June/July 2012 Contents
The June-July issue of the Monthly starts off with the number theory behind the arrangement of stars on the U.S. flag. Other articles look at the areas swept out by circles rolling on sinusoidal spirals, consider an analytic approach to Galileo’s Theorem on descent time, and more. Notes offer a divisibility condition that ensures a function is a polynomial, a method to optically measure the Gaussian curvature of a surface, and other tips. In our book review, Jennifer Quinn looks at Combinatorics the Rota Way, by Kung, Rota, and Yan. If all this doesn’t keep you busy enough, then our Problem Section will.
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ARTICLES
Arrangements of Stars on the American Flag
Dimitris Koukoulopoulos and Johann ThielIn this article, we examine the existence of nice arrangements of stars on the American flag. We show that despite the existence of such arrangements for any number of stars from 1 to 100, with the exception of 29, 69 and 87, they are rare as the number of stars increases.
Rolling Sinusoidal Spirals
Fred KuczmarskiA theorem of Apostol and Mnatsakanian states that as a circle rolls on a line, the area of the cycloidal sector traced by a point on the circle is always three times the area of the corresponding segment cut from the rolling circle. We generalize this result by showing that sinusoidal and logarithmic spirals rolling on lines have similar area ratio properties. We then extend our ideas to include one curve rolling on another. Such pairs of curves are a natural generalization of road-wheel pairs.
Analytic Approach to Galileo's Theorem on the Descent Time Along Two-Chord Paths in a Circle
Robert Mandl, Thomas Pühringer, and Maximilian ThalerWe present an analytic proof and extensions of Galileo's theorem comparing the descent time along one and two-chord paths in a circle.
A Prüfer Angle Approach to the Periodic Sturm-Liouville Problem
Paul Binding and Hans Volkmer
It is shown how to reduce the periodic or antiperiodic Sturm-Liouville problems to an analysis of the Prüfer angle. This provides a simple and flexible alternative to the usual approaches via operator theory or the Hill discriminant.
Mappings into the Euclidean Sphere
Raymond Mortini and Rudolf Rupp
We present short and elementary non-geometric analytic proofs of several standard results concerning extension of continuous mappings defined on compacta 
in with values in the unit sphere 
.
Is Napoleon's Theorem Really Napoleon's Theorem?
Branko Grünbaum
A result frequently attributed to Napoleon Bonaparte is the topic of this note; it has an interesting history, and there are a considerable number of papers devoted to it. Several relevant articles have appeared in this Monthly. Here we present additional information about the history of the result, supplementing and correcting some of the earlier publications.
NOTES
On Polynomials and Divisibility
Michael J. Dorfling and Johan H. Meyer
A divisibility condition, together with some kind of boundedness, ensure that a function 
is a polynomial function.
Gaussian Curvature, Mirrors, and Maps
Pedro Roitman
We present a method to optically measure the Gaussian curvature K of a surface and show how it can be used to establish a link between surfaces with constant K and area preserving maps between a sphere and a planar region. As an example, we show how Lambert's azimuthal equal area projection is naturally related to a sphere.
The Turán Number and Probabilistic Combinatorics
Alan J. Aw
In this short expository article, we describe a mathematical tool called the probabilistic method, and illustrate its elegance and beauty through proving a few well-known results. Particularly, we give an unconventional probabilistic proof of a classical theorem concerning the Turán number T(n, k, l).
Perfect Parallelograms
Walter Wyss
IA parallelogram, with its sides and diagonals being positive integers, is called a perfect parallelogram. Up to scaling we find a parameterization for all perfect parallelograms (quadruples). As a special case, we determine the perfect rectangles (Pythagorean triples).
The Equalization Probability of the Pólya Urn
Timothy C. Wallstrom
We consider a Pólya urn, started with b black and w white balls, where 
. We compute the probability that there are ever the same number of black and white balls in the urn, and show that it is twice the probability of getting no more than heads in 
tosses of a fair coin.
A Proof of Euler's Infinite Product for the Sine
Lars Holst
The product formula for the sine function is proved using the Gamma function and elementary probability theory. Some corollaries of the sine formula are also pointed out.
PROBLEMS AND SOLUTIONS
REVIEWS
Combinatorics the Rota Way. Joseph P. S. Kung, Gian-Carlo Rota, and Catherine H. Yan. Cambridge University Press, New York, 2009, xii+396 pp., ISBN 978-0-521-73794-4, $40.99.
Reviewed by Jennifer J. Quinn