A relation between Catalan Numbers and World Series type problems

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November is full of holidays worldwide, and, to celebrate, the *Monthly* brings you four outstanding articles and six terrific notes. Don’t miss the amazing two-dimensional objects constructed by stacking Cantor sets in Véhel and Mendivil’s "Christiane’s Hair." Ever wonder how fast the reciprocal sum of the prime numbers diverges? Find out in Bayless and Klyve’s "Reciprocal Sums as a Knowledge Metric: Theory, Computation, and Perfect Numbers." While the set of quotients of prime numbers is dense in the reals, is the same true for quotients of the Gaussian Primes in the complex plane? Stephan Garcia gives us a positive answer in "Quotients of Gaussian Primes." Betty Mayfield reviews *Calculus: Modeling and Application* by David A. Smith and Lawrence C. Moore, and, as always, our Problem Section will give you much to think about.

The *Monthly* will close out 2013 with a “bang” in December as David Pengelley shows us whether or not 23/67 equals 33/97. —*Scott Chapman*

Vol. 120, No. 9, pp.771-864.

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Jacques Lévy Véhel and Franklin Mendivil

We explore the geometric and measure-theoretic properties of a set built by stacking central Cantor sets with continuously varying scaling factors. By using self-similarity, we are able to describe its main features in a fairly complete way. We show that it is made of an uncountable number of analytic curves, compute the exact areas of the gaps of all sizes, and show that its Hausdorff and box-counting dimensions are both equal to 2. It provides a particularly good example to introduce and showcase these notions because of the beauty and simplicity of the arguments. Our derivation of explicit formulas for the areas of all of the gaps is elementary enough to be explained to first-year calculus students.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.771

Daniel J. Velleman and Gregory S. Warrington

The game of memory is played with a deck of $$n$$ pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one card and then another. The cards are removed from play if they match. Otherwise, they are flipped back over and the next move commences. A game ends when all pairs have been matched. We determine that, when the game is played optimally, as $$n\rightarrow\infty$$:

- The expected number of moves is $$(3-2\ln2)n+\frac{7}{8}-2\ln2\approx1.61n$$
- The expected number of times two matching cards are unwittingly flipped over is $$\ln2$$.
- The expected number of flips until two matching cards have been seen is $$2^{2n}/\left(\begin{array}{c}2n\\n\end{array}\right) \sim\sqrt{\pi n}$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.787

Antonín Slavík

We revisit the classical calculus problem of describing the flow of brine in a system of tanks connected by pipes. For various configurations involving an arbitrary number of tanks, we show that the corresponding linear system of differential equations can be solved analytically. Finally, we analyze the asymptotic behavior of solutions for a general closed system of tanks. It turns out that the problem is closely related to the study of Laplacian matrices for directed graphs.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.806

Jonathan Bayless and Dominic Klyve

We first provide a short survey of reciprocal sums. We discuss some of the history of their computation and application, show how they are applied in various modern contexts, and discuss some ways that their values are computed. We give an example of computing a reciprocal sum by providing (we believe) the first computation of the sum of the reciprocals of perfect numbers. Second, we introduce a new use for reciprocal sums; that is, they can be used as a knowledge metric to classify the current state of number theorists’ understanding of a given class of integers.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.822

Jorge Luis Cimadevilla Villacorta

In this paper, we prove certain inequalities of the type $$\sum_{i=1}^{n}d(si+t)\geq\sum_{i=1}^{n}d(ui+v)$$, where $$d(n)$$ is the divisor function and $$s,t,u,v\in\mathbb{Z}$$.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.832

Charles S. Kahane

**Editor’s Note**: Dr. Charles S. Kahane and his wife, Claire, were killed in an automobile accident in May of this year.

With $$\{n_{j}\}$$ an arbitrary subsequence of natural numbers, it is shown that $$\lim_{j\rightarrow\infty}e^{in_{j}\alpha}$$ does not exist for almost all $$\alpha\in\mathbb{R}$$. The result is then applied to provide an alternative proof that $$L_{1}(\mathbb{R})$$ is not weakly sequentially compact.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.837

Árpád Bényi and Branko Ćurgus

We prove a generalization of the well-known Routh’s triangle theorem. As a consequence, we get a unification of the theorems of Ceva and Menelaus. A connection to Feynman’s triangle is also given.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.841

David Pan

We prove a maximum principle for high-order derivatives under initial conditions.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.846

Gerhard J. Woeginger

We survey results on equiangular *n*-vertex polygons with edge lengths in arithmetic progression. Such a polygon exists if and only if *n *has at least two distinct prime factors.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.849

Stephan Ramon Garcia

It has been observed many times, both in the MONTHLY and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: “Is the set of all quotients of Gaussian primes dense in the complex plane?”

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.851

Problems 11733-11739

Solutions 11593, 11594, 11599, 11601, 11602, 11606, 11609

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.854

*Calculus: Modeling and Application*, by David A. Smith and Lawrence C. Moore

Reviewed by Betty Mayfield

full text (pdf)JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.862