Based on the notion of "arithmetic triangles," arithmetic...

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Reuben Hersh

"Over-representation" of Jews in the world of mathematics is a common observation, but rarely discussed in print. In response to a new book on the subject by Ioan James, this paper recounts the author's own interaction with Jewish over-representation at universities in the U.S., and also gives information about Hungary and Italy. The contrast with the drastic under-representation of Jews in U.S. mathematics in the 1920's and 30's is discussed and an explanation is given of how under-representation became over-representation and how this over-representation presently is fading away.

**Dogs Don't Need Calculus**

Mike Bolt

Many optimization problems can be solved without resorting to calculus. This article develops a new variational method for optimization that relies on inequalities. The method is illustrated by four examples, the last of which provides a completely algebraic solution to the problem of minimizing the time it takes a dog to retrieve a thrown ball, demonstrating that dogs don't need calculus.

**The Helen of Geometry**

John Martin

The cycloid has been called the Helen of Geometry, not only because of its beautiful properties but also because of the quarrels it provoked between famous mathematicians of the 17th century. This article surveys the history of the cycloid and its importance in the development of the calculus.

**Biangular Coordinates Redux: Discovering a New Kind of Geometry**

Brian Winkel, Michael Naylor

Biangular coordinates specify a point on the plane by two angles giving the intersection of two rays emanating from two fixed poles. This is a dual of Cartesian coordinates wherein a point on the plane is described by two distances. Biangular coordinates, first written about in 1803 in France, were subsequently studied in Britain at the end of the 19th century. Contemporary students will enjoy plotting biangular relations and discovering patterns in families of relations. The mathematical shapes which are given by biangular relations include standard shapes like lines, circles, hyperbolae and also unusual curves like.

**An Upper Bound for the Expected Range of a Random Sample**

James Marengo, Manuel Lopez

We consider the expected range of a random sample of points chosen from the interval [0, 1] according to some probability distribution. We then use the notion of convexity to derive an upper bound for this expected range which is valid for all possible choices of this distribution. Finally we show that there is only one distribution for which this bound is achieved.

**The Hardest Straight-in Pool Shot**

Rick Mabry

When playing pool or billiards, a player often has the opportunity to make a "straight-in" shot, that is, one in which the cue ball, the object ball, and the target (e.g., a pocket) are collinear. With the distance from the cue ball to the target assumed fixed, the relative difficulty is here explored of shots taken at varying positions of the object ball between the cue ball and target. What position of the object ball makes the shot the most difficult? This is the question addressed in the article. Proofs of the results, ranging from easy to challenging, are left as exercises and also posted on the CMJ web site.

**Computing Definite Integrals Using the Definition**

James L. Hartman

Students in a first semester calculus course are rarely asked to compute any integrals using only the definition of the Riemann integral. This article explains how to compute some definite integrals using only the definition and no appeal to auxiliary theorems.

**Waiting to Turn Left?**

Jody Sorenson, Elyn K. Rykken, Maureen T. Carroll

This article examines the rule used by the state of Pennsylvania to determine when the installation of a left-turn signal is justified. In creating a mathematical model, we encounter a natural application of the fundamental theorem of calculus.

**Series involving iterated logarithms **

J. Marshall Ash

The boundary between convergent and divergent series is systematically explored through sums of iterated logarithms.

**Sums of Integer Powers via the Stolz-Cesàro Theorem **

Sidney H. Kung

The Stoltz-Cesàro Theorem, a discrete version of l'Hôpital's rule, is applied to the summation of integer powers.

**Short Division of Polynomials **

Li Zhou

The familiar process of synthetic division is extended to much more complicated divisors and turns out, surprisingly, not to be as difficult as one might imagine.

**MOODY’S MEGA MATH CHALLENGE: A MODELING COMPETITION**

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