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. —*Scott Chapman*

Vol. 119, No. 6, pp.443-532.

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Dimitris Koukoulopoulos and Johann Thiel

In 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.

To purchase the article from JSTOR:http://dx.doi.org/10.4169/amer.math.monthly.119.06.443

Fred Kuczmarski

A 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.

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Robert Mandl, Thomas Pühringer, and Maximilian Thaler

We present an analytic proof and extensions of Galileo's theorem comparing the descent time along one and two-chord paths in a circle.

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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.

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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 $$\mathbb{R}^{n}$$ in with values in the unit sphere $$S_{n-1}$$.

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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.

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Michael J. Dorfling and Johan H. Meyer

A divisibility condition, together with some kind of boundedness, ensure that a function $$f:\mathbb{Z}\rightarrow\mathbb{Z}$$ is a polynomial function.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.06.502

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.

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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 $$T(n, k, l)$$.

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Walter Wyss

A 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).

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Timothy C. Wallstrom

We consider a Pólya urn, started with b black and w white balls, where $$b>w$$. 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 $$w-1$$ heads in $$b+w-1$$ tosses of a fair coin.

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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.

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Problems 11650-11655

Solutions 11527, 11531, 11534, 11535, 11536, 11541, 11542

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.06.522

*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

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.119.06.530