Everybody likes Ramsey Theory. This issue begins with some new results about a famous problem. Color the first $$n$$ integers with $$r$$ colors, and start looking for $$k$$-term arithmetic progressions...well, you can read what happens. This is a game that anyone can play, as many open problems remain.

This issue also revisits "self-tiling tile sets," introduces a magic trick, exhibits some elegant sphere packings, proves old identities in new ways, and reveals yet one more surprise in Pascal's triangle.—*Walter Stromquist*

Vol. 87, No. 2, pp. 82-160.

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## ARTICLES

### Unrolling Residues to Avoid Progressions

Steve Butler, Ron Graham, and Linyuan Lu

We look at the problem of coloring $$1, 2,\dots n$$ with $$r$$ colors to minimize the portion of monochromatic $$k$$-term arithmetic progressions. By using residues to color $$\mathbb{Z}_{m}$$ and then unrolling, we produce the best-known colorings for several small values of $$r$$ and $$k$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.83

### Math Bite: Indeterminate Forms

Mark Lynch

The indeterminate forms $$1^{\infty}$$, $$\infty^{0}$$, and $$0^{0}$$ are not automatically 1.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.94

### A 3-D Analog of Steiner’s Porism

Owen D. Byer and Deirdre L. Smeltzer

Steiner’s Porism, the classical result on chains of circles discovered by Jacob Steiner in the nineteenth century, is treasured for its beauty, for its simplicity, and as one of the great applications of inversion in the plane. In this note, we extend his result to a packing of spheres in 3-space, along with a surprising connection to regular polyhedra.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.95

### More On Self-Tiling Tile Sets

Lee Sallows

This is an extension of the author’s previous work on self-tiling tile sets. Hitherto, the only known method for discovering the latter was by means of a brute-force computer search. A new pencil and paper method for extracting such sets from rep-tiles is introduced, and a plethora of new examples presented.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.100

### The Geometry of Cubic Polynomials

Christopher Frayer, Miyeon Kwon, Christopher Schafhauser, and James A. Swenson

We study the critical points of a complex cubic polynomial, normalized to have the form $$p(z)=(z-1)(z-r_{1})(z-r_{2})$$ with $$|r_{1}|=1=|r_{2}|$$. If $$T_{\gamma}$$ denotes the circle of diameter $$\gamma$$ passing through 1 and $$1-\gamma$$, then there are $$\alpha,\beta\in[0,2]$$ such that one critical point of $$p$$ lies on $$T_{\alpha}$$ and the other on $$T_{\beta}$$. We show that $$T_{\beta}$$ is the inversion of $$T_{\alpha}$$ over $$T_{1}$$, from which many geometric consequences can be drawn. For example, (1) a critical point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a “desert” in the unit disk, the open disk $$\{z\in\mathbb{C}:|z-\frac{2}{3}|<\frac{1}{3}\}$$, in which critical points cannot occur.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.113

## NOTES

### A Reed–Solomon Code Magic Trick

Todd D. Mateer

A magic trick based on properties of a (4, 2, 3) extended Reed–Solomon code is introduced. We then discuss relevant concepts of the Reed–Solomon code to explain why the trick works and how it was designed. This magic trick can be used as a teaching device in introductory error-correcting codes courses.

Supplements are available here or here.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.125

### A Golden Product Identity for $$e$$

Robert P. Schneider

We prove an infinite product representation for the constant e involving the golden ratio, the Möbius function, and the Euler phi function—prominent players in number theory.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.132

### Dominance Orders, Generalized Binomial Coefficients, and Kummer’s Theorem

Tyler Ball, Tom Edgar, and Daniel Juda

We discuss the connections between a family of partial orders known as $$b$$-dominance orders, arithmetic in base $$b$$, and generalized binomial coefficients. In particular, we investigate theorems of Lucas and Kummer in relation to these orders and attempt to extend and explain these theorems using a family of generalized binomial coefficients derived from a simple integer sequence.

To purchase from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.135

### A Golden Connection

Robert P. Schneider

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.143

### A Proof of the Cosine Addition Formula Using the Law of Cosines

Patrik Nystedt

We give a proof of the cosine addition formula using the law of cosines.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.144

### A Dissection Proof of Leibniz’s Series for $$\pi/4$$

Mitsuo Kobayashi

Inspired by Lord Brouncker’s discovery of his series for $$ln 2$$ by dissecting the region below the curve $$1/x$$, Viggo Brun found a way to partition regions of the unit circle so that their areas correspond to terms of Leibniz’s series for $$\pi/4$$. Brun’s argument involved *ad hoc* methods which were difficult to find. We develop a method based on usual techniques in calculus that leads to Brun’s result and that applies generally to other related series.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.145

## PROBLEMS

Proposals 1941-1945

Quickies 1039 & 1040

Solutions 1916-1920

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.151

### REVIEWS

Twin primes; Fermi problems blog; Bridges

To purchase the article from JSTOR: http://dx.doi.org/10.4169/math.mag.87.2.159