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# Volume 43 - Issue 1 - June 2018

**• Utility theory and Deal or No Deal
** Michael A. Jones, Brittany Shelton and Jennifer Wilson

pp. 1–9

**Abstract**

*Deal or No Deal*was a game show on Channel 4 in the United Kingdom in which a Contestant selects one of 22 red boxes, each of which contains a different monetary amount. Then, in a series of rounds, the Contestant makes a decision about whether to take the deal offered by the Banker, thereby ending the game, or to continue playing the game. Versions of the show appeared in over 80 countries. After a primer on utility theory, we explain how utility theory can be used to generate the Banker's offer. We prove a proposition about the relationship of offers to the revealed amounts in the boxes. Motivated by counterintuitive offers by the Banker in an online version of the game, we apply a proposition to prove that utility theory was not used to make offers. However, many viewings of the U.S. televised version of

*Deal or No Deal*did not exhibit such counterintuitive offers, meaning that perhaps utility theory was used to generate the Banker's offer.

**• A mathematical approach to comply with ethical constraints in compassionate use treatments
** F. Thomas Bruss

pp. 10–22

**Abstract**

Patients who are seriously ill may ask doctors to treat them with unapproved medication, about which not much is known, or else with known medication in a high dosage. Apart from strict legal constraints, such cases may involve difficult ethical questions such as, for example, how long a series of treatments for different patients should be continued. Similar questions also arise in less serious situations. A physician trusts that a certain combination of freely available drugs is efficient against a specific disease and tries to help patients while at the same time following the

*primum-non-nocere*principle.

The objective of this paper is to contribute to the research on such questions in the form of mathematical models. Arguing in a step-by-step approach, we will show that certain sequential optimisation problems comply in a natural way with the true spirit of major ethical principles in medicine. We then suggest protocols and associate algorithms to find optimal, or approximately optimal, treatment strategies. Although the contribution may sometimes be difficult to apply in medical practice, the author thinks that the rational behind the approach offers a valuable alternative for finding decision support and should attract attention.

**• ****Newton's 501 jeans**

Andrew Simoson

pp. 23–31

**Abstract **

Let *R* and *p*, respectively, be the polar and equatorial radii of a homogeneously dense (nonrotating) Earth. In *The Principia*, Isaac Newton showed that if *R* is to *p* as 100 is to 101 then Earth's gravity at the North Pole is to gravity at the Equator as 501 is to 500. Although this latter ratio is certainly accurate, the ratio 505 to 504 is an order of magnitude more precise.

**• Simple models for inventories of perishables
** J. Gani and R. J. Swift

pp. 32–36

**Abstract**

This note examines a simple model for the inventory of perishables with a shelf life

*t*where

*e*<=

*t <=*2

*e*, with

*e*as the fixed reordering interval. The models presented are based upon the Poisson process and interpreted as a Markov chain. Some illustrative numerical examples are included.

**• Gibonacci extensions of a Fibonacci pleasantry
** Thomas Koshy

pp. 37–44

**Abstract**

We extend a charming third-order recurrence for Fibonacci squares to Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal, Jacobsthal–Lucas, Vieta, and Vieta–Lucas polynomials, and Chebyshev polynomials of both types. We then extract their numeric counterparts for Lucas, Pell, Pell–Lucas, Jacobsthal, and Jacobsthal–Lucas subfamilies.

**• A polynomial generator of three mutually disjoint sets****
** Martin Griffiths

pp. 45–51

**Abstract**

In this article we visit an area of mathematics associated with Diophantine equations. In particular, our interest here lies in the use of multivariable polynomials for generating well-known sets of numbers such as prime numbers and the Fibonacci numbers. This provides an appealing link between the fields of number theory and computability theory. In each of the aforementioned examples, however, just a single well-recognised set of numbers results. We show here that it is also possible for such polynomials to give rise to a union of well-known sets of numbers (or multiples of them in some cases). In fact, the main polynomial considered here generates the union of three mutually disjoint sets associated with the Fibonacci and Lucas numbers. We also provide brief details of the historical background to this aspect of mathematics.

**• ****A convenient expression for the Boys function ****
** Saralees Nadarajah and Stephen Chan

pp. 52–55

**Abstract**

A convenient closed analytical form is derived for the Boys function. It is elementary except for the distribution function of the normal distribution. A code in R software is provided to aid practitioners.

**• ****A random variable that does not belong to a domain of attraction, but its absolute value does ****
** Michael Grabchak

pp. 56–59

**Abstract**

In this paper we give an example of a distribution

*n*

_{a}, which does not belong to the domain of attraction of any stable distribution. However, if

*X*has distribution

*n*

_{a}, then the distribution of |

*X*| belongs to the domain of attraction of an

*a*-stable distribution. The distribution has a simple structure, and may be useful for pedagogical purposes.

**• Letter to the Editor: An inequality for birth**** –death processes with catastrophes
** R. J. Swift

pp. 60–61