , , , where , with rational. θ sin θ cos θ = 0, ±1, ±2 sec θ, cosec θ = ±1, ±2 tan θ, cot θ = 0, ±1 θ = rπ r In what follows, we first give a quick self-contained proof of this interesting fact; see also [4, p. 358]. It is clear that we may, without loss, prove this only for the cosine function since, for example,

102.18 The problème des ménages revisited

Abstract: Recurrence relations that compute the answer to Tait's question were soon given by Cayley and Muir [2, 3, 4, 5]. Almost fifteen years later, Lucas [6], evidently unaware of the work of Tait, Cayley and Muir, posed the problem in the formulation of husbands and wives, named it problème des ménages and supplied a recurrence relation already described by Cayley and Muir. But it was not until forty-three years later that an explicit formula for Tait's problem was given by Touchard [7], alas without a proof. (For historic accounts, see Kaplansky and Riordan [8] and Dutka [9].) The first proof for Touchard's explicit formula was given nine years later (sixty-five years later than Tait's question) by Kaplansky [10]. Specifically he showed that the number of permutations of that differ in all places from both the identity permutation and the cyclic permutation is {1, ... , n}

Emmental squares are tasty

Abstract: A shape is called equable if its area and perimeter are numerically equal relative to some given system of units. For example, if a square is equable, then its side, a, must satisfy 4a = a 2. So there is only one equable square, and it has side 4. It is easy to investigate this idea for other shapes. Though not connected with this problem, the work of Imre Lakatos suggested a generalisation to us. Lakatos showed that Euler's classical formula V + F = E − 2 for polyhedra could be extended when the notion of tunnels was introduced [1].

102.13 Distance from the incentre of the tangential triangle of an obtuse triangle to the Euler line

Abstract: 102.13 Distance from the incentre of the tangential triangle of an obtuse triangle to the Euler line The tangential triangle of a triangle is the triangle formed by the lines tangent to the circumcircle of the given triangle at its vertices. If is acute, then its circumcircle is the incircle of its tangential circle. It follows that the incentre of its tangential triangle is the circumcentre of , which lies on its Euler line. If, however, is obtuse, then its circumcircle is an excircle of the tangential circle, and is one of its excentres. ABC ABC ABC O ABC ABC O

102.51 Archimedes' quadrature of the parabola

Abstract: References 1. N. Lord, An unexpected appearance of the golden ratio, Math. Gaz. 101 (March 2017) pp. 98-99. 2. J. Molokach, A natural occurrence of the golden ratio, Math. Gaz. 101 (July 2017) p. 303. 3. S. Abu-Saymeh and M. Hajja, Equicevian points on the altitude of a triangle, Elem. Math. 67 (2012), pp. 187-195. 4. L. Yuan and R. Ding, Triangles in squares, Math. Gaz. 88 (July 2004) pp. 219-225.

The importance of characterisations in geometry

Abstract: When studying geometry, it is natural to choose an object and try to determine what properties it has. Take for example a trapezium (trapezoid in American English): a quadrilateral with at least one pair of opposite parallel sides. Then we probably want to know how to calculate its area, the length of the diagonals, what other properties the diagonals might have, if it has some angle properties, how to calculate the length of the midsegment and so on. All these are interesting investigations. But when they are done (and in many textbooks the story is not allowed to continue even this far), then it is very common to consider the exploration of the trapezium finished. The truth is that here we are only half way. The most interesting part of the journey is the second half. So far we only know necessary conditions in the trapezium, that is, given that we have a quadrilateral that is a trapezium, what properties it has. But what about the reverse question? Which of these properties are unique to the trapezium, that is, which are the sufficient conditions? Or in other words: which properties characterise trapezia and separate them from all the other quadrilaterals? To me that is the really interesting question!

102.19 A black hole conundrum

Abstract: As a fun problem when introducing parametric equations (and in particular epicycloids and hypocycloids) and under our black hole science fiction assumptions, suppose that an international research lab in Borneo (on the Earth's equator) plans to release a black hole. If the Earth is stationary, the black hole yo-yos through the Earth in simple harmonic motion along an Earth diameter. Yet because the Earth rotates, the black hole has initial tangential velocity with respect to the Earth's surface. To model the motion of the black hole we shall artificially assume that the Earth's radius is exactly 6400 km and that the acceleration of gravity at the Earth's surface is exactly 98 decimetres/sec2. Assuming that the Earth is a sphere and is uniformly dense, Isaac Newton in the Principia (Book I, Proposition 10, Corollary (i)) showed that the black hole's orbit is an ellipse, whose centre is the Earth's centre at the origin in the Earth's equatorial plane, and whose parametric equations are R g

102.06 A new elementary proof of Euler's continued fractions

Abstract: Continued fractions often reveal beautiful number patterns. The interested reader is referred to [1] for a collection of many interesting continued fractions of famous mathematical constants. Continued fractions also have applications in cryptography − the study of secret codes and data encryption [2, 3]. Euler was the first person who studied continued fractions systematically. In his foundational publication on the theory of continued fractions, De Fractionibus Continuis Dissertatio [4], Euler derived many interesting continued fraction identities. In this paper, we will present a new proof of the following two continued fractions of Euler's constant : e

Using high school algebra for a natural approach to derivatives and continuity

Abstract: The quadratic equation is a central topic in high school algebra. It provides the simplest generalisation of the familiar linear equation, and finding its roots introduces students to a non-trivial problem that requires the application of new techniques, such as completing the square and/or factorisation into linear factors involving the roots. It also introduces the student to the phenomenon of repeated roots, which opens the door to a discussion of multiplicities of roots. Furthermore, it naturally exposes the student to the case where the equation has no real roots, a phenomenon that could also be used to introduce the student to complex numbers.

Is a straight line the shortest path

Abstract: Is the shortest path from A to B the straight line between them? Your first response might be to think it's obviously so. But in fact you know that it's not quite that straightforward. Your sat-nav knows it's not that straightforward. It asks whether you would like it to find the shortest route or the fastest route, because finding the best path depends on knowing what exactly you mean by ‘long’. Likewise, if you're on a walk in the mountains, there's a good chance you'd rather follow the path around the head of the valley, rather than heading down the steep slope and up the other side. The same sorts of considerations apply in mathematical worlds. I use the mountainside image because it is my preferred way of thinking of a Riemannian metric. Pick an abstract surface S. A Riemannian metric on S gives a well-behaved distance function. By force of habit I tend to picture S as sitting somehow within the physical world. Probably, I'm looking at it from the outside. But if I change viewpoint, so that I am walking around on S, I can picture how the topography affects the idea of the ‘shortest path’.

Adventures in shape and space – and time

Abstract: As a youth entering the sixth form to study Mathematics, Further Mathematics and Physics I enjoyed the riches of the school's mathematics library and in particular three books which appealed to me, A mathematician's apology [1], A book of curves [2] and On growth and form [3].Hardy's book [1] is one that an impressionable, young mathematician should not read unguided. It left me with the impression that the proper pursuit of mathematics was as a pure subject, of no use or application, to be studied for its own sake; to my regret, I held to this view for several years before finally being able to shake it off through teaching Newtonian mechanics. Looking across mathematics teaching today I seem to observe great interest in geometry, number and algebra ‘curiosities’ that are rooted entirely in mathematics. This in itself is no bad thing, since it clearly draws us and our students into the fascinating world of mathematics. But what of the applications of mathematics? Might they be equally fascinating? Surely we do not want to lure our students into Hardy's trap?

Beyond the ratio test

Abstract: D'Alembert's ratio test, a very basic plank in the theory of infinite series, can be stated as follows:Suppose that an > 0 for all n ≥ 1. Then: (i) if for some n0 and some ρ < 1, we have for all n ≥ n0, then is convergent;(ii) if for some n0, we have for all n ≥ n0, then is divergent.