A relation between Catalan Numbers and World Series type problems

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ARTICLES

**Designing a Table Both Swinging and Stable**

Greg N. Frederickson

258

Howard Eves captured people's imagination by proposing a hinged table whose top could swing around to be either an equilateral triangle or a square. But is Eves's mathematically motivated design actually good furniture design? This paper identifies potential design flaws in that hinged table and proposes an intriguing new design. Both the original table and the new table are analyzed for factors contributing to stability, such as the size of pieces and the placement of legs.

**Centaurs: Here, There, Everywhere!**

Dimitri Dzlabenko and Oleg Ivril

267

Motivated by centaurs jumping around a circular stadium, we derive Kronecker's Approximation Theorem, which in turn provides elementary solutions to difficult problems in the theory of Diophantine approximations.

**How to Measure Angles with a Ruler**

Travis Kowalski

273

This article presents a neat way to approximate the measure of an angle using a ruler and discusses the accuracy of various versions of the method.

**The Truck Driver's Straw Problem and Cantor Sets**

Kevin Iga

280

A colleague was moving, and someone on the professional moving crew, upon hearing she was a mathematician, asked what happens when you repeatedly transfer water back and forth between two classes using a straw. The question is simple to solve if you alternate which glass you transfer from and to, but if more general patters are allowed, some surprises arise. Depending on the size of the straw, the set of achievable limit points may be an interval, or a Cantor set.

**A Helical Stairway Project**

Tom Farmer

291

We answer a geometric question that was raised by the carpenter in charge of erecting helical stairs in a 10-story hospital. The explanation involves the equations of lines, planes, and helices in three-dimensional space. A brief version of the question is this: If A and B are points on a cylinder and the line segment AB is projected radially onto the cylinder, then how non-helical is this curve?.

**CAPSULES**

**Graph Theory and Surface Reconstruction**

Darren A. Narayan

301

Traditional courses seldom offer concrete examples of real world applications of higher mathematics. As a result, students graduate asking the question, "What can I do with a mathematics degree besides teach?" This capsule will help students to answer this question: elementary tools from graph theory are used to solve a problem that arose at Microsoft Research.

**Fetching Water with Least Residues**

Herb Bailey

In a classic pouring problem, you are given two jugs with capacities *m* and *n* pints, where m and n are relatively prime integers. Given an unlimited supply of water you must obtain exactly *p* pints, where *p* is an integer, 0 ≤ *p* ≤ *m* + *n* . In this article we use some properties of least residues to show that there are two distinct pouring sequences to achieve this result. The more efficient sequence can be determined by solving a linear congruence.