Arimaa/Print version
From Wikibooks, the open-content textbooks collection
Arimaa is a two-player abstract strategy board game that can be played using the same equipment as chess.
Note: current version of this book can be found at http://en.wikibooks.org/wiki/Arimaa
Overview
Playing The Game
Introduction to Tactics
Introduction to Strategy
Relative Value of Pieces
Advanced Tactics
Positioning for an Attack
Lone Elephant Attacks
Elephant and Camel Attacks
Elephant and Horse Attacks
Elephant and Minor Piece Attacks
Multi-Piece Swarming Attacks
Camel and Horse Attacks
Double-Trap Attacks
Other Attacking Ideas
Arimaa Challenge History
Sample Games
Glossary
Overview
Arimaa is a two-player board game invented by Omar Syed, a computer engineer trained in artificial intelligence. Syed was inspired by Garry Kasparov's defeat at the hands of the chess computer Deep Blue to design a new game which would be difficult for computers to play well, but would have rules simple enough for his four-year-old son Aamir to understand. ("Arimaa" is "Aamir" spelled backwards plus an initial "a"). In 2002 Syed published the rules to Arimaa and announced a $10,000 prize, available yearly through 2020, for the first computer program able to defeat a top-ranked human player in a match six games or longer. The prize has not yet been won.
As of late 2007, Arimaa has proven so computer-resistant that top human players can no longer find much interest in games against Arimaa programs, even the top programs created by professional developers. This Wikibook therefore treats computer Arimaa only tangentially. The primary purpose of these pages is to give advice to humans on how to play well against other humans. (Some strategies for use specifically against computers can be found at the arimaa.com Wiki.)
Arimaa can be played in person with a chess set, or on-line at the arimaa.com gameroom. As of late 2007, no Arimaa-specific game pieces have been commercially produced, and only one face-to-face tournament has taken place (photos), but about 500 games per week are played on-line. Omar Syed hosts four events per year in the gameroom:
- The World Championship is an open tournament for human players from January to March. The current format is one round per week of an open Swiss qualifier followed by floating double-elimination among the top eight qualifiers. All four World Champions so far have been players who learned the game less than a year prior to the championship tournament, and who usurped more experienced players to win the title. This is a testament to how fast the knowledge of how to play well is expanding.
- The Computer Championship takes place in March, to determine which program earns the right to play in the Arimaa Challenge. The format is floating triple elimination, and participation is limited to the top eight programs. All four tournaments so far have been won by the program Bomb, by David Fotland.
- The Arimaa Challenge takes place in April. The top computer program attempts to win the $10,000 prize against human defenders. So far humans have dominated every challenge match.
- A Postal Mash begins in April and ends around October. The emphasis of this tournament is participation, rather than determining a champion. The objective is to advance the frontiers of strategic knowledge, as well as to spread around existing knowledge by pairing people to a variety of opponents.
United States Patent number 6,981,700 for Arimaa was filed on the 3rd of October 2003, and granted on the 3rd of January 2006. Omar Syed also holds a trademark on the name "Arimaa". Syed has released an experimental license called "The Arimaa Public License", with the declared intent to "make Arimaa as much of a public domain game as possible while still protecting its commercial usage". Items covered by the license are the patent and the trademark.
Playing The Game
Arimaa is played on a chessboard with four squares distinguished as trap squares, namely c3, f3, c6, and f6 in algebraic chess notation. The two players, Gold and Silver, each control sixteen pieces: these are, in order from strongest to weakest, one elephant, one camel, two horses, two dogs, two cats, and eight rabbits. The pieces may be represented by the chess king, queen, rooks, bishops, knights, and pawns respectively.
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The players begin by setting up their pieces however they choose on their home rows. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The objective of the game is to move a rabbit of one's own color onto the home rank of the opponent. Thus Gold wins by moving a gold rabbit to the eighth rank, and Silver wins by moving a silver rabbit to the first rank. However, because it is difficult to usher a rabbit to the goal line while the board is full of pieces, an intermediate objective is to capture opposing pieces by pushing or pulling them into the trap squares.
The game begins with an empty board. Gold places the sixteen gold pieces in any configuration on the first and second ranks. Silver then places the sixteen silver pieces in any configuration on seventh and eighth ranks. The diagram at right shows one possible initial placement.
After the pieces are placed on the board, the players alternate turns, starting with Gold. A turn consists of making one to four steps. With each step a friendly piece may move into an unoccupied square one space left, right, forward, or backward, except that rabbits may not step backward. The steps of a turn may be made by a single piece or distributed between several pieces in any order. A turn must make a net change to the position. Thus one may not, for example, take one step forward and one step back with the same piece, effectively passing the turn.
The second diagram, from the same game as the initial position above, helps illustrate the remaining rules of movement.
A player may use two steps of a turn to dislodge an opposing piece with a stronger friendly piece which is adjacent. For example, a friendly dog may dislodge an opposing rabbit or cat, but not a dog, horse, camel, or elephant. The stronger piece may pull or push the adjacent weaker piece. When pulling, the stronger piece steps into an empty square, and the square it came from is occupied by the weaker piece. The silver elephant on d5 could step to d4 (or c5 or e5) and pull the gold horse from d6 to d5. When pushing, the weaker piece is moved to an adjacent empty square, and the square it came from is occupied by the stronger piece. The gold elephant on d3 could push the silver rabbit on d2 to e2 and then occupy d2. Note that the rabbit on d2 can't be pushed to d1, c2, or d3, because those squares are not empty.
Friendly pieces may not be dislodged. Also, a piece may not push and pull simultaneously. For example the gold elephant on d3 could not simultaneously push the silver rabbit on d2 to e2 and pull the silver rabbit from c3 to d3. An elephant can never be dislodged, since there is nothing stronger.
A piece which is adjacent to a stronger opposing piece is frozen, unless it is also adjacent to a friendly piece. Frozen pieces may not be moved by the owner, but may be dislodged by the opponent. A frozen piece can freeze another still weaker piece. The silver rabbit on a7 is frozen, but the one on d2 is able to move because it is adjacent to a silver piece. Similarly the gold rabbit on b7 is frozen, but the gold cat on c1 is not. The dogs on a6 and b6 do not freeze each other because they are of equal strength. An elephant cannot be frozen, since there is nothing stronger, but an elephant can be blockaded.
A piece which enters a trap square is captured and removed from the game unless there is a friendly piece adjacent. Silver to move could capture the gold horse on d6 by pushing it to c6 with the elephant on d5. Also a piece on a trap square is captured if all adjacent friendly pieces move away. Thus if the silver rabbit on c4 and the silver horse on c2 move away, voluntarily or by being dislodged, the silver rabbit on c3 will be captured.
Note that a piece may voluntarily step into a trap square, even if it is captured thereby. Also, the second step of a pulling maneuver may be completed, even if the piece doing the pulling is captured on the first step. For example, Silver to move could step the silver rabbit from f4 to g4, step the silver horse from f2 to f3, which captures the horse, and still pull the gold rabbit from f1 to f2 as part of the horse's move.
In the diagrammed position, if it were Gold's turn to move, Gold could win in three steps: The dog on a6 can push the rabbit on a7 to a8, and when the dog is on a7, it unfreezes the rabbit on b7, which can step to b8 for the victory.
There are several ways for the game to end apart from a rabbit reaching its goal. (The frequency of each in the first thousand rated human vs. human games on the Arimaa server is shown in parentheses.)
- (1.3%) If, at the beginning of a player's turn, no steps are possible because all friendly pieces are frozen or blockaded, the player whose turn it is loses.
- (0.2%) If the same position occurs three times with the same player to move, the player whose turn caused it to occur the third time loses. Only positions at the end of each turn are considered by this rule, not positions at the end of each step. (This rule originally did not consider side to move, but Syed changed it to be more like the chess repetition rule on May 28, 2005.)
- (0.0%) If all sixteen rabbits are captured, the game is a draw. (In elimination tournaments where a draw would be inconvenient, Syed has ruled that a player who captures all eight opposing rabbits wins the game, although as of December, 2006, it has been a moot point.)
Finally, if an opposing rabbit is dislodged onto its goal line and dislodged off within the same turn, the game continues.
Introduction to Tactics
In Arimaa, a tactic is usually a one-move plan that results in tangible gain. When the opponent's options are constrained, it is sometimes possible to calculate precisely to two moves out, i.e. four steps for the player on move, four steps for the opponent, and four more steps for the first player. In this case a two-move plan may be considered a tactic as well. Plans which take three moves or longer to come to fruition are usually impossible to calculate out with precision, and therefore generally belong to the realm of strategy. This is in contrast to chess where precise tactics of three moves and longer (i.e. combinations) are quite feasible.
As of late 2006, the top computer program Bomb will occasionally announce a forced goal of 20 steps (two and a half moves) and once even announced at 24-step (three-move) goal against itself, but this is the outer limit of tactical competence. Given 30 seconds per move on a standard PC, Bomb will often overlook even a 16-step (two-move) forced goal against itself.
The most basic tactics to master are those in which a friendly rabbit can be brought to the goal, or an opposing piece captured, within four steps. If one can't achieve victory or capture in a single move, one's opponent will often be able to bring in reinforcements which change the situation dramatically, in which case the play becomes more strategic.
One-move goal
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Most simply, a rabbit may be able to make four steps to the goal, even if the path superficially appears blocked by a trap square and/or opposing pieces which would seem to freeze the rabbit. At left, the gold rabbit on b5 can step to victory via b6, c6, c7, and c8. The rabbit is safe on the trap due to the gold dog on c5, and is never frozen because it is always next to a friendly piece on the way home. True, the rabbit will be frozen once it reaches c8, but Gold will have won regardless.
Pulling away an opposing piece may allow a blocked rabbit to advance. At left, the gold rabbit on g6 can reach the goal after the gold cat on h7 retreats to h6, pulling the silver rabbit from g7 to h7. Note that the gold camel on f6 is captured when the rabbit advances, but reaching the goal is worth any sacrifice. Also note that while beginners seem to enjoy pushing more than pulling, a push is ineffectual in this case. If the cat pushes the silver rabbit, the cat will itself be in the way of the friendly rabbit, which won't then have time to reach the goal in the same move.
Near the goal line, a rabbit which is frozen but not blocked is a constant threat. Silver to move could push the gold horse on b3 to c3 with the camel on b4. This would unfreeze the silver rabbit on b2, which could then step to victory via a2 and a1. Occasionally a blocked and frozen rabbit may be unblocked and unfrozen at the same time. Silver to move could pull the gold rabbit from g2 to g3 with the silver elephant on g3 sliding to f3 to unfreeze the silver rabbit on f2, which would then move to g2 and g1. The silver elephant would be sacrificed on f3 in the process, which makes this tactic easy to miss in the heat of battle.
One-move capture
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
A piece which is two squares away from an undefended trap square can often be dislodged twice and captured. In the diagram at right, if it is Silver's turn, the silver dog on b6 could push the gold cat on b7 to c7 (or pull it to b6) and then push it to c6. A piece which is only one square away from an otherwise undefended trap square is even more vulnerable, because it is threatened even by non-adjacent enemy pieces. If it is Gold's turn, the gold elephant on d5 can step to c5, then b5, and still have time to push the silver dog on b6 to c6.
Sometimes one may need to push an obstructing piece out of the way to get at a vulnerable piece. Gold to move could push the silver cat on g7 to h7 with the gold camel on g6, then push the silver horse on f7 to f6, capturing it.
A piece on a trap square with only one adjacent friendly piece is extremely vulnerable. Silver to move could capture the gold horse on c3 by stepping the silver elephant on d4 to c4, then b4, and then dislodging the gold dog on b3.
If one has two friendly pieces near a trap square, and the opponent has only one, it may be possible to capture by stepping through the trap square and pulling in the opposing piece. Silver to move could step the silver camel on f4 south, where it is safe due to the silver dog on g3, then pull the gold dog on f2 north to f3, capturing it.
One-move capture defense
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The most common and fundamental capture defense is to station an elephant next to a trap square. Since nothing can dislodge an elephant, nothing can be captured in that trap square until the elephant voluntarily moves away. At left, no gold pieces can be captured in the northwest trap square as long as the gold elephant remains on c5. If all stronger enemy pieces are tied up elsewhere, a single piece such as a horse may defend a trap square alone, but must beware of a changing situation which liberates any stronger opposing piece to move nearer and threaten it.
The second-most common and fundamental capture defense, mutual protection, is to station two or more pieces next to a trap square. This allows weaker pieces with numerical superiority to defend against one, and sometimes two strong attackers, but not three because three attackers can surround three sides of a trap. At left, Silver has defended the c6 trap square with a dog on b6 and a rabbit on c7. If Gold pushes the dog away, there will not be enough steps left to capture the rabbit. Gold to move can at most set up the threat of a capture for the following move, which gives Silver time to defend.
Occasionally one may wish to defend a trap square without bringing a second piece adjacent to it. In this case one generally obstructs the path of the attacking piece with friendly pieces. The gold camel on g6 can't capture the vulnerable silver horse on f7, because the silver cat on g7 is in the way, and the silver rabbits on h7 and g8 prevent the cat from being pushed. Gold could pull the cat to g6, but that would only make it a second defender of the northeast trap square.
The situation around the southwest trap square is straightforward. Because of the gold rabbit on b4, the silver elephant can't get adjacent to the gold dog on b3 with two steps left to dislodge it.
One counter-intuitive way to block the path of an opposing piece is to station a friendly piece on a trap square. Silver to move could push or pull the gold cat on f3 in a variety of ways, but always to a square from which it defends the f3 trap square long enough to prevent the capture of the gold dog on f2 for at least the present move. The disadvantage of this strategy is that the cat itself will be captured if the dog can be dislodged.
A last resort capture defense is scattering, i.e. retreating threatened pieces away from a trap to the edges and corners of the board. Scattering usually only delays captures, because the weak pieces can be frozen and eventually pulled into the trap which has been stripped of its defense. Furthermore, scattering away from a home trap may leave a hole through which an opposing rabbit can march to the goal. On the other hand, delaying captures for a few moves may buy time which is critical to making progress elsewhere on the board. This defensive technique is most often useful late in the game when defenders are few, which makes mutual protection less feasible, and goal threats are imminent, which leaves less time to hunt down fleeing pieces.