The April *Monthly* opens with a tribute to William A. Hawkins, the 2013 winner of the Gung and Hu Award for Distinguished Service to Mathematics. In addition to articles by award-winning *Monthly* authors Tom M. Apostol, Mamikon A. Mnatsakanian, and Francis Su, the April issue also includes an interesting analysis by Vadim Ponomarenko and his 2012 REU students from San Diego State University of how the Fundamental Theorem of Arithmetic can fail in subsets of the natural numbers. Many readers will be pleased to see our Problem Section return from its one-month hiatus, and all should stay tuned for our May issue, which will feature a tribute to Nobel Prize Winner Lloyd Shapley. —*Scott Chapman*

Vol. 120, No. 4, pp.295-379.

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Ann E. Watkins

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.04.295

Tom M. Apostol and Mamikon A. Mnatsakanian

Archimedes' mechanical balancing methods led him to stunning discoveries concerning the volume of a sphere, and of a cylindrical wedge. This paper introduces new balancing principles (different from those of Archimedes) including a balance-revolution principle and double equilibrium, that go much further. They yield a host of surprising relations involving both volumes and surface areas of circumsolids of revolution, as well as higher-dimensional spheres, cylindroids, spherical wedges, and cylindrical wedges. The concept of cylindroid, introduced here, is crucial for extending to higher dimensions Archimedes' classical relations on the sphere and cylinder. We also provide remarkable new results for centroids of hemispheres in $$n$$-space. Throughout the paper, we adhere to Archimedes' style of reducing properties of complicated objects to those of simpler objects.

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Matthew Jenssen, Daniel Montealegre, and Vadim Ponomarenko

A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism). For monoids in this class, the important factorization property of finite elasticity is characterized.

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Kenneth S. Williams

Jacobi's four squares theorem asserts that the number of representations of a positive integer $$n$$ as a sum of four squares is 8 times the sum of the positive divisors of $$n$$, which are not multiples of 4. A formula expressing an infinite product as an infinite sum is called a product-to-sum identity. The product-to-sum identities in a single complex variable $$q$$ from which Jacobi's four squares formula can be deduced by equating coefficients of $$q^{n}$$ (the "parents") are explored using some amazing identities of Ramanujan, and are shown to be unique in a certain sense, thereby justifying the title of this article. The same is done for Legendre's four triangular numbers theorem. Finally, a general uniqueness result is proved.

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Kathryn L. Nyman and Francis Edward Su

We show that Fan's 1952 lemma on labelled triangulations of the $$n$$-sphere with $$n+1$$ labels is equivalent to the Borsuk-Ulam theorem. Moreover, unlike other Borsuk-Ulam equivalents, we show that this lemma directly implies Sperner's Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk-Ulam theorem implies the Brouwer fixed-point theorem, or that the Lusternik-Schnirelmann-Borsuk theorem implies the KKM lemma.

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Habib Bin Muzaffar

A new proof is given of the classical formula that the sum of the reciprocals of the squares converges to $$\pi^{2}/6$$, using the technique of differentiation under the integral sign. Some interesting definite integrals are also evaluated.

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Steve Alpern and Robbert Fokkink

A network is to be illuminated by placing lights at the nodes, so that each edge is bright enough and the total intensity is minimized. A folk result says this can always be done using lamps that have half or full intensity. We give a new elementary proof.

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Orr Moshe Shalit

A proof for the maximum modulus principle (in the unit disc) is presented. This proof is unusual in that it is based on linear algebra.

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Olivier Bernardi

Consider a walk in the plane made of $$n$$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its starting point is $$1/(n+1)$$. We give an elementary proof of this result. We also prove the following generalization, valid for any probability distribution $$\mu$$ on the positive real numbers: If two walkers start at the same point and make, respectively, $$m$$ and $$n$$ independent steps with uniformly random directions and with lengths chosen according to $$\mu$$, then the probability that the first walker ends farther away than the second is $$m/(m+n)$$.

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Problems 11698-11704

Solutions 11562, 11563, 11566, 11567, 11568, 11570, 11573

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*Linear and Nonlinear Programming*. By David G. Luenberger and Yinyu Ye. Reviewed by Marie Snipes.

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