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American Mathematical Monthly -JANUARY 2007

January 2007

Attainable Patterns in Alien Tiles
Peter Maier and Werner Nickel
This article considers generalizations of a board game called Alien Tiles, which is available on the internet at The aim of this game is to attain a given pattern from an initial pattern. A criterion for the attainability of patterns is presented together with an algorithm for obtaining a given pattern from the initial pattern. From this a general formula for the proportion of attainable patterns in the set of all patterns is derived.


Nonlinear Oscillators at Our Fingertips: Descriptive Summary
Tanya Leise and Andrew Cohen,
Did you know that you have nonlinearity in your fingertips? We describe a simple experiment that almost anyone can do and that reveals the essential nonlinearity of the human neuromuscular system. We analyze a differential equations model of the oscillatory finger motions in the experiment as an introduction to the magical world of coupled nonlinear oscillators and phase transitions. This modeling exercise leads to reflections on the roles of mathematical modeling in advancing our understanding of coordinated movement and, more generally, of pattern formation in complex systems.


Solving the Quartic with a Pencil
David Auckly,
It is well known that there is a formula that expresses the roots of a general quartic as a function of the coefficients using only addition, subtraction, multiplication, division, and extraction of roots. This paper uses the derivation of this formula as a place to introduce and describe the geometry of Lefschetz papers.


The Method of Coefficients
Donatella Merlini, Renzo Sprugnoli, and Maria Cecilia Verri,,
The paper gives an account of the "method of coefficients" due to G. P. Egorychev. The method is used, often without any explicit reference, in the practice of formal power series and generating functions, both in combinatorics and in the analysis of algorithms. Here we show how we can start with a restricted series of general rules and proceed to obtain many results in these fields. Special emphasis is given to the rules of convolution, composition, and inversion, and to the evaluation of combinatorial sums.



When Does Convergence in the Mean Imply Uniform Convergence?
William F. Ford and James A. Pennline

The Koch Curve: A Geometric Proof
Sime Ungar

A Characterization of Ellipses
Dong-Soo Kim and Young Ho Kim,

A Class of Dirichlet Series Integrals
Jonathan M. Borwein

Problems and Solutions


A First Course in Modular Forms.
By Fred Diamond and Jerry Shurman
Reviewed by Fernando Q. Gouvêa