American Mathematical Monthly October 2004

MONTHLY, October 2004

Lost in a Forest
by Steven R. Finch and John E. Wetzel
Steven.Finch@inria.fr, j-wetzel@uiuc.edu
Fifty years ago, Richard Bellman posed an interesting search problem that can be phrased as follows: A hiker is lost in a forest whose shape and dimensions (but not its orientation) are precisely known to him. What is the best path for him to follow to escape from the forest? Construing "best" as meaning "shortest," we survey what is known for regions of various shapes, we clarify the relationship with Leo Moser's well-known "worm" problem, and we consider some related questions.

Potter, Wielandt, and Drazin on the Matrix Equation AB = ∞ BA: New Answers to Old Questions
by Olga Holtz, Volker Mehrmann, and Hans Schneider
holtz@math.Berkeley.edu, hans@math.wisc.edu, mehrmann@math.TU-Berlin.de
In this partly historical and partly research-oriented note, as part of our continuing examination of the unpublished mathematical diaries of Helmut Wielandt we display a page dated 1951. There he gives a new proof of a theorem due to H. S. A. Potter on the matrix equation AB = ∞BA, which is related to the q-binomial theorem, and asks some further questions, which we mostly answer. We also describe results by M. P. Drazin and others on this equation.

Iterated Exponential
by Joel Anderson
anderson@math.psu.edu
In this article we discuss the following fascinating problem. Suppose a is a positive number and consider the sequence a, a^a, a^{(a^a)}, K . For which values of a does this sequence converge? This problem is remarkable both for its unexpected answer and the different threads that interweave in the development of its solution. We present a solution and provide some historical context.

The Sixty-Fourth William Lowell Putnam Mathematical Competition
by Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson

The First Sixty-Five Years of the Putnam Competition
by Joseph A. Gallian
jgallian@d.umn.edu
We survey the results of the Putnam Competition from its inception in 1938 through 2003. We include tables that provide the number of participants in each contest, the number of times each team has placed in top five, the number of Putnam Fellows each school has had, and the top five scores and medians for all competitions between 1967-2003. There is a section that identifies individuals who have excelled in the competitions and another section that identifies distinguished mathematicians and scientists who have performed well in the competitions.

Notes

A Simple Proof of the Hook Length Formula
by Kenneth Glass and Chi-Keung Ng
k.glass@qub.ac.uk, ckng@nankai.edu.cn

Playing Catchup with Iterated Exponentials
by R. L. Devaney, K. Josic, M. Moreno Rocha, P. Seal, Y. Shapiro, and A. T. Frumosu
bob@math.bu.edu, josic@math.uh.edu, mmoren02@tufts.edu, pseal@math.bu.edu, yshapiro@math.mit.edu, tais@math.bu.edu

An Intuitive Derivation of Heron’s Formula
by Daniel A. Klain
dklain@cs.uml.edu

A New Proof of Darboux’s Theorem
by Lars Olsen
lo@st-and.ac.uk

Evolution ofÂ…
Global Geometry
by M. F. Atiyah

Problems and Solutions

Reviews

A Companion to Analysis. A Second First and First Second Course in Analysis.
by T. W. Kouml;rner
Reviewed by Steven G. Krantz
sk@math.wustl.edu

Mathematics Elsewhere: An Exploration of Ideas Across Cultures.
by Marcia Ascher
Reviewed by Marion D. Cohen
mathwoman199436@aol.com