The beautiful representation of π by Wallis's product formula has attracted many authors who have given a variety of proofs (see [1, 2, 3, 4, 5, 6, 7]). The purpose of this note is to revisit some elementary derivations with a focus on probabilistic/statistical flavours of this formula. The famous Wallis's product formula for is given by

There are a number proofs available for this formula scattered in the literature. Here we will concentrate on the probabilistic nature of this formula.

Among the many astonishing formulae stated by Ramanujan we find in his Notebooks the following sequence (see [1], p 96, Entry 43):

which hold for any triple of complex numbers (a, b, c) such that a + b + c = 0.

Ramanujan concluded this list by ‘And so on’, which suggests that he had some kind of method or algorithm allowing him to extend this list to any even power of the quadratic form p(a, b, c) = ab + bc + ca, when a + b + c = 0. To explain this, Bruce Brendt details a theorem by S. Bhargava [2], which can be used to produce identities of the kind above, and infers that Ramanujan is likely to have used the same proof to establish his identities (see [1], pp. 97-100)

What is the most symmetric quadrilateral you can think of? Most people would say ‘a square’, but having given this (or any other) answer, can you create your own definition of a ‘quadrilateral’, and of ‘most symmetric’, so that you can justify your answer as a rigorous piece of mathematics? This is a challenging investigation because the answer may not be the obvious one.

Often the main difficulty in solving a problem lies in finding appropriate definitions; here we must first consider the questions ‘what is a quadrilateral?’, and ‘what is a symmetry of a quadrilateral?’.

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.

2. From Pascal to Leibniz

In Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.

Specifically, we define the sequence sn ; as follows [6]:

The forgotten continued fractions

Lord Brouncker's continued fraction for π is

(In this paper we will use the more convenient notation This fraction first appeared in the Arithmetica Infinitorum [1] by John Wallis. In this book, along with topics that lead to Newton's calculus, Wallis derives his famous product

Wallis tells us that he showed this product to Lord Brouncker, and in turn Brouncker converted it not only into (1), but into an infinite sequence of continued fractions.

You see a Martian walking on the street. What do you do?

(i) seek shelter

(ii) take some pictures

(iii) introduce yourself and offer him/her/it a ride

(iv) go online to buy lots of stocks.

In this article we analyse the long term behaviour of stock investments.The above unusual question will be answered too. In the discussion the notion of ‘almost surely’ will be often used. An event happens almost surely if it happens with probability 1. Note that it is still possible that an almost certain event A does not happen, but the probability of that (i.e. the probability of the complement of A) is zero.

Throughout the article we use areal (or barycentric) coordinates. The side lengths of BC, CA, AB are denoted as usual by a, b, c and for brevity and ease of reading we write a2 = p, b2 = q, c2 = r. The symmedian point K then has coordinates (a2, b2, c2 ) replaced by (p, q, r), it being the isogonal conjugate of the centroid, and the circumcentre O then has coordinates (a2(b2 + c2 − a2), b2(c2 + a2 − b2), c2(a2 + b2 − c2)) replaced by (p(q + r − p), q(r + p − q), r(p + q − r)). The construction of the triplicate ratio circle (also known as the Lemoine circle) and the Brocard circle is replicated but starting with the circumcentre O rather than the symmedian point K. It is found that the triplicate ratio circle is then replaced by a conic cutting internally each side of ABC twice and having centre X the midpoint of OK. The Brocard circle is replaced by a conic passing through O and K and also having centre X the midpoint of OK. These we define as the twin conics.

A common problem in modelling population change, whether of humans or animals, is estimating the values of the various parameters in the mathematical functions used to describe the population trajectory. Typically both the population's rate of growth and ultimate maximum size are unknown and so various numerical approximation methods have to be used (see [1, 2 and 3]). In this article a mathematical method of population projection is presented which avoids these difficulties and takes the baseline case that the only data available for computation are the populations from just three censuses. It is a method indicated, but not developed by Keyfitz [4].

Here is a result which will surely make one sit up. We start with a sequence t1 = (1, 1, 2, 2, 4, 4, 8, 8, 16, 16, … ) in which each power of 2 occurs twice in succession, and produce a second sequence whose n th term is the sum of the n th and (n + 1) th terms of the original one; we get the sequence t2 = (2, 3, 4, 6, 8, 12, 16, 24, 32, … ). We repeat the same operation on the resulting sequence, and continue this, iteratively. Here are the sequences t3, t4, t5:

t3 = (5, 7, 10, 14, 20, 28, 40, 56, 80, 1l2, 160, 224, 320, 448,…),

t4 = (12, 17, 24, 34, 48, 68, 96, 136, 192, 272, 384, 544, 768,…),

t5 = (29,41, 58, 82, 116, 164,232, 328, 464, 656, 928, 1312,…).

If we compute the ratio of the second term to the first term for each ti, we get the following sequence of rational numbers,

which converges to the square root of 2. (These fractions turn out to be successive convergents to the simple continued fraction for √2; see Section IV for details.)