The well-known Fibonacci numbers Fn are defined by the recurrence relation

Fn = Fn – 1 + Fn – 2. (1)

together with the starting values F0 = 0, F1 = 1, or equivalently F1 = F2 = 1.

We record the first few:

The recurrence relation can also be applied backwards in the form Fn = Fn + 2 – Fn + 1 to define Fn for n < 0. An easy induction verifies that F−n = (−1)n – 1Fn for n > 0.

Magic: the gathering is a trading card game published by Wizards of the Coast [1]. The aim of most variants of the game is to reduce your opponent's life total from twenty to zero, thus winning the game. As it may take several turns to reduce a player's life total to zero, players need a mechanism to keep track of their current life total. For this purpose, players often use a device called a spindown life counter, shown in Figure 1. A spindown life counter is an icosahedron with a special labelling of the faces, such that − starting with 20 life total − a player can reduce their life total in decrements of one by rolling the icosahedron onto an adjacent face each time. A spindown life counter appears similar to a standard icosahedral die, known in gaming as a d20; however, the labelling of faces is different, as also shown in Figure 1.

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.

The title of this Article pays homage to Greg Frederickson's widelyknown book on geometrical dissections, Dissections: Plane & Fancy [1]. I enjoyed the book so much, I built an honours geometry course around it − a terminal mathematics course for honours students needing a mathematics credit for their general education requirements. Not only is there significant geometry involved, but there is a fair amount of number theory relevant to finding infinite families of dissections of, for example, m squares to n squares.

However, although many dissections from one regular polygon to another are quite elegant, the mathematics involved in specifying the dissections is often formidable.

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]).

The trapezium (trapezoid in American English) is one of the seven most well-known quadrilaterals that is studied in school geometry. However, it is always assumed that it is convex. In this note we shall derive metric formulas for the most important quantities in crossed trapezia and compare them to the similar formulas for convex trapezia. The corresponding investigation regarding crossed cyclic quadrilaterals was conducted in [1]. Any crossed quadrilateral has a pair of opposite sides intersecting and both of its diagonals are outside of itself.

The circles C1, … , Cn form a chain of length n, or an n-chain, if Ci touches Ci + 1, for i = 1, … , n − 1, and the chain is closed if also Cn touches C1. If Ci touches Ci + 1 at Qi, for i = 1, … , n (subscripts being interpreted modulo n), then Q1, … , Qn (assumed distinct) are the contact points of the chain. A cyclic chain is a chain for which all the circles touch another circle S, the base circle of the chain, and if Ci touches S at Pi, then P1, … , Pn are the base points of the chain.

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.

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:

1. (i) if for some n0 and some ρ < 1, we have for all n ≥ n0, then is convergent;

2. (ii) if for some n0, we have for all n ≥ n0, then is divergent.

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.

Mathematicians know that the path from adolescent competence to adult professional life is full of unexpected twists and turns, and is fraught with potential pitfalls. Even when faced with very able adolescents, it is not at all clear which of them will flourish in the long run, or how one can best support their future development (when they eventually become young adults) in a way that takes their mathematical ability and inclinations seriously – whether in pure or applied mathematics, in computer science or software development, in engineering, mathematical biology, financial mathematics or linguistics, in management, or in some totally different profession.

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!