Mathematics Magazine - December 2009

ARTICLES

Casting Light on Cube Dissections
by Greg N. Frederickson
323-331
A decade ago, Michael Boardman created a beautifully symmetrical 5-piece dissection of squares to demonstrate the identity 3² + 4²= 5²and then extended it to a symmetrical 5n-piece dissection of squares for (x-n)² + ... + (x-1)² +x² = (x+1)² + ... + (x+n)² for each natural number n.. In this article we identify an analogous family of identities based on cubes and produce analogously symmetrical 9n-piece dissections. We then show how to trade away symmetry to get 8n-piece dissections, the first infinite class of cube dissections that are arguably minimal.

The Monty Hall Problem, Reconsidered
by Stephen Lucas, Jason Rosenhouse, and Andrew Schepler
332-342
The Monty Hall problem is one of the most frustrating brainteasers in all of mathematics. Despite its seemingly simple game-show format, most people, even those with mathematical training, find it very challenging. Part of the difficulty lies in the counter-intuitive nature of conditional probability. We offer an intuitive guide to focusing on what is important in the problem. We then show how to apply this new-found intuition to numerous, increasingly complicated, variations on the basic set-up.

Modeling a Diving Board
by Michael A. Karls and Brenda M. Skoczelas
343-353
The beam equation is a classic partial differential equation that one may encounter in an introductory course on boundary value problems or mathematical physics, which can be used to describe the vertical displacement of a vibrating beam. A diving board can be thought of as a cantilever beam, which is a bar with one end fixed and the other free to move. Using a video camera and physics demonstration software to record displacement data from a vibrating cantilever beam, we verify a modified version of the beam equation that incorporates damping and a forcing term.>

LETTER TO EDITOR

Correction to  “Fibonacci Clock ”
by Edward Dunne
353
Corrects relatively minor errors in the Note  “Long Days on a Fibonacci Clock, ” which appeared in the April, 2009 issue of the Magazine.

NOTES

Self-Curvature Curves
by Steven V. Wilkinson
354-359
If you graph the curvature function for a curve, its shape may look nothing like the original curve. Are there any curves that are the same or close to the same as the graphs of their curvature functions? If so, what are their attributes? We show there are such curves, but they have only limited shapes.

PROOF WITHOUT WORDS

Squares in Circles and Semicircles
by Roger B. Nelsen
359
We wordlessly prove that a square inscribed in a semicircle has 2/5 the area of a square inscribed in a circle of the same radius.

Cars, Goats, π , and ℯ
by Alberto Zorzi
360-363
Monty Hall’s problem (known also as “the three-door problem ”) has become very popular since 1990 because of its counterintuitive solution. We propose two extensions of the classical version, which solutions respectively involve the numbers π , and ℯ .

MATH BITE

A Graph Theoretic Summation of the Cubes of the First n Integers
by Joe DeMaio and Andy Lightcap
363-370
In this Math Bite we provide a combinatorial proof of the sum of the cubes of the first n integers by counting edges in complete bipartite graphs.

Elementary Proofs of Error Estimates for the Midpoint and Simpson’s Rules
by Edward C. Fazekas, Jr. and Peter R. Mercer
365-370
Students usually meet the Trapezoid and Midpoint Rules in Calculus 2 but they see how to obtain their error estimates much later, if at all. D. Cruz-Uribe and C. Neugebauer showed in Mathematics Magazine 76 (2003) how a Calculus 2 student can obtain Trapezoid Rule error estimates via integration by parts. We extend their idea to obtain error estimates for the Midpoint Rule and for Simpson’s Rule—while keeping things accessible to any Calculus 2 student.

PROOF WITHOUT WORDS

A Reciprocal Pythagorean Theorem
by Roger B. Nelsen
370
We wordlessly prove that if a and bs are the legs and h the altitude to the hypotenuse c of a right triangle, then .

Sines and Cosines of Angles in Arithmetic Progression
by Michael P. Knapp
371-372
In a recent Math Bite, Judy Holdener gave a physical argument for certain sums of sines and cosines. She comments that “It seems that one must enter the realm of complex numbers to prove this result.” In this note, we present a more general result whose proof does not require complex numbers.

Black Holes through The Mirrour
by Andrew Simoson
372-381
This note shows that the path of a free falling pebble—a Stephen Hawking small black hole—when dropped at the equator of a spinning rotating homogeneous earth is a hypocycloid. Furthermore, given any trochoid (any epitrochoid or hypotrochoid) T, the pebble can be thrown at a tangent from the equator of a nonrotating earth, after which the earth can be given a fixed rotation rate so that the path of the pebble through the earth, with respect to the earth, is T. Also analyzed is the path of the pebble when dropped from northern climes.

POEM

The Scholar's Song
by Richard Cleveland
381