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March 2005 Contents

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ARTICLES

**Saari, with no Apologies**

*Deanna Haunsperger*

90-100

Proselytizing the beauty, power, and ubiquity of mathematics is not an obligation for Donald Saari — it's his passion. This article is the result of an interview with him at MathFest one year.

**Breaking the Holiday Inn Priority Club: CAPTCHA**

*Edward Aboufadel, Julia Olsen, and Jesse Windle*

101-108

CAPTCHAs are images used on the internet by companies that want to block computer programs from their site. This article presents a method for solving some of these puzzles.

**Another Broken Symmetry**

*C. W. Groetsch*

109-113

Resistance destroys symmetry. In this note, a graphical exploration serves as a guide to a rigorous elementary proof of a specific asymmetry in the trajectory of a point projectile in a medium offering linear resistance.

**Phoebe Floats!**

*Ezra Brown*

114-122

The narrative in this article begins with a simple remark about a heavenly body to a class, and then winds its way through a lot of mathematical topics.

**The Golden Ratio — A Contrary Viewpoint**

*Clement Falbo*

123-134

Many assertions about the occurrence of the golden ration Φ in art, architecture, and nature have been shown to be false, unsupported, or misleading. For instance, we show that the spirals found in sea shells, in particular the *Nautilus pompilius*, are not in the shape of the golden ratio, as is often claimed. Some of the most interesting properties of Φ turn out to be shared by entire families of numbers. On the other hand, Φ is not without interest: it is often the simplest number that has a given property, and we look at examples of this.

**Taking a Whipper —The Fall-Factor Concept in Rock Climbing**

*Dan Curtis*

135-140

Most serious rock climbers are familiar with a counter-intuitive fact about their sport: The force experienced by a falling climber due to the rope as it arrests his fall does not depend simply on the length of the fall, but rather on a ratio called the fall-factor. This article explains, using elementary physics and simple differential equations, why this is so. Implications for techniques for climbing and for the design and testing of climbing gear are discussed.

**Fallacies, Flaws, and Flimflam**

*Ed Barbeau, editor*

141-143

**CAPSULES**

*Michael Kinyon, editor*

144-159

**Leapfrogs: The Mathematical Details**

*Matt Wyneken, Steve Althoen, and John Berry*

144-146

This note analyzes the following game: There are 2*n*+1 chairs in a row, with girls occupying the *n* on the left and boys the *n* on the right. The object is to get the boys on the left and the girls on the right in the fewest moves, where a move consists either of sliding to an adjacent open seat or jumping over one person to the open seat.

**Approaching ln ***x*

*James V. Peters*

146-147

The connection between the integrals of *x*^{-1 }and *x*^{p} for the other values of *p* are explored using a graphing calculator.

**An Elementary Proof of the Monotonicity of (1 + 1/***n)*^{n} and (1 + 1/*n)*^{n}+1

*Duane W. DeTemple*

147-149

This proof uses integrals and is more elementary than many other proofs of these monotonicity results.

**Spraying a Wall with a Garden Hose**

*James Alexander*

149-152

This note discusses the motion of the "splash point" where water from a rotating sprayer hits a wall.

**Snapshots of a Rotating Water Stream**

*Steven L. Siegel*

152-154

Here, the author looks at the curve of the stream of water as a sprinkler rotates.

**The Computation of Derivatives of Trigonometric Functions via the Fundamental Theorem of Calculus**

*Horst Martini and Walter Wenzel*

154-158

The derivatives of the sine and cosine functions are found in a non-traditional way, using the fundamental theorem of calculus.

**An Upper Bound on the ***n*^{th} Prime

*John H. Jaroma*

158-159

The author uses induction and a result of Nagura that, for *n* ≥ 25, there is a prime between *n* and 6*n*/5 to show that the *n*^{th} prime is less than (1.2)^{n }for *n* ≥ 25.

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Problems and Solutions

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Media Highlights