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Mathematical Games, Toys, and Puzzles
Look, my first award! Many thanks to
Planet Science for picking this as their
Site of the
Day for April 26, 1996. (Unfortunately, you need to
to see their stuff.) While you're here, take a look at
some of my other pages, too.
Sorry about the dust. Several of these pages have moved or died in the last few
months, and there's lots of new stuff I need to add, but I've been busy with
my real job. Thanks for your patience.
Mathematicians use the words "game theory" in two different senses. The more common
sense of the word includes things like the Prisoner's Dilemma, poker, the New York Stock
Exchange, or other "imperfect-information" games. Research in this type of game theory
makes a lots of money for a lot of people, and has even resulted in a few Nobel prizes.
sketch is available at Drexel University.
Those aren't the kind of mathematical games I like. Combinatorial game theory tries to
analyze the outcomes of "perfect-information" games like Nim and Tic-tac-toe. More
difficult real-world examples are Hex, Reversi, Mancala, Checkers, Chess, Shogi, and Go.
Nancy Casey has a page full
of examples. (Unfortunately, all of the beautiful mathematics has been omitted,
presumably because it wouldn't be understandable to a
For a more thorough introduction, see On Numbers and Games by John Conway
or Winning Ways for your Mathematical Plays by Elwyn Berlekamp, John Conway,
and Richard Guy. ONAG sets the foundations and wanders further into the
mathematics; Winning Ways focuses more on actual games.
- David Applegate, Guy Jacobson, and Daniel Sleator wrote a
Computer Alanysis of Sprouts.
- Achim Bachem's
research group at the University of Köln studies
combinatorial games arising from prbolems in theoretical computer science.
- Here's an article from The Sciences by Charles Seife about
Fraenkel has a number of online papers on combinatorial games, including an
bibliography. You can also find his bibliography in the
Dynamic Surveys section of the
Electronic Journal of Combinatorics.
- David Eppstein has a
combinatorial game theory page
with lots of Usenet excerpts, Web pointers, and other stuff.
- Dan Garcia master's thesis is a
meta-game program called
- Richard Guy maintains a list of
Unsolved Problems in Combinatorial Games, available courtesy of David
- Daniel Loeb, a researcher in
theory, has translated Don Knuth's play Surreal Numbers into
- David Moews analyzes
sliding coins and Go.
- Following Elwyn Berlekamp and David
Wolfe, Matrin Müller analyzes
Go as the sum of smaller games in his
- Dave Rusin describes a number of
mathematically-oriented games, and offers some source code useful for analyzing
them, in his
- David Wolfe (with help from a
few other people) has written some
software for evaluating combinatorial game positions.
- After taking
Elwyn Berlekamp's combinatorial games class, I wrote a couple of
papers, one on Toads and Frogs and
another on Sowing Games. These papers
appear in the proceedings of the
games workshop held at
MSRI in July 1994.
What? Isn't all math fun?
- FAQ and archives of the
Usenet newsgroup rec.games.abstract
- Clever Games for
Clever People from
Nancy Casey's Mega-Math
- Keith Pomakis'
includes pointers to several Web implementations including his own
Connect! (Connect-4 for arbitrary values of 4).
Crescent is the second simplest combinatorial game in existence, with value
*. The simplest, which as far as I know is not represented on the Web, is the
empty game, also known as "I Resign" or "You Win".
- Othello, by
Mikael Roos and others
- Pegs, by
Rock-Paper-Scissors-Spock-Lizard is really an "economics" game.
Quarto, a Java applet by
- Turnablock, one
of many games invented and analyzed by John Conway, implemented as a
Java applet by
- Nobody can agree on the rules of Mancala.
- This Java version from
the entropy libration front lets you
choose whichever variant you know best. The rules I first learned were
asymmetric, capture to home, replay, no continue, 6 stones, except that
the capturing stone also went to the capturing player's store (home pit).
- This WWW version from
Touchstone Games follows the rules I first learned, except that
they start with three stones per pit instead of six.
- Another WWW version from
Imagiware doesn't skip the opponent's store.
(These two versions appear to be written by the
- A Java
version from Herky
Gottfried starts with four seeds per pit, doesn't let you capture
empty pits, and makes you move from the wrong side.
- Another Java
version from Intermetrics
obviously isn't finished yet, since it doesn't know about capturing or
repeating moves, and can't play against you.
- Here's a rather low-tech version of
the "other" African sowing game.
- Here are the rules for Wari, which for some reason the author calls
The Game Cabinet. (According to
the rules I learned, if it's your move, your opponent doesn't have any
stones on his side of the board, and you can't put any there, then the
game ends, and your opponent gets all your stones.)
- Brent Ridley can't decide if Mancala is
a game, a toy, a teaching tool, or a philosophical boondoggle
- Daniel Loeb and Duane M. Broline sneak some
math into the stew!
- The Games Domain is
mostly about PC video games (yawn), but there are a few pointers to other
"abstract" games pages.
- Val Kartchner's
The Game Room has rules for a vast
number of real-world abstract games, and pointers to
other abstract games
- Zarf's List of Interactive Games on the
Web has tons of pointers to actual Web games.
- For people who prefer human opponents, The
kNights Of the Square Table is a friendly postal/correspondence gaming club
primarily (but not exclusively) oriented towards chess and its
- Hans Bodlaender maintans a page of
- Nomic, a meta-game popularized by Douglas Hofstadter in his book/column
Metamagical Themas, isn't a combinatorial game, but I like it
anyway. The whole point of Nomic is to change the rules of the game. There are
several Nomic or Nomic-ish games running by email. Michael Norrish has a
list of running Nomic games.
- Find the Spam! is the
exact opposite of Nomic. (This appears to be dead. Food poisoning perhaps?)
- I once won in a level-3 game of
which Hofstadter called "Hruska" in Metamagical Themas. In the
version we played, there are three players. A level-zero games consists of
choosing an integer between 1 and 5. The player who chooses the second-highest
number wins, and adds that many points to his level-1 score. A level-n
game consists of five level-(n-1) games. The player with the second-highest
total level-(n-1) score wins the level-n game, and adds that score
to his level-n score. We had some rather elaborate rules for avoiding
ties, which involved adding or subtracting 1/3 from certain player's scores
depending on which round it was. Needless to say, the 125th level-zero game took
us a long time!
What? Isn't all m... Never mind.
Yoshiaki Araki has more tilings than you can shake a stick at.
- Thomas Banchoff provides the Web
with a copy of
- David Bell's treatise
"Spaceships in Conway's Game of Life", courtesy of
- Paul Callahan's
includes among its many treasures a
animator and descriptions of new results like
fillers and Tim Coe's
(It does not, however, include the phrase "well-separated pair
a substitution tiling program written in Tcl/Tk by Roger Evans Critchlow Jr.,
available courtesy of the entropy liberation
- Tony Davie keeps a page
Topics in Mathematical Recreations.
- David Eppstein has a
mathematics page, where he collects
geometry "junk". (He's
obviously thinking of the sort of junkyard where my dad used to find
hundred-year old microscopes.)
- Martin Gardner,
perhaps the world's best-known mathematician, was the author of
Scientific American's "Mathematical Games" column, many of which have
been collected in
book form. He is also the author of
several other books on recreational mathematics, pseudoscience, and
- The Geometry Center has some
really cool toys!
- The Geometry Forum folks at Swarthmore keep
of the geometry.* newsgroups.
- Chaim Goodman-Strauss plays with
and polyhedral fractal foam,
among other things.
- David Griffeath whips up some
- George Hart collects hundreds of
Pavilion of Polyhedreality presents plenty of pointers to other polyhedron
- Douglas Hofstadter's
home page doesn't contain a single
self-reference! How utterly disappointing! Here's the
reference to GEB:EGB in
The Jargon File,
and here's a somewhat less glowing
- David Joyce has a number of
cool geometric pages,
including the Web's only decent
explorer, pages about
"Millefiori" patterns (symmetric chaos), and even a
Java-illustrated verison of
- Speaking of self-reference and reviews, John
"Sloppy" Millington collects
reviews of this page.
- John Mount
Genetic Art and
Genetic Movies. He also has some
- Kari Nurmela and Patric Östergård
describe how to pack
cans into boxes. The picture to the right shows their packing of 49 cans,
which disproves a conjecture that the obvious square packing is optimal. (I added
colors showing locally hexagonal parts of the packing.) See also
Graham and B. D. Lubachevsky's paper
Electronic Journal of Combinatorics.
- Yadda yadda Cliff
Pickover blah blah chaos yadda yadda fractals blah blah graphics yadda yadda
- Jim Propp plays with arctic circles,
ants, cookies, and games with three players.
- Rudy Rucker is the
William S. Burroughs of
- Rob Scharein
- Undergrads at the University of Toronto publish a "wrecreational
math" newsletter called
Math 007 I News, which has little or nothing to do with
- Stan Wagon publishes a
Problem of the Week. Stan also organizes the
- Last, but not least, here's Eric
Trove of Mathematics.
All mortals must pay homage
to Eve Andersson,
green-skinned Web goddess from the planet Gragel. Nobody loves pi like Eve, but
people try anyway. [Note hidden in Eve's homepage: "This photo was taken with
a Polaroid Captiva camera which doesn't do a very good job of capturing true
colors. In reality my skin is a lighter, more feminine shade of green."]
My Erdös number
is three, but I don't have a
I'll give a dollar to the first person who can name someone other than Steven
Hawking with both a finite Erdös number and a finite Bacon number.
I'd really love to see a page of
(Larry Marder is the nexus of all comic book realities. Unfortunately, he also appears to
be the nexus of all ugly Web pages.)
Here's everything you always wanted to know about
everybody's favorite random
yellow pigs. Don't
believe Douglas Adams - the real answer is
Archimedes Plutonium (also known as Ludwig
Plutonium) believes that the entire universe is contained in the 94th electron of a
Plutonium-231 atom. He also claims to have proved or disproved
Fermat's Last Theorem,
The Four-Color Theorem,
the Riemann Hypothesis, the Poincaré Conjecture,
Kepler's sphere-packing conjecture,
the Goldbach conjecture,
the existence of odd perfect numbers,
the infinitude of twin primes,
Cantor's continuum hypothesis, and
Gödel's incompleteness theorem. You can read his regular diatribes/
rants/ prayers/ hymns on several Usenet newsgroups, including
27 Jun 96