American Mathematical Monthly - January 2003

JANUARY 2003

Hilbert’s Twenty-Fourth Problem
by Rüdiger Thiele
thieler@medizin.uni-leipzig.de

Hilbert and his twenty-three problems have become proverbial. However, there is a twenty-fourth problem that Hilbert included neither in his Paris address nor in any printed version thereof. I found this canceled twenty-fourth problem in Hilbert’s unpublished mathematical notebooks, in which he wrote down mathematical and philosophical remarks, questions, and problems. To the best of my knowledge this problem has remained hidden for a century and is discussed in this paper for the first time in print. The canceled problem belongs to the realm of foundations of mathematics: in a nutshell, it asks for the simplest proof of any mathematical problem. The paper goes into some detail on the canceled problem and attempts to place it in historical, logical, mathematical, and philosophical context.

Candy Sharing
by Glen Iba and James Tanton
mathcircle@earthlink.net

A candy game, first described in the 1962 Beijing Mathematical Olympiad, has recently resurfaced as a stimulating mathematical activity for students of all ages to explore. We examine generalizations of the game and find conditions that ensure the game terminates after finitely many iterations of play.

Pick’s Theorem—What’s the Big Deal?
by John E. McCarthy
mccarthy@math.wustl.edu

In 1916, Georg Pick solved the problem of determining whether there was a holomorphic function mapping the unit disk of the complex plane to itself that also had prescribed values at certain points. Yet dozens of articles are still written every year about Pick’s interpolation problem and its generalizations. Why is it still interesting? We explain how certain problems in automatic control theory lead to Pick problems, and describe how operator theory can be used to solve them.

Problems and Solutions

Notes

Fermat Primes and Heron Triangles with Prime Power Sides
by Florian Luca
fluca@matmor.unam.mx

An Extension of the Results of Servais and Cramer on Odd Perfect and Odd Multiply Perfect Numbers
by Jennifer T. Betcher and John H. Jaroma
jjaroma@gw.lssu.edu

A Generalized Parallelogram Law
by Alan Nash
anash@math.ucsd.edu

Rearrangement of a Conditionally Convergent Series
by Uri Elias
elias@tx.technion.ac.il

Reviews

Geometry: Euclid and Beyond
by Robin Hartshorne

Reviewed by Victor Pambuccian
pamb@math.west.asu.edu

Rockefeller and the Internationalization of Mathematics Between the Two World Wars: Documents and Studies for the Social History of Mathematics in the 20th Century.
by Reinhard Siegmund-Schultze

Reviewed by Della Fenster
dfenster@richmond.edu

Telegraphic Reviews

Editor’s Endnotes