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Volume 4, Number 28

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12 July 1999                                        Vol. 4, No. 28


Discrete Math Games - Ensley | Hexagonal Honeycomb Conjecture - Hales |
MathML DISCRETE MATH GAMES - Douglas Ensley Javascript explorations from a course designed to introduce discrete mathematics through the analysis of puzzles and games, given at the Pennsylvania Governor's School for the Sciences (PGSS): - The Josephus Problem Arrange people in a circle and choose one person to be number one. Proceed clockwise, removing every 3rd - or 7th, etc. - person. Which person is the last one left? - Josephus Shuffles Named for their resemblance to the Josephus Problem, these shuffles mix up a deck of cards in a peculiar way. Here an interactive page allows you to investigate the final order of the cards for various values of n and k. - Dice Game Two players roll dice repeatedly, noting who has the higher roll each time. You may set the number of rolls and the original number of dice for each person. - Hank and Ted On-line A simple gambling game in which two people flip a coin over and over, at a wager of one marker, until one of them is out of markers. - Series Simulator A program that lets you use a random number generator to simulate a series. You may customize team names and game probabilities, and specify the number of games to be played. - Water Puzzle Starting with a pail of water holding some unknown quantity greater than 8 quarts, and small (3 qt) and large (5 qt) vases as measuring devices, explore how to split 8 quarts of water originally in the pail equally between two people. - Grid Game Play a two-player game against the computer: players alternate marking squares in a grid until all the squares have been marked. - Nim Two people play using several piles of stones. On each move, a player removes as many stones as he or she desires, but forms only one pile. The player who takes the last stone wins. Douglas Ensley is an Associate Professor of Mathematics and Computer Science at Shippensburg University: -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- HEXAGONAL HONEYCOMB CONJECTURE PROVED - The Hexagonal Honeycomb Conjecture Prof. Thomas C. Hales of the Department of Mathematics, University of Michigan, whose joint proof with Samuel Ferguson of the Kepler Conjecture we highlighted in April, has announced a proof of the Hexagonal Honeycomb Conjecture, which asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. This paper in a variety of formats (postscript, PDF, DVI) gives the first general proof, with some historical background. - Hales Proves Hexagonal Honeycomb Conjecture (Math Chat) - Frank Morgan, for MAA Online Prof. Thomas Hales of the University of Michigan has announced a proof of the Hexagonal Honeycomb Conjecture, which says that regular hexagons provide the least- perimeter way to enclose infinitely many unit areas in the plane. This may partly explain why hexagons occur in nature, as in the bees' honeycomb... -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- MATHEMATICAL MARKUP LANGUAGE - MATHML MathML is an application for describing mathematical expression structure and content. Its goal is to serve, receive, and process mathematics on the Web, just as HTML has enabled this functionality for text. A number of groups are working on its implementation. For progress reports and other information, visit: - Mathematical Markup Language (MathML): W3C Recommendation from the World Wide Web Consortium: about interactions between MathML content and presentation tags, how MathML renderers might be implemented and should interact with browsers, and MathML entities (extended characters) and their relation to fonts - MathML Information Center - Geometry Technologies, Inc. an extensive resource list, sample documents, and a Web-based discussion group for MathML developers. Includes A Gentle Introduction to MathML: Also from Geometry Technologies: WebEQ: Putting Math on the Web - a suite of Java applets that allow Web authors to include mathematics in their pages: - MathML: Mathematical Markup Language - Pankaj Kamthan, Computational Mathematics Lab., Concordia Univ., Canada a comprehensive source of information about MathML, with links to authoring tools; mailing lists and usenet groups; companies and organizations supporting MathML; documentation; information on the history and possible future of MathML; examples and applications; interoperability; information on links in MathML code and its syntax in general; news; lists of compilers, editors, renderers, and translators; and other Web resources - MathML - What's In It For Us? - Janus Boye an article presenting a brief history of the standard; descriptions of the elements and attributes, tokens and basic layout schemata, and containers; a demo of MathML in a browser; the EzMath plug-in; the future of MathML; Web references; and content and presentation element reference tables -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- CHECK OUT OUR WEB SITE: The Math Forum Ask Dr. Math Problems of the Week Mathematics Library Teacher2Teacher Discussion Groups Join the Math Forum Send comments to the Math Forum Internet Newsletter editors _o \o_ __| \ / |__ o _ o/ \o/ __|- __/ \__/o \o | o/ o/__/ /\ /| | \ \ / \ / \ /o\ / \ / \ / | / \ / \

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