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Mathematics Magazine Contents - December 2015

This issue offers diverse cross sections of mathematics: cooperative play in the card game Hanabi is related to hat-guessing games; a fair division algorithm is proposed and analyzed; and surprising ratios between volumes and surfaces areas are considered for different solids of revolutions. Cross sections of tori are shown to be related to foci of families of ellipses. There is also the first in a series of interviews with mathematical artists or artistic mathematicians. Amy and David Reimann interview George Hart in George Hart: Troubadour for Geometry.  —Walter Stromquist, editor


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Vol. 88, No. 5, pp 320 – 388

Letter from Editor


How to Make the Perfect Fireworks Display: Two Strategies for Hanabi

Christopher Cox, Jessica de Silva, Philip Deorsey, Franklin H. J. Kenter, Troy Retter and Josh Tobin

The game of Hanabi is a multiplayer cooperative card game that has many similarities to a mathematical “hat guessing game.” In Hanabi, a player does not see the cards in her own hand and must rely on the actions of the other players to determine information about her cards. This article presents two strategies for Hanabi. These strategies use different encoding schemes, based on ideas from network coding, to efficiently relay information. The first strategy allows players to effectively recommend moves for other players, and the second strategy allows players to determine the contents of their hands. Results from computer simulations demonstrate that both strategies perform well. In particular, the second strategy achieves a perfect score more than 75 percent of the time.

To purchase the article from JSTOR: 10.4169/math.mag.88.5.323

Proof Without Words: Half Issues in the Equilateral Triangle and Fair Pizza Sharing

Grégoire Nicollier

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How to Divide Things Fairly

Steven J. Brams, D. Marc Kilgour, and Christian Klamler

We propose an intuitively simple sequential algorithm (SA) for the fair division of indivisible items that are strictly ranked by two or more players. We analyze several properties of the allocations that it yields and discuss SA's application to real-life problems, such as dividing the marital property in a divorce or assigning people to committees or projects.

To purchase the article from JSTOR: 10.4169/math.mag.88.5.338

Volume/Surface Area Ratios for Globes, with Applications

Tom M. Apostol and Mamikon A. Mnatsakanian

We introduce families of solids called globes, having an invariant ratio of volume to surface area. An application determines the lateral surface area of an elliptical wedge in terms of its volume. We also relate surface areas and volumes of corresponding zonal slices of a spheroid and sinoid via the eccentricity of an ellipse.

To purchase the article from JSTOR: 10.4169/math.mag.88.5.349

The Parallelogram with Maximum Perimeter for Given Diagonals Is the Rhombus—A Proof Without Words and a Corollary

Angel Plaza

By the Law of Cosines and the arithmetic mean-root mean square inequality it is proved without words that The Parallelogram with Maximum Perimeter for given Diagonals is the Rhombus. As a corollary it also proved that for two positive numbers, their arithmetic mean is greater or equal than the arithmetic mean of their geometric mean and their root mean square.

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Crossword Puzzle: Joint Mathematics Meetings 2016

Brendan W. Sullivan

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The Focus Locus Problem and Toric Sections

Michael Gaul and Fred Kuczmarski

We investigate curves (foci loci) traced by the foci of one-parameter families of ellipses. The elliptical families are shadows cast by a circle. The focal paths turn out to be cross sections of tori.

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George Hart: Troubadour for Geometry*

Amy L. Reimann and David A. Reimann

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Math’s effects on art; burn math class!; the hot hand is hot; system gaps

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To purchase the article from JSTOR: 10.4169/math.mag.88.5.387