# Volume 38 - Issue 1 - June 2013

A simple formula for De La Loubère's magic squares method
Livinus U. Uko, Jennifer L. Sinclair
pp. 16

Abstract
De La Loubère's method is the most popular rule used to generate magic squares. In this paper we derive a simple formula for the method that is much easier to use and to code than the traditional rule. We also show that the arrays generated from our formula are magic squares, thereby providing a proof of De La Loubère's method.

Distant parents in complete binary trees
S. P. Glasby
pp. 716

Abstract
There is a unique path from the root of a tree to any other vertex. Every vertex, except the root, has a parent: the adjoining vertex on this unique path. This is the conventional definition of the parent vertex. For complete binary trees, however, we show that it is useful to define another parent vertex, called a distant parent. The study of distant parents leads to novel connections with dyadic rational numbers. Moreover, we apply the concepts of close and distant parent vertices to deduce an apparently new sense in which continued fractions are 'best' rational approximations.

Some connections between the formulas for pi of Vieta, Wallis, and
Brouncker
Thomas J. Osler
pp. 1722

Abstract
Vieta's product for 2pi has factors that are nested radicals. The Wallis product for 2pi has factors that are rational numbers. Brouncker gave continued fractions for  4pi. By summarizing some recently published results in this expository paper, we show that these seemingly unrelated results are connected. We give a general formula in which the products of Vieta and Wallis are special cases. We give another general formula of which the results of Brouncker and Wallis are special cases. Both formulas allow us to 'morph' from one result to the other.

Group-theoretic aspects of a knockout tournament allowing for
first-round byes
Martin Griffiths
pp. 2333

Abstract
In this paper we consider the group-theoretic aspects of a certain type of unseeded single-elimination knockout tournament. The fact that the initial number of participants is not restricted to be of the form 2k means that there is the potential for some players to be given byes in the first round. Such tournaments turn out to exhibit a wonderfully rich structure, allowing us to place a number of abstract notions within a concrete setting.

An alternative estimator for the shape parameter in the negative
binomial distribution
Benn Robertson, Thomas Fung, Neville Weber
pp. 3442

Abstract
Best and Gipps (1974) showed that the negative binomial distribution can be approximated closely by a gamma distribution. Using the moment estimators for the parameters in the gamma distribution, we use the approach of Best and Gipps to propose an alternative estimator for the shape parameter alpha which, while biased, appears to perform better in many situations, both in terms of mean square error and percentile measure, than its key competitor, namely the method of moments estimator.

The mathematics of retirement investing
Marlena Herman, Paul J. Laumakis
pp. 4349

Abstract
The purpose of this paper is to illustrate both the power and utility of mathematics in helping people devise savings plans for retirement. In addition to demonstrating how spreadsheets can be useful in retirement planning, the development of an analytical model based on discrete dynamical systems is provided. A discussion of the fundamentals of mutual funds investing is also included, along with a list of retirement investing planning resources.

A Leslie matrix approach to an age-structured epidemic
Joe Gani, Linda Stals
pp. 5060

Abstract
We consider a Leslie-type matrix approach to an SIR epidemic in discrete time. We give examples of the population of susceptibles, infectives, and removals for different birth rates and two different infection rates. Finally, when the infection rate depends on the number of infectives, we derive conditions for a steady state.

Letter to the Editor: An extension of the Pythagorean theorem and its
application to Fermat's last theorem
Mels Sluyser
pp. 6162

Letter to the Editor: A note on the consecutive sums problem
Greisy Winicki-Landman, Randall Swift
pp. 6364

Letter to the Editor: A first integral of a nonlinear integro-differential
equation of Volterra