12 July 1999 Vol. 4, No. 28
THE MATH FORUM INTERNET NEWS
Discrete Math Games  Ensley  Hexagonal Honeycomb Conjecture  Hales  MathML
DISCRETE MATH GAMES  Douglas Ensley
http://www.ship.edu/~deensl/pgss/games.html
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 Online
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 twoplayer 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:
http://www.ship.edu/~deensl/
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HEXAGONAL HONEYCOMB CONJECTURE PROVED
 The Hexagonal Honeycomb Conjecture
http://www.math.lsa.umich.edu/~hales/countdown/honey/
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
http://www.maa.org/features/mathchat/mathchat_6_17_99.html
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...
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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
http://www.w3.org/TR/RECMathML/
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.
http://www.webeq.com/mathml/
an extensive resource list, sample documents, and a
Webbased discussion group for MathML developers.
Includes A Gentle Introduction to MathML:
http://www.webeq.com/mathml/gitmml/
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:
http://www.webeq.com/webeq/
 MathML: Mathematical Markup Language  Pankaj Kamthan,
Computational Mathematics Lab., Concordia Univ., Canada
http://indy.cs.concordia.ca/mathml/main.html
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
http://irt.org/articles/js081/
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 plugin; the future of MathML;
Web references; and content and presentation element
reference tables
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