Much of this issue of the Magazine demonstrates the connections between different areas of mathematics. Barnes, Burry, Russell, and Schaubroeck visually examine Julia sets that emerge when complex functions are iterated. Brown offers more names for the (7,3,1) design through its connections to codes, finite geometries, difference sets, and real normed algebras. Berele and Catoiu relate the rationalizing of denominators to symmetric function theory, field theory, and algebraic number theory. Swart and Shelton generalize a card trick from The Ellen DeGeneres Show. Byer and Smeltzer extend a geometric result about three mutually tangents circles in the plane to higher dimensions. The issue rounds out with two proofs without words, a crossword puzzle, Reviews and Problems. —*Michael A. Jones, Editor*

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Vol. 88, No. 2, pp 90 – 160

## Articles

### Emerging Julia Sets

Julia A. Barnes, Clinton P. Curry, Elizabeth D. Russell, and Lisbeth E. Schaubroeck

Functions from the complex plane to itself are difficult to visualize; we consider the real and imaginary projections. In this paper, we explore the connections between the graphs of the real and imaginary parts of various complex functions and their corresponding filled Julia sets. We begin by examining the family of complex quadratic functions.We then expand our results to a broader collection of rational maps, including functions whose Julia sets form a Cantor set of simple closed curves, checkerboards, and a perturbed rat.

To purchase from JSTOR: 10.4169/math.mag.88.2.91

### Many More Names of (7, 3, 1)

Ezra Brown

The (7, 3, 1) block design is an object that shows up in many areas of mathematics. In fact, (7, 3, 1) seems to appear again and again in unexpected places. A 2002 paper described (7, 3, 1)'s connection with such areas as graph theory, number theory, topology, round-robin tournaments, and algebraic number fields. In this paper, we show how (7, 3, 1) makes appearances in the areas of error-correcting codes, n-dimensional finite projective geometries, difference sets, normed algebras, and the three-circle Venn diagram.

To purchase from JSTOR: 10.4169/math.mag.88.2.103

### Rationalizing Denominators

Allan Berele and Stefan Catoiu

We present several techniques for rationalizing the denominators of fractions which involve radical expressions of rational numbers. Our algorithms are based on prime numbers, indeterminate coefficients, symmetric polynomials, and Galois theory.

To purchase from JSTOR: 10.4169/math.mag.88.2.121

## Notes

### Revelations and Generalizations of the Nine Card Problem

Breeanne Baker Swart and Brittany Shelton

The nine card problem is a magic trick performed by shuffling nine playing cards according to a set of rules. The magic is that a particular card will always reappear. The success of this trick can be easily explained by considering the lengths of the words in the names of playing cards, which define the shuffling rules. In this paper, we use permutations to prove that the trick will always work. We then use this methodology to generalize the trick to any number of cards with shuffles according to different rules

To purchase from JSTOR: 10.4169/math.mag.88.2.137

### Proof Without Words: President Garfield and the Cauchy-Schwarz Inequality

Claudi Alsina and Roger B. Nelsen

We prove wordlessly the Cauchy–Schwarz inequality (for n = 2) using a trapezoid partitioned into three right triangles.

To purchase from JSTOR: 10.4169/math.mag.88.2.144

### Mutually Tangent Spheres in *n*-Space

Owen D. Byer and Deirdre L. Smeltzer

In this note we prove that the points of tangency of n + 1 mutually tangent spheres in n-dimensional space lie on a generalized sphere. Coxeter's observation that for each of five mutually tangent spheres there is a sphere passing through the six points of mutual contact of the remaining four is a corollary of this result in the *n* = 3 case.

To purchase from JSTOR: 10.4169/math.mag.88.2.146

### Proof Without Words: A Trigonometric Identity for sec *x* + tan *x*

Rober B. Nelsen

We prove wordlessly the identity sec *x* + tan *x* = tan(*π*/4 + *x*/2).

To purchase from JSTOR: 10.4169/math.mag.88.2.151

### Unifying Two Proofs of Fermat's Little Theorem

Massimo Galuzzi

A new simple proof of Fermat's little theorem is given that generalizes the proofs given in this MAGAZINE by Levine (1999) and Iga (2003).

To purchase from JSTOR: 10.4169/math.mag.88.2.152

### Books for a Math Audience

Brendan W. Sullivan

To purchase from JSTOR: 10.4169/math.mag.88.2.155

## Problems and Solutions

Proposals, 1966-1970

Quickies, 1049-1050

Answers, 1049-1050

To purchase from JSTOR: 10.4169/math.mag.88.1.72

## Reviews

Why do you do mathematics?; twin primes on film; constitutional mathematics; fun!

To purchase from JSTOR: 10.4169/math.mag.88.2.159