American Mathematical Monthly - November 2003

NOVEMBER 2003

The Mylar Balloon Revisited
by Ivaïlo M. Mladenov and John Oprea
mladenov@bgcict.acad.bg, oprea@math.csuohio.edu
An explicit parametrization in terms of elliptic functions for the Mylar balloon is found, which then is used to calculate various geometric quantities as well as to study various kinds of geodesics on this surface.

Tetrahedron ABCD of Width 1 with Minimum AB+BC+CD
by K. S. Sarkaria
sarkaria_2000@yahoo.com
A woodborer, freshly hatched at a random point within a slab of timber of unit thickness, wants to tunnel its way out by making at most two changes in its randomly chosen initial direction, in such a way that the worst case distance is minimized. What strategy does it adopt? We present the solution of this problem, also partial results about the analogous higher-dimensional problems.

Bell’s Theorem and the Demise of Local Reality
by Stephen McAdam
mcadam@math.utexas.edu
Einstein believed that the behavior of entangled particles could be explained only by the existence of local variables that determined their behavior. John Bell devised a clever argument showing that the reverse was true: the behavior of entangled particles proved that local variables cannot exist. We present an explanation of Bell's insight that stresses a mathematical point of view.

The Mathematics of Survival: From Antiquity to the Playground
by Chris Groër
cgroer@math.uga.edu
Nearly two thousand years ago, the historian Flavius Josephus allegedly saved his own life by quickly solving a combinatorics problem now named in his honor. The "Josephus Problem" involves arranging n people around a circle and then "counting out" every qth person until only one remains. In this article, we consider a variation on this classic problem where our n people are instead arranged in a line and then systematically removed via a back and forth procedure---sort of like the old playground rhyme "One Potato, Two Potato." We present a solution to this problem in a particular case and explore several surprising properties of this variation. We show a relationship between this new problem and fractal-like images and explore some properties of an unusual function related to these images. After presenting the reader with anecdotal proof that the number thirteen is truly unlucky, we conclude with a historical assessment of the somewhat dubious association between Josephus and mathematics

Problems and Solutions

Notes

Two Proofs of Graves’s Theorem
by Kamal Poorrezaei
kamalporeza@mehr.sharif.edu

A Characterization of the Unit Sphere
by Jeongseon Baek, Dong-Soo Kim, and Young Ho Kim
jsbaek@chonnam.chonnam.ac.kr, dosokim@chonnam.chonnam.ac.kr, yhkim@knu.ac.kr

An Elementary Proof of Lebesgue’s Differentiation Theorem
by Michael W. Botsko
mike.botsko@email.stvincent.edu

Almost Every Number Has a Continuum of ?-Expansions
by Nikita Sidorov
Nikita.A.Sidorov@umist.ac.uk

Reviews

A New Kind of Science.
by Stephen Wolfram
Reviewed by Rudy Rucker
rucker@cs.sjsu.edu

Ramanujan: Essays and Surveys.
Edited by Bruce C. Berndt and Robert A. Rankin
Reviewed by Krishnaswami Alladi
alladi@math.ufl.edu

Telegraphic Reviews