Solving an expected value problem without using geometric series

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**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 3^{2} + 4^{2}= 5^{2}and then extended it to a symmetrical 5*n*-piece dissection of squares for (*x*-*n*)^{2} + ... + (*x*-1)^{2} +*x*^{2} = (*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 9*n*-piece dissections. We then show how to trade away symmetry to get 8*n*-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.>

**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.

**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 .

**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 RuleÂ—while keeping things accessible to any Calculus 2 student.

**A Reciprocal Pythagorean Theorem**

by Roger B. Nelsen

370

We wordlessly prove that if *a* and *b*s 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 Â“It seems that one must enter the realm of complex numbers to prove this result.Â” 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 pebbleÂ—a Stephen Hawking small black holeÂ—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.

The Scholar's Song

by Richard Cleveland

381