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American Mathematical Monthly -October 2007

October 2007

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A New Approach to Hilbert's Third Problem
By: David Benko
Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? This is Hilbert's third problem, which was solved by Max Dehn. We give a simple new solution that has been overlooked for a century.

Classical and Alternative Approaches to the Mersenne and Fermat Numbers
By: John H. Jaroma and Kamaliya N. Reddy,
The objective of this paper is to illustrate nine propositions that pertain to either the Mersenne or Fermat numbers. Each of the properties is demonstrated in a manner that compares a standard textbook-type proof with an approach that uses the theory of the Lucas sequences. A secondary goal of this paper is to show how the Mersenne numbers may be used as the foundation for testing the primality of certain Sophie Germain primes. The result had been addressed by Euler in 1750 and was later proved by both Lagrange in 1775 and Lucas in 1878.

The Uncertainty Principle of Game Theory
By: Gábor J. Székely and Maria L. Rizzo,
In 1927-1928 two fundamental papers changed our view on nature and life. Heisenberg's uncertainty principle (1927) led to a probabilistic description of nature (unlike Newton's deterministic laws), and von Neumann's minimax theorem on zero-sum games (1928) led to a probabilistic description of optimal decisions in life (in games, economics and other social sciences) even if the rules of the game do not involve chance. Heisenberg's uncertainty principle gives an explicit lower bound for the randomness in terms of the commutator of two linear operators that describe "observables.'' In this paper we give an explicit lower bound for the randomness of optimal decisions in terms of the commutator of two nonlinear operators: minimum and maximum.

Multimagic Squares
By: Harm Derksen, Christian Eggermont, and Arno van den Essen,,
More than one hundred years ago the first 2-multimagic square appeared in print. This paper establishes for the first time the existence of n-multimagic squares for all n. Multimagic squares are a very special class of magic squares. A magic square is a square of numbers for which each row, column, and (main) diagonal sums to the same constant. An n-multimagic square of order m is an m-by-m magic square consisting of the consecutive integers 1, 2, 3, Â…, m2 that remains a magic square if each of its entries is raised to the power p for each p in the range from 1 to n.

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


The Cauchy Integral Theorem
By: Peter Lax

Taming a Hydra of Singularities
By: Folkmar Bornemann and Thomas Schmelzer,

Pick's Theorem via Minkowski's Theorem
By: M. Ram Murty and Nithum Thain

A Characterization of Real C(K)-Spaces
By: F. Albiac and N. J. Kalton

Problems and Solutions


Mathematics and Social Utopias in France: Olinda Rodrigues and His Times.
Edited by Simon Altmann and Eduardo L. Ortiz
Reviewed by: Warren P. Johnson

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