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College Mathematics Journal Contents—September 2014

The annual College Mathematics Journal issue dedicated to puzzles and games focuses this year on Rubik's Cube, invented forty years ago. David Joyner offers a profile of Tomas Rokicki and his work leading to the 2010 result that it takes at most 20 moves to solve any cube position. That result is for one of multiple ways of measuring moves; in the lead article, Rokicki offers work towards determining "God's number" under another metric. Continuing the theme, there is a review of Liberty Science Center's Beyond Rubik's Cube exhibit and an analysis of Rubik's Slide by Jones, Shelton, and Weaverdyck. Among articles on other topics including hexagonal chess, the issue presents four puzzles: crossword, "imbalance," logic, and "permudoku."—Brian Hopkins

Vol. 45, No. 4, pp. 242-333

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ARTICLES

Towards God’s Number for Rubik’s Cube in the Quarter-Turn Metric

Tomas Rokicki

A difficult problem in computer cubing is to find positions that are hard—positions that are as far from solved as possible. An effective way to find such positions is to examine positions exhibiting symmetry; empirically, we see that hard positions have a higher frequency in the subset of positions exhibiting symmetry than in general. While there are many symmetric positions, they are a small fraction of all positions and belong to a few large subgroups, which enables us to solve them effectively with a coset solver.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.242

The Man Who Found God’s Number

David Joyner

This is a tale of two problems. For years, Tom Rokicki worked to determine the exact value of God’s number for the Rubik’s Cube (the smallest number of moves needed to solve the cube in the worst case), a very difficult problem. By the time he solved this, Tom was completely deaf. Digitizing human hearing, and then implementing that into a medical device, is also a very difficult problem. Thanks to recent medical advances, Tom’s hearing was restored about the same time that he discovered God’s number.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.258

On God’s Number(s) for Rubik’s Slide

Michael A. Jones, Brittany C. Shelton, and Miriam E. Weaverdyck

Rubik’s Slide consists of a 3 × 3 grid of squares reminiscent of a face of Rubik’s Cube. Squares may light up in one of two colors or remain unlit, and the goal is to use a series of moves to change a given initial pattern to a given end pattern. Viewing these moves as permutations, we use algebraic and graph-theoretic tools to analyze a simpler version of the puzzle and the three difficulty levels of Rubik’s Slide. We determine the maximum number.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.267

Math Frenzy

By Charlie Smith, Park University

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.276

Graph Theory Problems from Hexagonal and Traditional Chess

Stan Wagon

Chess pieces of various sorts give rise to intriguing graphs and studying their properties can yield nice conjectures, and sometimes simple proofs. This paper examines some problems related to traditional queens and bishops, and also some pieces arising in a hexagonal version of chess. Using powerful algorithmic methods such as integer-linear programming is critical to discovering various patterns.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.278

Imbalance Puzzles

Paul Salomon

The diagrams indicate scales that tip down to the heavier side of hanging weights.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.288

Chutes and Ladders with Large Spinners

Darcie Connors and Darren Glass

We prove a conjecture from a 2011 College Mathematics Journal article addressing the expected number of turns in a Chutes and Ladders game when the spinner range is close to the length of the board. While the original paper approached the question using linear algebra and the theory of Markov processes, our main method uses combinatorics and recursion.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.289

Story Puzzles

Oscar Levin

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Knights, Knaves, Normals, and Neutrals

Jason Rosenhouse

Puzzles about knights and knaves, in which knights only make true statements and knaves only make false statements, are a fun and instructive way of introducing principles of classical logic. There are, however, many systems of non-classical logic as well. We consider what knight/knave puzzles might look like with respect to one such non-classical system: three-valued logic. In this system, all statements are either true, false, of neutral, with the third value applying to statements that are vague, or neither true nor false.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.297

Permudoku Puzzle

David Nacin

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.307

Book Review

Games and Mathematics: Subtle Connections by David Wells

Reviewed by Michael Henle

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Exhibit Review

Beyond Rubik’s Cube at the Liberty Science Center, Jersey City NJ.

Reviewed by Calvin Armstrong and Susan Goldstine

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.254

Classroom Classrooms

A Topological Definition of Limit for Use in Elementary Calculus

Charles L. Cooper and Michael S. McClendon

We describe a topological definition of the limit that can be used as an alternative to the standard definition in elementary calculus. In particular, we replace intervals centered about the relevant quantities being approached in the domains and ranges of functions with arbitrarily small general open intervals about those quantities. In many cases, this simplifies the work of verifying specific limits and the verification of many of the basic limit properties.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.313

Stretched Circles are Conic Sections, a Geometric Proof

Stephan Berendonk

We give a geometric proof for the fact that a conic section, if it is a closed curve, satisfies the equation of an ellipse.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.316

Problems and Solutions

Problems: 1031 – 1034

Solutions: 1006 – 1010

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.318

Media Highlights

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.4.324

Dummy View - NOT TO BE DELETED