A relation between Catalan Numbers and World Series type problems

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**Christiaan Huygens and the Problem of the Hanging Chain**

John F. Bukowski

2-11

The seventeen-year-old Christiaan Huygens was the first to prove that a hanging chain did not take the form of the parabola, as was commonly thought in the early seventeenth century. We will examine Huygen's geometrical proof, and we will investigate the later history of the catenary.

**Hermit points on a Box**

Richard Hess, Charles, Grinstead, Marshall Grindstead, and Deborah Bergstrand

12-23

Suppose that we are given a rectangular box in 3-space. Given any two points on the surface of this box, we can define the surface distance between them to be the length of the shortest path between them on the surface of the box. This paper determines the pairs of points of maximum surface distance for all boxes. It is often the case that these pairs of points are not the diagonally opposite corners of the box that many expect.

**The Right Right Triangle on the Sphere **

William Dickinson and Mohammed Salmassi

24-33

The question explored here is whether having a 90° angle is the most fruitful analogue in spherical geometry to right triangles in Euclidean geometry. A strong case is made for the property of a triangle with one angle equal to the sum of the other two.

**Summing Up the Euler φ Function**

Paul Loomis, Michael Plytage, and John Polhill

34-42

The Euler φ function counts the number of positive integers less than and relatively prime to a positive integer *n*. Here we look at perfect totient numbers, number for which φ(n) + φ(φ(*n*)) + φ(φ(φ(*n*))) + ... + 1 = *n*.

**The Depletion Ratio**

Charles W. Groetsch

43-48

How fast does a tank drain? Of course this depends on the shape of the tank and is governed by a physical principle known as Torricelli's law. This note investigates some connections between tank shape and a mathematical function related to the time required for the tank to drain completely. The techniques employed provide some interesting illustrations of the fundamental theorem of calculus, l'Hôpital's rule and the mean value theorem for integrals.

**CAPSULES**

**Pairs of Equal Surface Functions**

Daniel Cass and Gerald Wildenberg

51-54

The well known fact that the surface areas of a sphere slice and a cylinder of the same thickness and radius is generalized by giving a procedure that generates pairs of functions such that on any subinterval the surfaces of revolution of the pair have equal areas. A variety of examples are given and a proof of the impossibility of polynomial examples is also provided.

**A Tricky Linear Algebra Example**

David Sprows

54-56

In this article a classroom "trick" involving *n*-by-*n* square arrangements of natural numbers is used to motivate a discussion of a special class of matrices. In particular, a basis is obtained for those *n*-by-*n* matrices with the property that if *n* entries are selected from the matrix so that no two values are in the same row or the same column, then the sum of these *n* entries will always be the same.

**A Quick Change of Base Algorithm for Fractions**

Michael Weiner and Juan Gil

56-59

This note is on the digital (floating-point) representation in various arithmetic bases of the reciprocal of an integer, 1/*m*. We give a surprisingly simple algorithm to change the representation of 1/*m *in base b to its representation in base *b*+*mt* for any integer *t*."

**A Waiting-Time Surprise**

Richard Parris

59-63

Consider the sum of n random real numbers, uniformly distributed in the unit interval. Although the expected value of this sum is *n*/2, the value of n for which this sum first exceeds a given target value t is expected to be more than 2*t*, by an amount that is asymptotically constant.

**The Pearson and Cauchy-Schwarz Inequalities**

David Rose

In this note, the Cauchy-Schwarz inequality is derived from Pearson's inequality. It follows that the two are equivalent.

**Columns**

**Fallacies, Flaws, and Flimflam**

Ed Barbeau

**Problems and Solutions**

Jim Bruening and Shing So

**Media Highlights**

Warren Page

**Pólya Award Winners**