(2021). Letter from the Editor. The American Mathematical Monthly: Vol. 128, No. 1, pp. 3-3.

Abstract Numerous versions of the question “what is the shortest object containing all permutations of a given length?” have been asked over the past fifty years: by Karp (via Knuth) in 1972; by Chung, Diaconis, and Graham in 1992; by Ashlock and Tillotson in 1993; and by Arratia in 1999. The large variety of questions of this form, which have previously been considered in isolation, stands in stark

Abstract The geometry of the deltoid curve gives rise to a self-map of C 2 that is expressed in coordinates by f ( x , y ) = ( y 2 − 2 x , x 2 − 2 y ) . This is one in a family of maps that generalize Chebyshev polynomials to several variables. We use this example to illustrate two important objects in complex dynamics: the Julia set and the iterated monodromy group.

(2021). Sums of Proper Powers. The American Mathematical Monthly: Vol. 128, No. 1, pp. 40-40.

Abstract We add another brick to the large building comprising proofs of Pick’s theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick’s theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.

Abstract The mean value theorem of calculus states that, given a differentiable function f on an interval [ a , b ] , there exists at least one mean value abscissa c such that the slope of the tangent line at ( c , f ( c ) ) is equal to the slope of the secant line through ( a , f ( a ) ) and ( b , f ( b ) ) . In this article, we study how the choices of c relate to varying the right endpoint b. In

(2021). 100 Years Ago This Month in The American Mathematical Monthly. The American Mathematical Monthly: Vol. 128, No. 1, pp. 61-61.

Abstract The story of the question “must commuting maps of the unit interval have a common fixed point” is used to illustrate strategies that advance mathematical research.

(2021). Perfect Metric Spaces of Increasingly Large Cardinality. The American Mathematical Monthly: Vol. 128, No. 1, pp. 69-69.

Abstract A probabilistic proof of Girard’s angle excess formula for the area of a spherical triangle emerges from the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of 2-dimensional orthogonal projections: a 2-dimensional convex cone with one vertex, a 2-dimensional half-plane (an outcome with probability zero), and a 2-dimensional plane

Abstract In this note, we prove a graph inequality based on the sizes of the common neighborhoods. We also characterize the extremal graphs that achieve equality. The result was first discovered as a consequence of Foster’s classical theorem about electrical networks. We also present a short combinatorial proof that was inspired by a similar inequality related to the Turán’s celebrated theorem.

(2021). Fibonacci’s Bunny-pocalypse! The American Mathematical Monthly: Vol. 128, No. 1, pp. 78-78.

Abstract A fundamental result in cake cutting states that for any number of players with arbitrary preferences over a cake, there exists a division of the cake such that every player receives a single contiguous piece and no player is left envious. We generalize this result by showing that it is possible to partition the players into groups of any desired sizes and divide the cake among the groups

Abstract We prove that a metric space is separable if and only if there exists a Borel measure such that the measure of open balls is positive and finite.

(2021). What’s Special about the Perfect Number 6? The American Mathematical Monthly: Vol. 128, No. 1, pp. 87-87.

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by May 31, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions. An

Abstract We give an elementary treatment of the curvature of surfaces of revolution in the language of vector calculus, using differentials rather than an explicit parameterization. We illustrate some basic features of curvature using embedding diagrams, and then use such a diagram to analyze the geometry of the Schwarzschild black hole.

Abstract We consider the following game played in the Euclidean plane: There is any countable set of unit speed lions and one fast man who can run with speed 1 + ε for some value ε > 0 . Can the man survive? We answer the question in the affirmative for any ε > 0 .

Abstract A sphere that is tangent to all four face-planes (i.e., the affine hulls of the faces) of a tetrahedron is called a tangent sphere of the tetrahedron. Two tangent spheres are called neighboring if exactly one face-plane separates them. Grace’s theorem states that for a pair of neighboring tangent spheres S and T of a tetrahedron there is a unique sphere Θ such that (1) Θ passes through the

Abstract Motivated by strong versions of Green’s theorem, we give an example of a differentiable vector field for which the curl is continuous but not all the partial derivatives are continuous.

Abstract In his recent note in this Monthly, Glasser derived a formula transforming a double integral of Beukers’s type to a single integral. The purpose of this article is to generalize his result to multiple integrals and to present a short new proof of this relation.

(2020). Two Uniqueness Theorems at Work. The American Mathematical Monthly: Vol. 127, No. 10, pp. 927-927.

Abstract A decomposition of an integer n as a sum of several fractions is called faithful if no positive integer less than n can be obtained by replacing some of the numerators with smaller nonnegative integers. In this note, we prove the existence of faithful decompositions of all positive integers n, and determine the minimum number of fractions of the faithful decomposition. Here we also illustrate

Abstract We consider an old problem of partitioning a triangle into three similar triangles and three parallelograms by lines parallel to the sides of the triangle through a given interior point M. In this note, we show that the locus of those points M for which the sum of the triangle areas is equal to the sum of the parallelogram areas is precisely the Steiner inellipse of the triangle.

Abstract We compare the number of finite groups of order n with the number of finite rings of order n, with some surprising results.

(2020). Another Proof of the Infinitude of Primes. The American Mathematical Monthly: Vol. 127, No. 10, pp. 938-938.

Abstract The Smarandache function of a positive integer n, denoted by S(n), is defined to be the smallest positive integer j such that n divides the factorial j ! . In this note, we prove that for any fixed number k > 1, the inequality n k < S ( n ) ! holds for almost all positive integers n. This confirms Sondow’s conjecture which asserts that the inequality n 2 < S ( n ) ! holds for almost all positive

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by April 30, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2020). An Introduction to Undergraduate Research in Computational and Mathematical Biology Edited by Hannah Callender Highlander, Alex Capaldi, and Carrie Diaz Eaton; and A Project-Based Guide to Undergraduate Research in Mathematics. Edited by Pamela Harris, Erik Insko, and Aaron Wootton. The American Mathematical Monthly: Vol. 127, No. 10, pp. 953-957.

(2020). EDITOR’S ENDNOTES. The American Mathematical Monthly: Vol. 127, No. 10, pp. 958-960.

(2020). Monthly Referees for 2020. The American Mathematical Monthly: Vol. 127, No. 10, pp. i-iii.

Abstract Suppose the only thing we know about a matrix is which of its entries are zero. What can we say about its rank? We develop a framework for applying matroid theory to this question. One consequence is a generalization of the question to the setting of matroids; over an infinite field, we recover the original question by considering only matroids representable over that field. This framework

Abstract We give an elementary cut-and-paste proof of results of Bott and Shchepin: the third finite subset space of the circle is homeomorphic to the 3-sphere and the inclusion of the first finite subset space in it is a trefoil knot. Moreover, we give an explicit simplicial decomposition of the space which is isomorphic to the Barnette sphere.

(2020). Counting Pairs to Find a Finite Field. The American Mathematical Monthly: Vol. 127, No. 9, pp. 806-806.

Abstract Do examples as in the title exist? It depends on how the term Riemann sum is understood. For the standard, left, or right Riemann sums such examples do not exist. However, as we will see, they do exist for the lower and upper Riemann sums. Nevertheless, there are only a few examples of such pairs and they have a very simple structure. In this article, we describe all such pairs among Riemann

Abstract In 1987, R. B. Paris introduced an analytic function g to estimate the rate of convergence of nested square root radicals to the golden ratio. The function g is nonentire and, perhaps, cannot be expressed in terms of some standard known functions. We show that the inverse f=g−1 is an entire function satisfying the Poincaré equality. We provide some explicit expansions of f based on exact formulas

Abstract It is easy to see that the number of automorphisms of a finite group of order n cannot exceed n!. Ledermann and Neumann proved conversely that the order of a finite group G can be bounded by a function depending only on the number of automorphisms of G. While their proof is long and complicated, the result was rediscovered by Nagrebeckiĭ 14 years later. In this article, we give a short and

(2020). A Quick Proof of the Order-Extension Principle. The American Mathematical Monthly: Vol. 127, No. 9, pp. 835-835.

Abstract Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the vertex set of Penrose tilings and other regular model sets, which serve as a mathematical model for quasicrystals. In this note, we show that the point

Abstract We give a short construction of the exponential function using only the simplest properties of convergent sequences and two well-known elementary inequalities. It could be used at an early stage in introductory analysis courses, before studying convergent series, continuity, or differentiability.

Abstract In this note, we show that each positive rational number can be written as φ(m2)/φ(n2), where φ is Euler’s totient function and m and n are positive integers.

(2020). Bases for Second Order Linear ODEs. The American Mathematical Monthly: Vol. 127, No. 9, pp. 849-849.

Abstract In this note, we present a new short proof of the classical Hermite–Hadamard inequalities.

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by March 31, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2020). Mathematics for Human Flourishing. The American Mathematical Monthly: Vol. 127, No. 9, pp. 861-864.

(2020). The Eightieth William Lowell Putnam Mathematical Competition. The American Mathematical Monthly: Vol. 127, No. 8, pp. 675-687.

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse generalizing a classic problem about circles. We give a brief history of the circle problem an account of Price’s ellipse proof and a reorganized proof with some new ideas designed to situate the result within a dense web of connections to classical mathematics. It is inspired

In Conway’s game, Sylver Coinage, the set of legal plays forms the complement of a numerical semigroup after a finite number of turns. Our goal is to show how the tools and techniques of numerical semigroups can be brought to bear on questions related to Sylver Coinage. We begin by formally connecting the definitions and concepts related to the game of Sylver Coinage with those of numerical semigroups

(2020). Wald’s Identity and Geometric Expectation. The American Mathematical Monthly: Vol. 127, No. 8, pp. 716-716.

Zero divided by zero is arguably the single most important concept underlying calculus. For functions of more than one variable, methods of proof for indeterminate limits are not as familiar as for functions of a single variable. We present a l’Hôpital’s rule that provides a way to simplify and resolve a wide variety of zero-over-zero limits in terms of quotients of their derivatives.

We solve a very classical problem: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter, or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles, and (an) edge length, the remaining parts being derivable by symmetry or

We give a simple proof of the Siebeck–Marden theorem. We use only basic facts of geometry and complex numbers.

We provide an elementary derivation of the Green function for Poisson’s equation with Neumann boundary data on balls of arbitrary dimension. Surprisingly, until very recently this Green function was only known in dimensions up to three, and an explicit construction (even in low dimensions) on the level of an undergraduate PDE class was lacking. This changes if one derives the Green function for Poisson’s

(2020). Null Subsets of All Sizes Inside Vitali Sets. The American Mathematical Monthly: Vol. 127, No. 8, pp. 743-743.

A numerical semigroup S is an additive subsemigroup of the nonnegative integers, containing zero, with finite complement. Its multiplicity m is its smallest nonzero element. The Apéry set of S is the set of elements Ap(S)={n∈S:n−m∉S}. Fixing a numerical semigroup, we ask how many elements of its Apéry set have nonunique factorization and define several new invariants.

Using the group consisting of the eight Möbius transformations x, – x, 1/x,−1/x, (x−1)/(x+1),(x+1)/(1−x), (x+1)/(x−1), and (1−x)/(x+1), we present an enumerative proof of the classical result for when the element 2 is a quadratic residue in the finite field Fq .

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by February 28, 2021, via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions

(2020). A Formula for the Stirling Numbers of the Second Kind. The American Mathematical Monthly: Vol. 127, No. 8, pp. 762-762.

(2020). Discrete Morse Theory by Nicholas Scoville. The American Mathematical Monthly: Vol. 127, No. 8, pp. 763-768.

We say that a simple, closed curve γ in the plane has bounded convex curvature if for every point x on γ, there is an open unit disk Ux and εx>0 such that x∈∂Ux and Bεx(x)∩Ux⊂Int γ. We prove that the interior of every curve of bounded convex curvature contains an open unit disk.