*This project uses a sampling problem to compute certain...*

**Read the latest issue of the Monthly online. (This requires MAA membership.) **

**A New Approach to Hilbert's Third Problem**

By: David Benko

dbenko2007@yahoo.com

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

jhjaroma@loyola.edu, kreddy@austincollege.edu

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

gabors@bgnet.bgsu.edu, mrizzo@bgnet.bgsu.edu

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

hderksen@umich.edu, c.eggermont@science.ru.nl, essen@math.ru.nl

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…, *m*^{2} 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

**Notes**

**The Cauchy Integral Theorem**

By: Peter Lax

lax@cims.nyu.edu

**Taming a Hydra of Singularities**

By: Folkmar Bornemann and Thomas Schmelzer

bornemann@ma.tum.de, thomas.schmelzer@balliol.ox.ac.uk

**Pick's Theorem via Minkowski's Theorem**

By: M. Ram Murty and Nithum Thain

murty@mast.queensu.ca

**A Characterization of Real C(K)-Spaces**

By: F. Albiac and N. J. Kalton

albiac@math.missouri.edu nigel@math.missouri.edu

**Problems and Solutions**

**Reviews**

Mathematics and Social Utopias in France: Olinda Rodrigues and His Times.

Edited by Simon Altmann and Eduardo L. Ortiz

Reviewed by: Warren P. Johnson

wpj002@bucknell.edu

Monthly Homepage

**Read the October issue of the Monthly online. (This requires MAA membership.) **