The television quiz program Letters and Numbers, broadcast on the SBS network, has recently become quite popular in Australia. This paper considers an implementation in Excel 2010 and its potential as a vehicle to showcase a range of mathematical and computing concepts and principles.

In two earlier papers, an intricate Jackpot structure and analysis of pseudo-random numbers for Keno in the Australian state of Queensland circa 2000 were described. Aspects of the work were also reported at an international conference . Since that time, many aspects of the game in Australia have changed. The present paper presents more up-to-date details of Keno throughout the states of Queensland, New South Wales and Victoria. A much simpler jackpot structure is now in place and this is described. Two add-ons or side-bets to the game are detailed: the trivial Heads or Tails and the more interesting Keno Bonus, which leads to consideration of the subset sum problem. The most intricate structure is where Heads or Tails and Keno Bonus are combined, and here, the issue of independence arises. Closed expressions for expected return to player (ERTP) are presented in all cases.

The potential for simple linear relationships arising from a computer game to build student modelling and "word problem" skills is explored. The fundamental capability of the spreadsheet to tabulate and graph possible solutions is used to lay bare the problem structure for the students.

Spreadsheet Investigations in Modular Arithmetic Steve Sugden, Bond University, Australia Abstract Modular arithmetic has sometimes been regarded as a bit of a curiosity, at least by those unfamiliar with its importance to both abstract algebra and number theory, and with its numerous applications. However, with the ubiquity of fast digital computers, and the need for reliable digital security systems such as RSA, knowledge of this important branch of mathematics is now considered almost essential for IT professionals. Indeed, computer arithmetic is, ipso facto, modular. This paper describes the use of a modern graphical spreadsheet (Microsoft’s Excel) to clearly illustrate the basics of modular arithmetic, and to solve certain classes of problems. Via conditional formatting and observation of patterns in Excel, students may gain structural insight, form and test conjectures, and solve problems.

The basic principles and equations are developed for elementary finance, based on the concept of compound interest. The five quantities of interest in such problems are present value, future value, amount of periodic payment, number of periods and the rate of interest per period. We consider three distinct means of computing each of these five quantities in Excel 2007: (i) use of algebraic equations, (ii) by recursive schedule and the Goal Seek facility, and (iii) use of Excel's intrinsic financial functions. The paper is intended to be used as the basis for a lesson plan and contains many examples and solved problems. Comment is made regarding the relative difficulty of each approach, and a prominent theme is the systematic use of more than one method to increase student understanding and build confidence in the answer obtained. Full instructions to build each type of model are given and a complete set of examples and solutions may be downloaded (Examples.xlsx and Solutions.xlsx).

This is an update of an earlier paper, and is written for Excel 2007. A series of Excel 2007 models is described. The more advanced versions allow solution of f(x)=0 by examining change of sign of function values. The function is graphed and change of sign easily detected by a change of colour. Relevant features of Excel 2007 used are Names, Scatter Chart and Conditional Formatting. Several sample Excel 2007 models are available for download, and the paper is intended to be used as a lesson plan for students having some familiarity with derivatives. For comparison and reference purposes, the paper also presents a brief outline of several common equation-solving strategies as an Appendix.

This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

Recurrence relations in mathematics form a very powerful and compact way of looking at a wide range of relationships. Traditionally, the concept of recurrence has often been a difficult one for the secondary teacher to convey to students. Closely related to the powerful proof technique of mathematical induction, recurrences are able to capture many relationships in formulas much simpler than so-called direct or closed formulas. In computer science, recursive coding often has a similar compactness property, and, perhaps not surprisingly, suffers from similar problems in the classroom as recurrences: the students often find both the basic concepts and practicalities elusive. Using models designed to illuminate the relevant principles for the students, we offer a range of examples which use the modern spreadsheet environment to powerfully illustrate the great expressive and computational power of recurrences.

Aspects of Keno modelling throughout the Australian states of Queensland, New South Wales and Victoria are discussed: the trivial Heads or Tails and the more interesting Keno Bonus, which leads to consideration of the subset sum problem. The most intricate structure is where Heads or Tails and Keno Bonus are combined, and here, the issue of independence arises. Closed expressions for expected return to player are presented in each case.

This paper is intended to be used in the classroom. It describes essentially every step of the construction of an Excel model to help solve Sudoku puzzles. For those up to moderate difficulty, it will usually solve the puzzle to completion. For the more difficult ones, it still provides a platform for decision support. The paper may be found useful for a lesson in which students, who, having some basic knowledge of Excel, are learning some of its lesser-known features, such as conditional formatting. It also generates a useful tool for working with Sudoku puzzles, from the very easiest right up to the ones often labelled as fiendish or diabolical. Fundamental mathematical concepts such as set intersection, set partition and reduction of set partition to singletons are very graphically illustrated by the present Excel model for Sudoku. Prominent spreadsheet concepts presented here are conditional formatting, names, COUNTIF, CONCATENATE. The paper is accompanied by a completed Excel model, constructed by using the steps described herein. No VBA code is employed; the whole thing is done with Excel formulas and conditional formatting.

In telecommunications network design, nodes need to be linked in an economical way to handle expected traffic. Capacity constraints, degree constraints and hop limits are to be respected. A genetic algorithm with some novel features is described. The crossover method generates an optimal child solution for the parents selected.

Modern telecommunication networks are characterized by a heterogeneous mix of traffic classes, ranging from traditional telephone calls to video and data services. Therefore, there is a need to solve the traffic allocation problem with different quality-of-service (QoS) requirements, wherein each traffic demand is fulfilled. This paper investigates the possibility of allocating loaded traffic in a low earth orbital (LEO) satellite network by considering multiservice traffic. Both Poisson and Markov models are used for the incoming traffic, which includes multiservice requirements. By introducing a privilege parameter, a quantum of bandwidth can be reserved for high priority traffic and a better QoS can be given for this traffic. The algorithm performs well for both traffic patterns tested. Simulation results are presented.

This paper studies the capacitated minimum spanning tree (CMST) problem, which is one of the most fundamental and significant problems in the optimal design of local computer networks. A solution method using a node-oriented branch and bound technique is introduced and its performance is presented. We show the advantages of the algorithm while illustrating the process of searching for the optimal solution. Techniques for finding tighter lower bounds are emphasized. Computational experiences demonstrate the algorithm's effectiveness.

Traffic allocation strategy becomes a significant factor in optimization of bandwidth usage of telecommunication resources, especially with increasing use of broadband applications. Allocation strategy in dynamic LEO (Low Earth Orbital) satellite communication network is studied, to improve their Quality of Service (QoS). Traffic allocation control is performed to provide a near optimal utilization of their Inter Satellite Links (ISLs). A combination of two algorithms is used to allocate traffic in ISLs. Empirical analysis is performed to examine performance of the proposed algorithm, GALPEDA. Result shows that the proposed algorithm is useful for traffic allocation of multiclass traffic in LEO satellite communication.

This paper studies the capacitated minimum spanning tree problem (CMST), which is one of the most fundamental and significant problems in the optimal design of communication networks. CMST has a great variety of applications, such as in the design of local access networks, the design of minimum cost teleprocessing networks, the vehicle routing and so on. A solution method using branch and bound technique is introduced. Computational experiences demonstrate the algorithm’s effectiveness.

This paper explores the use of the meta-heuristic search algorithm Simulated Annealing for solving a minimum cost network synthesis problem. This problem is a common one in the design of telecommunication networks. The formulation we use models a number of practical problems with hop-limit, degree and capacity constraints. Emphasis is placed on a new approach that uses a knapsack polytope to select amongst a number of pre-computed traffic routes in order to synthesise the network. The advantage of this approach is that a subset of the best routes can be used instead of the whole set, thereby making the process of designing large networks practicable. Using simulated annealing, we solve moderately large networks (up to 30 nodes) efficiently.

Discrete Mathematics is a fundamental subject for Information Technology (IT) majors. For students whose primary orientation is not to mathematics, but computing, it is important to relate fundamental mathematical principles to practical computing realities. At Bond University, Microsoft Excel is the primary practical vehicle used to illustrate these connections, and has proved to be an extremely valuable tool for this purpose. No programming or macro creation is required to implement most of the basic principles taught in Discrete Mathematics. In particular, the topic of Mathematical Induction (MI) poses a considerable challenge to many students, and there are a number of ways in which a spreadsheet such as Excel can assist with the learning process.

Gambling and probably theory have a long intertwined history, with early publications dating to the 17th century. Fair games of chance, such as Blackjack or Baccarat, are exact: because the relevant elements of randomness within these games are known, probability theory can be applied to deduce outcomes, notably the house advantage. Application of probability theory can, however, prove unwieldy for computing the outcome of everyday gambling behaviours, such as using betting systems, like the Martingale, or enacting gambling biases such as the Gambler? Fallacy or its converse? Allow the Herd Mentality’s Simulation is one proven way to assess possible outcomes in such situations. Within this presentation we share insights from two gaming simulations we have created. The first simulates the behaviour of Optimal blackjack players, all of whom intend to play approximately two hours, or one hundred hands. This simulation shows the effect of setting monetary loss/win limits. As should be expected, regardless of the width of the limit or the limit’s symmetry, the house advantage does not change; however, the length of play does, which in turn affects the cumulative losses and the handle, a relevant insight for gaming establishments. Asymmetrical betting limits, such as? Leave if I lose $500, but stay until I win $1000 increases the number of players who finish with losses relative to players with symmetrical bet limits, which could affect player psychology. A second simulation explores the effect of the herd mentality when applied to Baccarat (i.e., contrary to the gambler? fallacy bets escalate on to the winning side of a streak). The able differential is the difference between total banker bets to total player bets or vice versa, thereby limiting the casino’s exposure to loss. Intuitively, as the differential increases, so does the collective player handle, hence house winnings, thus suggesting that high differentials are beneficial for the house? Regardless of differential, the house advantage does not change; but the greater the differential, the greater the number of shoes that end with players losing and the more extreme are the spikes in the positive domain for players (the house paying out for a hand of play). This simulation is therefore intended to help casino managers set betting limits that maximize total winnings while bearing in mind both the probability and magnitude of negative outcomes of increased differentials.