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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In this partly historical and partly research-oriented note, as part of our continuing examination of the unpublished mathematical diaries of Helmut Wielandt we display a page dated 1951.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^{(aa)}, 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