# 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 register 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.

## Combinatorial Games

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. An introductory 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 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.

### 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!

## Other Fun Math

What? Isn't all math fun?

## Silly Math!

What? Isn't all m... Never mind.
 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