We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

In [1], the following theorem is stated and proved as Proposition XXXII, using the Principle of Inclusion and Exclusion:

The number of ways of deranging a series of n terms so that no term may be followed by the term which originally followed it is

(Note that the conventional definition of a derangement of a series of terms is an arrangement such that no term occupies its original position. However, the usage in this paper allows some terms to remain in their original positions.) This may most easily be visualised by considering a string of n distinct characters, abc … n say. This string has a sequence of n − 1 pairs of consecutive characters, namely, ab, bc, … mn. This article derives three theorems that can be deduced from the above theorem. The first theorem determines the number of permutations of a character string that are free from a random subset comprising a given number m of the n − 1 pairs, but contain all the remaining pairs. The second theorem sums the results of the first theorem for m, m + 1, …, n − 1 pairs.

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formula

with its alternative form

These can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].

However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

The Lucas-Lehmer (LL) test is the most efficient known for testing the primality of Mersenne numbers, i.e. the integers Ml = 2l − 1, for l ≥ 1. The Mersenne numbers are so-called in honour of the French scholar Marin Mersenne (1588-1648), who in 1644 published a list of exponents l ≤ 257 which he conjectured produced all and only those Ml which are prime, for l in this range, namely l = 2,3,5,7, 13, 17, 19,31,67, 127 and 257 [1]. Mersenne's list turned out to be incorrect, omitting the prime-producing l = 61, 89 and 107 and including the composite-producing l = 67 and 257, although this was not finally confirmed until 1947, using both the LL test and contemporary mechanical calculators [2]. The LL test is based on the following theorem.

Every student of calculus learns that one typically solves a differential equation by integrating it. However, as Euler showed in his 1758 paper (E236), Exposition de quelques paradoxes dans le calcul intégral (Explanation of certain paradoxes in integral calculus) [1], there are differential equations that can be solved by actually differentiating them again. This initially seems paradoxical or, as Euler describes it in the introduction of his paper:

Here I intend to explain a paradox in integral calculus that will seem rather strange: this is that we sometimes encounter differential equations in which it would seem very difficult to find the integrals by the rules of integral calculus yet are still easily found. not by the method of integration. but rather in differentiating the proposed equation again; so in these cases, a repeated differentiation leads us to the sought integral.

“The ambitious learner should carefully study a new fact; he should turn it over and over, consider it under various aspects, scrutinize it from all sides … Moreover, he should try to expand and enlarge any newly acquired knowledge by application, generalization, specialization, analogy, and in all other ways.” [1]

Vincenzo Viviani, a 17th century mathematician, proved that the sum of the (perpendicular) distances from a point within an equilateral triangle to its sides is constant as shown in Figure 1. The theorem, named after him, generalises to polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel. Viviani's theorem is easily proved by summing the areas of triangles APB, BPC and CPA, equating to the area of the triangle ABC, and then simplifying to obtain h1 + h2 + h3 = H, where H is the altitude of ABC.

Problems concerning the set

of numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identity

which shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).

Suppose n (fair) dice each having m faces marked with numbers 1 to m, are thrown at random. The problem of determining the number of ways in which the sum of the numbers exhibited by the dice will be equal to a given number k has a very long history. In particular, the three dice problem (i.e. the case where m = 6, n = 3), goes back to the 13th century (see [1]). Later, it was Cardano and Galileo who solved it (see [2, 3]). The general case was stated without proof in A. de Moivre's first work on probability, De Mensura Sortis, (1712, p.220). De Moivre was the first who published a proof (using generating functions) in Miscellanea Analytica (1730, p. 196). Furthermore, Montmort had also the solution (via inclusion-exclusion) by 1713, and independently of De Moivre (see [4, 2]).

Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. Thus

This constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].

Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).

This article is concerned with two sequences and the matrices they induce. The first, the Derangement sequence is well known to readers of the Gazette. It has been written about and its properties derived in many ways and on many previous occasions [1 – 11]. In contrast the second sequence, the Deadend sequence, is not well known – and certainly not the name since I made that up while writing this article. However, I want to explore these sequences in a single, systematic way. For the former it gives a fresh account of its properties, and for the latter it sets out corresponding results. The surprise is that they tum out to be closely related; surprising since their very different definitions mask such a connection. More than this, most of these properties may be established in the most fundamental way, based on elegant enumerative arguments. So they are free of specialised expertise or technique and may be enjoyed just as they are.

This paper is about the classification of frieze groups. A frieze is a decorative strip of paper (or wood, stone,…) on which a pattern is produced by the periodic repetition of a picture along the strip. A symmetry of a frieze is an isometry of the plane that leaves the pattern unchanged, and a frieze group is the group of symmetries of some frieze. A popular exercise is to start with a given frieze and then try to identify its frieze group. However, in order to do this we need to know that there are only seven possible frieze groups, and what these groups are. Seven friezes, with different frieze groups, are illustrated below, in such a way that each pattern is invariant under the same translation, namely x → x + 1. Our task is to show that (up to a change in the motif, and a simple change of coordinates in the plane) these are the only frieze patterns.

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.

For non-negative integers n, the recurrence relation defined by

with initial conditions

can be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

When I was teaching calculus recently, an exercise in the textbook [1, p. 687, qn. 3] asked students to find the Taylor series of cosx centred at π/4. My students were able to find the series

but wondered how to take care of the sign changes, in order to express the series in I.-notation. They were quite puzzled after finding that the answer given in [I] is

The exponent n (n + 1)/2 seemed a rabbit out of the hat! This article is the result of trying to answer their question satisfactorily, and to have some serious fun. Readers are invited to try two teasers first before reading for their solutions.

As I write (Spring 2012) I am aware that I have been a member of the Mathematical Association for fifty years. I joined during the annual conference in 1962 when Dr Combridge was the main figure in the society. The highlight was a session at the conference (at King's College, London) by Geoffrey Matthews showing how matrices might be introduced into the school curriculum. Some treated this as hilarious, others as a signpost to the future. My friend Frank Budden and I regarded it as a premature ploy, which, we thought, might not tum out to be such a good idea. We subsequently wrote a successful book Mathematics through Geometry (1964) arguing in detail why the modem mathematics revolution—which was then gathering pace—might end in tears. We identified spatial imagination as the heartland of mathematics, and contrasted it with a denial of spatial imagination in the abstract calculi which were all the vogue at the time.

Let T be a triangle and let A be the area of T. Given a point p inside the triangle and a number r that satisfies 0 < r ≤ ½, we seek to count the number N (p, r) of straight lines that pass through p and cut T into two pieces so that one piece has area rA. As the point p and the ratio r vary, the value of N (p, r) ranges from 0 to 6. Given a ratio r, we want to determine the regions within T for which the function N (p, r) assumes various integer values. For example, a region inside T for which N (p, r) = 6 only exists for values of r between and ½, and we would like to quantify the area of this region (as a fraction of the total area of the triangle) as r varies.