Ataxx
Ataxx (also known as SlimeWars and Frog Cloning) is a board game which first appeared in 1990 as an arcade video game by The Leland Corporation. The game was invented by Dave Crummack and Craig Galley in 1988 and was called Infection. It was first programmed on Amiga, Commodore 64, and Atari ST. The game is purportedly in the public domain now.[citation needed]
Game play
Ataxx is an abstract strategy board game which involves play by two parties on a seven-by-seven square grid. The object of the game is to make your pieces constitute a majority of the pieces on the board at the end of the game, by converting as many of your opponent's pieces as possible.
Each player begins with two pieces, white and black, for the first player and second player respectively. The game starts with the four pieces on the four corners of the board, with white in the top left and bottom right and black on the other two. White moves first.
During their turn players must first select the piece they wish to move, the source, and then select and empty square, the destination, that is either adjacent to, or having a distance of less than three, from the source. Diagonal distances are equivalent to orthogonal distances, i.e. it is legal move to a square who's relative position is two squares away both vertically and horizontally. If the destination is adjacent to the source, a new piece is created on the empty destination square. Otherwise the piece on the source moves to the destination. After the move, all of the opponent player's pieces adjacent to the destination square are converted to the color of the moving player. Players must move unless no legal move is possible, in which case they must pass.
The board configuration may be altered by having certain squares filled (not playable by either player), or using different board sizes.
The game ends when all squares have been filled or one of the players has no remaining pieces. The player with the most pieces wins. A draw may occur when the number of squares are even, either from non playable squares or non standard sizes having an even number of squares. Some versions also implement the threefold repetition rule from chess.
Ataxx complexity
Ataxx has between 2^n and 3^n possible end game positions, where n represents the board area. Attempts to calculate the complexity of games are normally made by raising the branching factor to the power of the number of plies. In Ataxx, the average number of plies is about 100 and the average branching factor about 60, giving 60^100 games. However, this greatly underestimates the true complexity. In Ataxx, nearly every combination of the 4 original pieces on the board can be obtained. Thus C = ( n! / r! (n – r)! ), where n is the board area, r is the number of slimes, and C is the number of possible positions. Even this value is much too small, as each combination without repetition having r slimes may have any permutation with repetition of red or blue slimes (with the exception of 0 red or blue). Therefore, C = ( n! / r! (n – r)! )*( 2^r – 2 ) approximates the number of possible positions for a given number of slimes. To calculate the state space complexity, i.e. the number of different positions which may arise in a game, one calculates the summation of C for all values of r where n > r >= 4. Note that the game is over when n = r, so it would be incorrect to add ( 2^r – 2 ) as only one of these possible positions will occur. Thus:
S = Σ ( n! / (r! (n – r)!) )*( 2^r – 2 )
For a 7 by 7 board this gives 5.982483 * 10^22
where r represents to the number of slimes and goes from 4 to n – 1, n represents the board area, and S is the state space complexity. The formula calculates the summation of the combinations without repetition for r slimes multiplied by the permutations with repetition for r slimes, or the total number of positions which can be achieved for every number of slimes possible without ending the game. The – 2 present in 2^r – 2 is necessary to avoid the two situations where all slimes are either blue or red. In fact, not all permutations with repetition of red or blue slimes may be achieved; for example, if r = 4 and the four slimes are adjacent, then they must necessarily be the same colour. Even so, it is probable that the above formula is fairly accurate, and at least serves as an upper bound.
To calculate the game tree complexity, i.e. the total number of possible games that can be played, the summation must be replaced with a product. This gives the following formula:
G = Π ( n! / (r! (n – r)!) )*( 2^r – 2 )
For a 7 by 7 board this gives 1.371545 * 10^896
where r represents to the number of slimes and goes from 4 to n (not n – 1 as above), n represents the board area, and G is the game tree complexity. The reason multiplication is used rather than addition is that, for each position obtained with r slimes, all possible positions with r + 1 slimes (except the restrictions mentioned above) may be obtained. The – 2 again adds some ambiguity; including it implies that a player cannot have zero slimes, excluding it implies that once a player has zero slimes the game may continue (neither of which are true). This answer is probably less accurate than the approximation of the state-space complexity.
Other versions
The arcade version is timed, similarly to blitz chess. Each player's timer counts down while it is their turn to move; they lose if it reaches zero, but may buy more time (100 seconds per quarter) while the game is in progress. After completing a game (whether they won or lost), a player may choose whether or not to play another game with the time remaining.
Ataxx can be found all over the internet in numerous variations including board size. The GNOME desktop environment for Linux features an Ataxx game.
Ataxx can be played by email, using Richard Rognlie's Play-By-eMail Server.
Hexxagōn is a variant of Ataxx that is played on a hexagonal board instead of a square one. It was released as a PC game in 1993 by Software Creations. The rules of the game are the same as in Ataxx.
Ataxx clones
The most popular of the Ataxx clones is the Laboratory Puzzle in The 7th Guest, where the player faces off with the AI in the game, in this case the antagonist "Stauf", who is incredibly difficult to beat - all the while laughing and ridiculing the player with phrases like "Ooooooooo Baaaaaaaad Mooooooooove." and "We'll all be dead by the time you solve this!" The pieces were seen through a microscope on a slide, interpreted as viral infections, two sets blue and two sets green. Passing over this puzzle in the game requires two trips to the library to the hint book. Winning this puzzle causes Stauf to let out a growl, then say "Don't think you'll be so lucky next time!"
Another popular Ataxx clone was Boogers, an online 4 player version hosted by the ImagiNation Network, a pre-internet dial-up game network. Two to four players fought on a large board (9×9). The pieces, called boogers, were colored green, yellow, purple and blue. Blocks were randomly placed on the board to alter game play. Jumping one's booger over an opponent's booger would change the victim's color, resulting in more boogers for the aggressor. Chatting both openly and privately was integrated into the game: this promoted shady non-aggression pacts, ganging up, and stabbing opponents in the back.
Spot for NES, PC, and Game Boy implements the same rules as Ataxx.
The 11th Hour also features a puzzle similar to Ataxx, but with a board of hexagons.
The Sierra educational title Mega Math included a clone of Ataxx where the player must answer math questions in order to move. [1]
Buzzlabs has a freeware version of Ataxx called Aquataxx for Mac OS X, written in the Java language with Apple's Cocoa Frameworks. It has online chat and network play. [2]
Trivia
- In an episode of Mystery Science Theater 3000 (612- The Starfighters), Crow T. Robot sets up a PC in an attempt to get on the "Information Super Highway". After numerous setbacks throughout the episode, he finally succeeds, only to find himself invited to play "Four-person Boogers". He accepts the offer.