In 1840 C. L. Lehmus sent the following problem to Charles Sturm: ‘If two angle bisectors of a triangle have equal length, is the triangle necessarily isosceles?’ The answer is ‘yes’, and indeed we have the reverse-comparison theorem: Of two unequal angles, the larger has the shorter bisector (see [1, 2]).

Sturm passed the problem on to other mathematicians, in particular to the great Swiss geometer Jakob Steiner, who provided a proof. In this paper we give several proofs and discuss the old query: ‘Is there a direct proof?’ before suggesting that this is no longer the right question to ask.

We go on to discuss all cases when an angle bisector (internal orexternal) of some angle is equal to one of another.

In [1] Deshpande and Shiwalkar explore the number of double heads that may appear in sequences of coin tosses. They sought to answer questions such as,

‘If a (fair) coin is tossed r times, how many possible outcomes result in 0 double heads, how many result in 1 double head, …, how many result in (r – 1) double heads?’

Their answer involves Fibonacci expressions. Hirschhorn [2] extended these results using generating functions which on expansion led to some closed form expressions and the statistics of their distribution. Griffiths [3] continues this theme. Another article that touches on the basic idea – a run of particular outcomes – is [4] which is also the source of other references; [5] is also concerned with the same idea.

The perimeter of the ellipse x2/a2 + y2/b2 = 1 is 4J (a, b), where J (a, b) is the ‘elliptic integral’

This integral is interesting in its own right, quite apart from its application to the ellipse. It is often considered together with the companion integral

Of course, we may as well assume that a and b are non-negative.

Consider the problem of defining a continuous function f(x) which agrees with factorials at integers. There are many possible ways to do this. In fact, such a function can be constructed by taking any continuous definition of f(x) on [0,1] with f(0) = f(1) = 1 (such as f(x) = 1), and then extending the definition to all x > 1 by the formula f(x + 1) = (x + 1)f(x). This construction was discussed by David Fowler in [1] and [2]. For example, the choice f(x) = ½x(x − 1) + 1 results in a function that is differentiable everywhere, including at integers.

However, this approach had already been overtaken in 1729, when Euler obtained the conclusive solution to the problem by defining what we now call the gamma function. Among all the possible functions that reproduce factorials, this is the ‘right’ one, in the sense that it is the only one satisfying a certain smoothness condition which we will specify below. Admittedly, Euler didn't know this. It is known as the Bohr-Mollerup theorem, and was only proved nearly two centuries later.

First, a remark on notation: the notation Γ (x) for the gamma function, introduced by Legendre, is such that Γ (n) is actually (n − 1)! instead of n!. Though this might seem a little perverse, it does result in some formulae becoming slightly neater. Some writers, including Fowler, write x! for Γ (x + 1), and refer to this as the ‘factorial function’. However, the notation Γ (x) is very firmly entrenched, and I will adhere to it here.

We came across the ‘two girls’ version of the children's gender problem nearly 35 years ago. How we came to it we cannot remember, but Martin Gardner had published a variant of it in the Scientific American in 1959. It re-emerged for us in the summer of 2010, following the publication of an article in Science News [1]. Subsequently Keith Devlin wrote about how this re-emergence impacted on him, and noting that ‘Probability Can Bite“ [2]. The mathematics herein reflects and extends that in Devlin's article.

In case the reader has not encountered the problem before, we first pose four problems.

1. A family has two children. One of them is a girl. What is the probability that they are both girls?

2. A family has two children. The younger is a girl. What is the probability that they are both girls?

3. A family has two children. One of them is a girl, and she was born on a Tuesday. What is the probability that they are both girls?

4. A family has two children. One of them is a girl, and she has green hair. What is the probability that they are both girls?

In a recent paper [1] Michael Hirschhorn and the present author had to prove that the polynomial

is positive for all integers n ≥ 4. We did so by writing down the identity

Since all the coefficients of the quotient polynomial are positive as well as the remainder, this shows by inspection that f(n) > 0 for n ≥ 5 and computing f(4) = 12025 completes the proof. This example illustrates a useful and efficient method for proving that polynomials have strictly positive values for all real numbers exceeding a given one. We leamed this method from a paper by Chen [2], (see [3] and the references cited there), and we have subsequently used it in our own researches [1,4].

We also conjectured in [1] that the theoretical basis of the method is a true theorem and the present paper is dedicated to proving this conjecture.

Theorem 1: Let f(x) be a polynomial with real coefficients whose leading coefficient is positive and with at least one positive root x = a. Then there exists an x = b ≥ a such that

where f(b), and all the coefficients of the quotient polynomial g(x), are non-negative.

The well-known Fibonacci and Lucas numbers continue to faxcinate the mathematical community with their beauty, elegance, ubiquity, and applicability. After several centuries of exploration, they are still a fertile ground for additional activities, for Fibonacci enthusiasts and amateurs alike.

Fibonacci numbers Fn and Lucas numbers Ln belong to a large integer family {xn}, often defined by the recurrence xn = xn−1 + xn−2, where x1 = a, x2 = b, and n ≥ 3. When a = b = 1, xn = Fn; and when a = 1 and b = 3, xn = Ln. Clearly, F0 = 0 and L0 = 2. They satisfy a myriad of elegant properties [1,2,3]. Some of them are:

In this article, we will give a brief introduction to the Q-matrix, employ it in the construction of graph-theoretic models [4, 5], and then explore some of these identities using them.

In 1960 C.H. King studied the Q-matrix

In [1], W. D. Munn proved the following result.

Theorem 1: Infinitely many ellipses pass through the four vertices of a given convex quadrilateral.

Much of the geometry that I studied as an undergraduate in the 1950s concerned complex projective space, in which convexity plays no part. So I found Theorem 1 especially piquant and sought to understand it better. This article is the result. After examining the convexity of quadrilaterals in general, especially those inscribed in conics, I consider the following problem. Let P be a variable point in the plane, distinct from the vertices of a given convex quadrilateral ABCD. It is well known that there is a unique conic, S (P), through the five points A, B, C, D and P. How does the nature of this conic depend on the position of P? As a spin-off, we get a very short proof of Theorem 1. Finally I look at what happens when the quadrilateral ABCD is not convex. In this case, S (P) is always a hyperbola, but the distribution of A, B, C and D on its branches is still of interest.

The article ‘Reflections on the Woolwich Ferry’ [1] brought back some childhood memories. I remember being whisked away with my siblings from the verdant pastures of north-west Kent by bus and tram to Woolwich for a ride on one of the ‘Free Ferry’ paddle steamers; there were four of them and their passing in the mid 1960s was not universally welcomed. It was indeed a real treat that consumed just a little more than a full working day. The depths of a winter school holiday could find us huddled, privileged below the ferry decks on a gantry near the fire hole to receive the welcoming hot blast as its doors opened to accommodate the hand-stoked coke. It was hand-stoked to reduce smoke output, and each steamer could make about eight knots. Presumably health and safety considerations now would not allow for such a luxury of positioning.

However, it was not always wine and roses. To borrow a phrase from one of Kipling's poems, it was sometimes a case of ‘not with this wind blowing, and this tide’, for there were rare occasions when journeys were temporarily delayed by exceptionally windy weather. The elements below (river velocity effect) conspired with the elements above (wind velocity effect) to delay ferry movements. The earlier article proposed a velocity model to deal with effects due only to the river flow and it is the purpose of this note to extend that argument by considering also a velocity effect due to the wind, because the superstructure on a ferry can be as susceptible to the wind as is the keel to the tide.

A well-known exercise in classical differential geometry [1, 2, 3] is to show that the set of all points (x, y, z) ∈ ℝ3 which satisfy the cubic equation

is a surface of revolution.

The standard proof ([2], [3, p. 11]), which, in principle, goes back to Lagrange [4] and Monge [5], is to verify that (1) satisfies the partial differential equation (here written as a determinant)

which characterises any surface of revolution F (x, y, z) = 0 whose axis of revolution has direction numbers (l, m, n) and goes through the point (a, b, c). This PDE, for its part, expresses the geometric property that the normal line through any point of must intersect the axis of revolution (this is rather subtle; see [6]). All of this, though perfectly correct, seems complicated and rather sophisticated just to show that one can obtain by rotating a suitable curve around a certain fixed line. Moreover, to carry out this proof one needs to know a priori just what this axis is, something not immediately clear from the statement of the problem. Nor does the solution give much of a clue as to which curve one rotates.

A search of the literature failed to turn up a treatment of the problem which differs significantly from that sketched above (although see [1]).

The polynomial (1) is quite famous and has been the object of numerous algebraical and number theoretical investigations. See the delightful and informative paper [7].

What are (themed) cryptic crossnumber puzzles and where do they appear?

I'll start by answering the second question first, as that is the easier. This rather unusual genre of puzzle appears quarterly as part of the series of Listener crosswords, which found a new home in The Times newspaper following the demise of The Listener magazine in 1991. There is also a monthly on-line crossword magazine called The Magpie, which has a numerical puzzle in each issue. As a result, fans of crossnumber puzzles have sixteen a year to tackle. Themed cryptic crosswords appear much more frequently – perhaps ten times as often.

All Listener puzzles have a title and the setter is given a pseudonym. They are also numbered, and in what follows this reference is given in square brackets following the letter L. Puzzles from the The Magpie are referenced according to the issue of the magazine, using M.

Many mathematical puzzles involve using a scale to compare the weights of two objects, or two groups of objects. The present work studies a similar but different type of question involving a scale.

We are given a set of marbles, whose weights are all different. The only way to distinguish them is to use a special kind of scale. The scale has three trays and each can accept exactly one marble. The scale then indicates which is the heaviest, the lightest and consequently the middle one of the three marbles.

The paper studies the question of ordering 15 different marbles with as few weighings as possible. The problem is taken from the website Enigmes, casse-têtes, curiosités et autres bizarreries [1] and Toppuzzle [2] where the best reported strategy requires 23 weighings. The problem can also be found at the website Trick of Mind [3] where a strategy requiring 22 weighings is proposed. These websites contain many challenging mathematical puzzles, some are classical problems, others are original creations. The author is not aware of any other work on this problem.

The present paper describes a strategy that improves on the value of 22 weighings, and is structured as follows. Section 2 introduces some useful definitions and provides strategies to sort up to 9 marbles. Section 3 proves our main result, that 20 weighings is always sufficient to sort 15 marbles.