Vol. 120, No. 9, pp.771864.
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ARTICLES
Christiane's Hair
Jacques Lévy Véhel and Franklin Mendivil
We explore the geometric and measuretheoretic properties of a set built by stacking central Cantor sets with continuously varying scaling factors. By using selfsimilarity, 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 boxcounting 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 firstyear calculus students.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.09.771
What to Expect in a Game of Memory
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 $$(32\ln2)n+\frac{7}{8}2\ln2\approx1.61n$$

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 $$2^{2n}/\left(\begin{array}{c}2n\\n\end{array}\right) \sim\sqrt{\pi n}$$.
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Mixing Problems with Many Tanks
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.
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Reciprocal Sums as a Knowledge Metric: Theory, Computation, and Perfect Numbers
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.
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NOTES
Certain Inequalities Associated with the Divisor Function
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 $$s,t,u,v\in\mathbb{Z}$$.
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On the Nonexistence of Certain Limits for the Complex Exponential
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.
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A Generalization of Routh's Triangle Theorem
Árpád Bényi and Branko Ćurgus
We prove a generalization of the wellknown 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.
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A Maximum Principle for HighOrder Derivatives
David Pan
We prove a maximum principle for highorder derivatives under initial conditions.
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Nothing New about Equiangular Polygons
Gerhard J. Woeginger
We survey results on equiangular nvertex polygons with edge lengths in arithmetic progression. Such a polygon exists if and only if n has at least two distinct prime factors.
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Quotients of Gaussian Primes
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?”
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PROBLEMS AND SOLUTIONS
Problems 1173311739
Solutions 11593, 11594, 11599, 11601, 11602, 11606, 11609
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REVIEWS
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