# College Mathematics Journal Contents—September 2015

The annual games and puzzles issue of The College Mathematics Journal explores the mathematics in many popular recreations, from Candy Crush combinatorics to Settlers of Catan probabilities.  Cookie Monster returns to make new combinatorial games and we learn the number of possible extensions of Rock-Paper-Scissors-Lizard-Spock with a sixth option.  Articles on One-Round War and colorful pairings (with many open questions) grew from New York Times NumberPlay blog posts.  David Nacin supplies four MAA Centennial puzzles that mix filling in grids with letters and numerals with some repetition with adding up numbers or pen strokes!  The playful theme continues into a Classroom Capsule on a permutation-based card trick and a review of Siobhan Robert's biography of John Horton Conway. -Brian Hopkins

Vol 46 No 4, pp 242-323

## ARTICLES

### How to Win at (One-Round) War

Richard E. Chatwin and Dana Mackenzie

One-Round War is a card game with a variable deck size that is easy enough to teach to young children. For any given number of cards in the deck, the authors find an optimal strategy, in the sense of maximizing the expected number of tricks won. The threecard version of this problem is, in effect, nearly 2500 years old and was “solved” by Sun Bin, a legendary Chinese military strategist. The general solution calls upon a surprising variety of techniques from combinatorics and analysis, including the central limit theorem and the Hungarian algorithm.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.242

### MAA 100th Anniversary CMJ Puzzle A

David Nacin

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.254

### Candy Crush Combinatorics

Dana Rowland

In the popular game Candy Crush, differently colored candies are arranged in a grid and a player swaps adjacent candies in order to crush them by lining up three or more of the same color. At the beginning of each game, the grid cannot have three consecutive candies of the same color in a row or column, but it must be possible to swap two adjacent candies in order to get at least three consecutive candies of the same color. How many starting configurations are there? We derive recurrence relations to answer this question for a single line of candy, and also for a pair of lines in the 2-color version of the game.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.255

### MAA 100th Anniversary CMJ Puzzle C

David Nacin

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.263

### Square–Sum Pair Partitions

Gordon Hamilton, Kiran S. Kedlaya, and Henri Picciotto

We present a middle school problem and generalize it in several ways, including many new variations for readers to explore. The problems should be attractive to students, as they have few prerequisites and lend themselves to beautiful visual representations.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.264

### The Uniqueness of Rock-Paper-Scissors-Lizard-Spock

Brian J. Birgen

We show that the extension of Rock-Paper-Scissors to include Lizard and Spock is the unique such five move fair game up to isomorphism. We also categorize all analogous six move fair games.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.270

### MAA 100th Anniversary CMJ Puzzle J

David Nacin

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.274

### The Settlers of Catan: Using Settlement Placement Strategies in the Probability Classroom

Jathan Austin and Susanna Molitoris-Miller

The Settlers of Catan, a property-building and trading board game, contains many opportunities for mathematical exploration. In this paper we discuss Catan settlement placement strategies suitable for teaching basic concepts of probability and expected value to undergraduate students.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.275

Tanya Khovanova and Joshua Xiong

We research a combinatorial game based on the Cookie Monster problem called the Cookie Monster game that generalizes the games of Nim and Wythoff. We also propose several combinatorial games between this Cookie Monster game and Nim, and discuss the winning positions of these games.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.283

### MAA 100th Anniversary CMJ Puzzle M

David Nacin

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.294

## Classroom Capsules

### A Magic Trick Leads to an Identity: Some Induction Fun

Robert W. Vallin

A magic trick based on a mathematical principle and popularized by Martin Gardner leads us to an interesting identity.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.295

### Area of a Circle via the Second Fundamental Theorem of Calculus

Denis Bell

We provide an easy argument, using the second fundamental theorem of calculus, for computing the area of the unit circle.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.299

## Problems and Solutions

Problems and Solutions: 300-308

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.300

## Book Review

### Genius at Play: The Curious Mind of John Horton Conway by Siobhan Roberts

Reviewed by Joseph O'Rourke

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.309

## Media Highlights

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.46.4.315