A relation between Catalan Numbers and World Series type problems

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This issue of *The College Mathematics Journal* begins with two articles on disease: Ronald Mickens formulates an SIR epidemiological model that, unlike the usual version, is exactly solvable and in which the epidemic has a finite lifetime; and Matthew Glomski and Edward Ohanian discuss disease eradication and why smallpox is, so far, the only infectious disease to be eliminated world-wide.

Continuing January's celebration of Martin Gardner's Mathematics, the March issue contains five articles based on ideas popularized by Gardner in his influential Scientific American columns. Featured topics are Bulgarian solitaire (Brian Hopkins), the Catalan numbers (Thomas Koshy and Zhenguang Gao), Conway's Life (Yossi Elran), non transitive dice (Jorge Moraleda and David G. Stork), and combinatorial games (Aviezri S. Fraenkel).

The issue also contains a Student Research Project (on optimizing the amount of attic insulation), two Classroom Capsules, Problems and Solutions, a Book Review, and Media Highlights.

**An Exactly Solvable Model for the Spread of Disease**

*Ronald E. Mickens*

We present a new SIR epidemiological model whose exact analytical solution can be calculated. In this model, unlike previous models, the infective population becomes zero at a finite time. Remarkably, these results can be derived from only an elementary knowledge of differential equations.

**Teaching Tip: How to Manipulate Test Scores**

*Colin Foster*

**Eradicating a Disease: Lessons from Mathematical Epidemiology**

*Matthew Glomski and Edward Ohanian*

Smallpox remains the only human disease ever eradicated. In this paper, we consider the mathematics behind control strategies used in the effort to eradicate smallpox, from the life tables of Daniel Bernoulli, to the more modern susceptible-infected-removed (SIR)-type compartmental models. In addition, we examine the mathematical feasibility of the eradication of polio and certain other infectious diseases.

**Teaching Tip: Consider a Circular Cow**

*Ezra Halleck*

**MORE MARTIN GARDNER MATHEMATICS:
30 Years of Bulgarian Solitaire **

Bulgarian solitaire is a puzzle popularized by Martin Gardner that can be considered an operation on integer partitions. We explore its history in Russia, Bulgaria, Sweden, and the United States before revisiting Gardner’s treatment and considering subsequent research. The article concludes by introducing a related combinatorial game.

**MORE MARTIN GARDNER MATHEMATICS:
Convergence of a Catalan Series**

This article studies the convergence of the infinite series of the reciprocals of the Catalan numbers. We extract the sum of the series as well as some related ones, illustrating the power of the calculus in the study of the Catalan numbers.

**MORE MARTIN GARDNER MATHEMATICS:
Retrolife and The Pawns Neighbors**

One of Martin Gardner’s most famous columns introduced John Conway’s game of Life. The inverse problem, finding a previous generation in the Game of Life given some extra constraints, was introduced a few years ago and is referred to as Retrolife. In this paper we present a puzzle played on a chessboard that is isomorphic to a variation of Retrolife, in memory of Martin Gardner.

**MORE MARTIN GARDNER MATHEMATICS:
Lake Wobegon Dice**

We introduce Lake Wobegon dice, where each die is “better than the set average.” Specifically, these dice have the paradoxical property that on every roll, each die is more likely to roll greater than the set average on the roll, than less than this set average. We also show how to construct minimal optimal Lake Wobegon sets for all n ≥ 3.

**MORE MARTIN GARDNER MATHEMATICS:
RATWYT**

WYTHOFF is played on a pair of nonnegative integers, (M, N). A move either subtracts a positive integer from precisely one of M or N such that the result remains nonnegative, or subtracts the same positive integer from both M and N such that the results remain nonnegative. The first player unable to move loses. RATWYT uses rational numbers instead, transformed using a generalization of the rules of WYTHOFF. Using the Calkin-Wilf tree, we show how to play RATWYT, and any other rational take-away game.

**STUDENT RESEARCH PROJECT:
The Optimal Level of Insulation in a Home Attic**

The project models the conductive heat loss through the ceiling of a home. Students are led through a sequence of tasks from measuring the area and insulation status of a home to developing several functions leading to a net savings function where the depth of insulation is the input. At this point students use calculus or a graphing utility to determine the optimal savings. Extensions to air conditioning savings and a present values of future savings are also included.

**CLASSROOM CAPSULES**

**The Spider and the Fly**

*Keith E. Mellinger and Raymond Viglione*

The Spider and the Fly puzzle, originally attributed to the great puzzler Henry Ernest Dudeney, and now over 100 years old, asks for the shortest path between two points on a particular square prism. We explore a generalization, find that the original solution only holds in certain cases, and suggest how this discovery might be used in the classroom.

**A Real Proof of the Principal Axis Theorem**

In this capsule we give an elementary proof of the principal axis theorem within the real field, i.e., without using complex numbers.

**PROBLEMS AND SOLUTIONS**

**BOOK REVIEW**

*The Shape of Inner Space*, by Shing-Tung Yau and Steve Nadis, Basic Books, 2010, 377 + xx pp., ISBN 978-0-465-02023-2, $30.

Reviewed by David A. Cox

**MEDIA HIGHLIGHTS**