March 2005 Contents
Saari, with no Apologies
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
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
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.
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
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
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
Michael Kinyon, editor
Leapfrogs: The Mathematical Details
Matt Wyneken, Steve Althoen, and John Berry
This note analyzes the following game: There are 2n+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
The connection between the integrals of x-1 and xp 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
This proof uses integrals and is more elementary than many other proofs of these monotonicity results.
Spraying a Wall with a Garden Hose
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
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
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 nth Prime
John H. Jaroma
The author uses induction and a result of Nagura that, for n ≥ 25, there is a prime between n and 6n/5 to show that the nth prime is less than (1.2)n for n ≥ 25.
Problems and Solutions