The October issue begins with four papers on recreational mathematics including a commutative algebra-inspired generalization of Nim (by Haley Dozier and John Perry); an attempt to fix Instant Insanity II (by Robert Beeler and Amanda Justus Bentley); a look at the effect of adding wild cards in poker (by Ashley Fielder, Carlie Maasz, Monta Meirose, and Christopher Spicer); and fuzzy logic puzzles in a Smullyan style (by Jason Rosenhouse). Stephen Kackowski provides three proofs of a limit. Amy and Dave Reimann interview Dick Termes about six-point perspective and spheres. There is also a math and elections themed crossword, details from the USA and USA Junior Mathematical Olympiads, the announcement of the Allendoerfer Award winners, as well as problems/solutions and reviews.

—*Michael A. Jones, Editor*

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Vol. 89, No. 4, pp 233 – 315

## Articles

### Androids Armed With Poisoned Chocolate Squares: Ideal Nim and Its Relatives

Haley Dozier and John Perry

We describe two new combinatorial games. The first, Ideal Nim, both generalizes the well-known game Nim and its relative Chomp, and provides a recreational perspective on some important ideas of commutative algebra; for instance, the fact that the game is guaranteed to end is equivalent to Dickson’s lemma, a well-known fact of commutative algebra. This relationship leads to a game-based proof of Dickson’s lemma. The second game, Gröbner Nim, is really a variant of Ideal Nim that illustrates Buchberger’s algorithm to compute a Gröbner basis. We conclude by describing the relationship between Gröbner Nim and polynomial rings.

To purchase from JSTOR: 10.4169/math.mag.89.4.235

### Proof Without Words: The Parallelogram With Maximum Perimeter for Given Diagonals Is the Rhombus

Ángel Plaza

By using the ellipse with foci at the extreme points of the shortest diagonal and the minor axis being the longest diagonal, it is proved without words that the parallelogram with maximum perimeter for given diagonals is the rhombus.

To purchase from JSTOR: 10.4169/math.mag.89.4.251

### Curing Instant Insanity II

Robert A. Beeler and Amanda Justus Bentley

Instant Insanity II is a 4 by 4 sliding tile puzzle designed by Philip Orbanes. The packaging indicates that there is a unique solution to the puzzle, up to rotations of the columns and permutations on the rows. However, a recent paper by Richmond and Young shows that there are in fact two solutions to the puzzle. This paper presents several attempts at ``fixing'' the puzzle to guarantee a unique solution. Of these, the only one that guaranteed a unique solution was removing a color to create a 3 by 3 puzzle.

To purchase from JSTOR: 10.4169/math.mag.89.4.252

### Proof Without Words: An Identity for a Recurrence Satisfied by the Fibonacci and Lucas Numbers

José Ángel Cid Araujo

We present a visual proof of an identity for three consecutive terms of a recurrence relation satisfied by the Fibonacci and Lucas numbers.

To purchase from JSTOR: 10.4169/math.mag.89.4.262

### Extreme Wild Card Poker

Ashley Fiedler, Carlie Maasz, Monta Meirose and Christopher Spicer

Rankings of hands in traditional five-card poker are based on the relative frequency of each type of hand occurring. When wild cards are added to the standard deck, these rankings can go awry quite quickly. In this paper, wild card poker is taken to the extreme. We find the minimum number of wild cards needed to ensure five-of-a-kind is the most common hand.

To purchase from JSTOR: 10.4169/math.mag.89.4.263

### Proof Without Words: The Automorphism Group of the Petersen Graph Is Isomorphic to S5

Japheth Wood

The automorphism group of the Petersen graph is known to be isomorphic to the symmetric group on 5 elements. This proof without words provides an insightful and colorful image that proves this fact, without words. The image represents the Petersen graph with the ten 3-element subsets of {1, 2, 3, 4, 5} as vertices, and two vertices are adjacent when they have precisely one element in common. This representation of the Petersen Graph is similar to the Kneser graph *K G*_{5,2}, a nice picture of which can be found in John Baez’s Visual Insight blog.

To purchase from JSTOR: 10.4169/math.mag.89.4.267

### Fuzzy Knights and Knaves

Jason Rosenhouse

Puzzles about knights, who only make true statements, and knaves, who only make false statements, have long been a mainstay of classical logic. They are valuable not just as recreational puzzles, but as a pedagogical device for exploring fundamental issues in logic. However, the possibilities for puzzles based on nonclassical logics have been mostly unexplored. In this paper we consider knight/knave dialogs based on fuzzy logic, in which a truth value can be any real number between zero and one inclusive. Following the example of classical logic, our purpose is both recreational and educational.

To purchase from JSTOR: 10.4169/math.mag.89.4.268

### Proof Without Words: Alternating Row Sums in Pascal’s Triangle

Ángel Plaza

Based on the Pascal’s identity, we visually demonstrate that the alternating sum of consecutive binomial coefficients in a row of Pascal’s triangle is determined by two binomial coefficients from the previous row.

To purchase from JSTOR: 10.4169/math.mag.89.4.281

### The Limiting Value of a Series With Exponential Terms

Stephen Kaczkowski

A sequence defined by a series of exponential terms was discussed by Spivey [*Math.Mag*. **79** (2006) 61–65] and Holland [*Math. Mag.* **83** (2010) 51–54]. In this paper, a simple variation of this sequence is discussed, and three proofs are given to establish that both sequences have the same limiting value. A proof by comparison uses an asymptotic approximation of an improper integral that models the difference between the sequences. Then a direct proof of this fact is obtained using well known inequalities involving exponential functions along with basic properties of geometric series. Finally, Tannery’s theorem is defined and applied to efficiently demonstrate that the sequence converges to the ratio of *e* and (*e* – 1).

To purchase from JSTOR: 10.4169/math.mag.89.4.282

### Mathematics and Elections

Michael A. Jones and Jennifer Wilson

To purchase from JSTOR: 10.4169/math.mag.89.4.289

### Dick Termes: Art of the Sphere

Amy L. Reimann and David A. Reimann

To purchase from JSTOR: 10.4169/math.mag.89.4.290

## Problems and Solutions

Proposals, 2001-2005

Quickies, 1063-1064

Solutions, 1971-1975

Answers, 1063-1064

To purchase from JSTOR: 10.4169/math.mag.89.4.293

## Reviews

Mathematics as common sense; mathematical depth; learning to count; mathematics as sex

To purchase from JSTOR: 10.4169/math.mag.89.4.301

## News and Letters

### 45th United States of America Mathematical Olympiad and 7th United States of America Junior Mathematical Olympiad

Evan Chen and Doug Ensley

To purchase from JSTOR: 10.4169/math.mag.89.4.303

### Carl B. Allendoerfer Awards

To purchase from JSTOR: 10.4169/math.mag.89.4.312