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Mathematical Games, Toys, and Puzzles

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.

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Contents

• Contents
• Combinatorial Games
  • Game Theory (and Game Theorists)
  • Actual Games
• Other Fun Math
• Silly Math!

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Combinatorial Games

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 nine-year old!)

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.

Game Theory (and Game Theorists)

• 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 John Conway.
• Aviezri Fraenkel has a number of online papers on combinatorial games, including an extensive 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 GAMESMAN.
• Richard Guy maintains a list of Unsolved Problems in Combinatorial Games, available courtesy of David Wolfe.
• Daniel Loeb, a researcher in combinatorial game theory, has translated Don Knuth's play Surreal Numbers into French.
• 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 Ph.D. Thesis.
• Dave Rusin describes a number of mathematically-oriented games, and offers some source code useful for analyzing them, in his games directory
• 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.

Actual Games

• FAQ and archives of the Usenet newsgroup rec.games.abstract

• Clever Games for Clever People from Nancy Casey's Mega-Math
• Keith Pomakis' Connect-4 page includes pointers to several Web implementations including his own Connect! (Connect-4 for arbitrary values of 4).
• Mornington 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 Glenn Bresnahan
• Rock-Paper-Scissors-Spock-Lizard is really an "economics" game.
• Quarto, a Java applet by Dan Hutchings
• Turnablock, one of many games invented and analyzed by John Conway, implemented as a Java applet by Ken Shirriff.

• 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 same person!)
  • 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 Wari, the "other" African sowing game.
  • Here are the rules for Wari, which for some reason the author calls Trysse, from 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 actual 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 resources.
• 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 mutant offspring.
• Hans Bodlaender maintans a page of Chess Variants.

• 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 Mediocrity, 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!

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Other Fun Math

• Yoshiaki Araki has more tilings than you can shake a stick at.
• Thomas Banchoff provides the Web with a copy of Edwin Abbott's Flatland.
• David Bell's treatise "Spaceships in Conway's Game of Life", courtesy of Jörg Heitkötter.
• Paul Callahan's Life page includes among its many treasures a Java animator and descriptions of new results like space fillers and Tim Coe's c/5 spaceship. (It does not, however, include the phrase "well-separated pair decomposition".)
• Subtile, a substitution tiling program written in Tcl/Tk by Roger Evans Critchlow Jr., available courtesy of the entropy liberation front.
• Tony Davie keeps a page full of Topics in Mathematical Recreations.
• David Eppstein has a recreational 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 magic.
• The Geometry Center has some really cool toys!
• The Geometry Forum folks at Swarthmore keep archives of the geometry.* newsgroups.
• Chaim Goodman-Strauss plays with aperiodic tilings and polyhedral fractal foam, among other things.
• David Griffeath whips up some primorial soup.
• HAKMEM
• George Hart collects hundreds of virtual polyhedra. His Pavilion of Polyhedreality presents plenty of pointers to other polyhedron pages.
• 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 review.
• David Joyce has a number of cool geometric pages, including the Web's only decent Mandelbrot explorer, pages about wallpaper groups and hyperbolic tilings, "Millefiori" patterns (symmetric chaos), and even a Java-illustrated verison of Euclid's Elements.
• Speaking of self-reference and reviews, John "Sloppy" Millington collects reviews of this page.
• John Mount breeds Genetic Art and Genetic Movies. He also has some hyperbolic tilings.
  
• 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 Ron L. Graham and B. D. Lubachevsky's paper in the Electronic Journal of Combinatorics.
• Yadda yadda Cliff Pickover blah blah chaos yadda yadda fractals blah blah graphics yadda yadda yadda.
• Jim Propp plays with arctic circles, ants, cookies, and games with three players.
• Rudy Rucker is the William S. Burroughs of mathematics.
• Rob Scharein plots knots.
• Undergrads at the University of Toronto publish a "wrecreational math" newsletter called Math 007 I News, which has little or nothing to do with James Bond.
• Stan Wagon publishes a Problem of the Week. Stan also organizes the Konhauser Problemfest.
• Last, but not least, here's Eric Weisstein's Treasure Trove of Mathematics.

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Silly Math!

	All mortals must pay homage to Eve Andersson, green-skinned Web goddess from the planet Gragel. Nobody loves pi like Eve, but many foolish 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 Bacon number. 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 Marder numbers! (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 number of yellow pigs. Don't believe Douglas Adams - the real answer is 17!
	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 sci.math and alt.sci.physics.plutonium.

Jeff Erickson (jeffe@cs.uiuc.edu) 27 Jun 96