Arimaa/Print version

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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

Table of Contents

  1. Overview
  2. Playing The Game
  3. Introduction to Tactics
  4. Introduction to Strategy
  5. Relative Value of Pieces
  6. Advanced Tactics
  7. Positioning for an Attack
  8. Lone Elephant Attacks
  9. Elephant and Camel Attacks
  10. Elephant and Horse Attacks
  11. Elephant and Minor Piece Attacks
  12. Multi-Piece Swarming Attacks
  13. Camel and Horse Attacks
  14. Double-Trap Attacks
  15. Initial Piece Placement
  16. Other Attacking Ideas
  17. Arimaa Challenge History
  18. Sample Games
  19. Glossary
  20. Resources
  21. GNU Free Documentation License

Overview

Arimaa is a two-player board game invented by Omar Syed, a computer engineer trained in artificial intelligence. Inspired by Garry Kasparov's defeat at the hands of the chess computer Deep Blue, Syed wanted 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. In fact, "Arimaa" is "Aamir" spelled backwards plus an initial "a." In 2002 Syed published the rules to Arimaa and announced a $10,000 prize, available 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.

Arimaa was specifically designed so that it could be played using a chess set—an 8×8 board is used, and each player has sixteen pieces, in a 1-1-2-2-2-8 distribution. It can also be played on-line at the arimaa.com gameroom. In 2009, Z-Man Games began producing a commercial Arimaa set. Only one face-to-face tournament has taken place, but about 900 games are played on-line every week. Omar Syed hosts four events per year in the gameroom:

  • The World Championship is a tournament for human players which typically runs from January to March. Jean Daligault of France was the Arimaa world champion five times, in 2007, 2009, 2010, 2011 and 2013. On a scale of ratings comparable to Elo chess ratings, Daligault is rated near 2400.
  • The Computer Championship matches the top Arimaa bots in an elimination tournament. David Fotland's program Bomb placed first every year from 2004 to 2008. Jeff Bacher's Clueless prevailed in 2009, Mattias Hultgren's Marwin in 2010, and David Wu's Sharp in 2011; Marwin repeated in 2012; and Ricardo Barreira's Ziltoid won in 2013.
  • The Arimaa Challenge takes place following the Computer Championship. The top two bots are for a time available to play. The bot with the better record against humans who have played two games against each moves on to face three ultimate human defenders, who have been selected beforehand and prohibited from playing in the initial round. If this bot can defeat all three human defenders, its developer wins the $10,000 prize. So far, humans have dominated every challenge match.
  • The yearly Postal Mixer 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.

In addition to these events, the 1st Arimaa Online Festival has been organized for 11 September 2010. This event included Arimaa matches, a strategy workshop, and an interview with Omar Syed.

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

Understanding gameplay

Arimaa, like chess, is played on an 8×8 square board. The two players, Gold and Silver, control sixteen pieces each: one elephant (Elephant), one camel (Camel), two horses (Horse), two dogs (Dog), two cats (Cat), and eight rabbits (Rabbit), in descending order of strength. If Arimaa is played using a chess set, the pieces may be represented by the king, queen, knights, bishops, rooks, and pawns respectively. Unlike their chess counterparts, however, Arimaa pieces move only in cardinal directions, and must all step from square to square. The relative strength of each piece lies in its power to push, pull, or immobilize weaker enemy pieces. Four squares of the game-board are distinguished as trap squares, on which a piece can potentially be lost—these are the squares designated as c3, f3, c6, and f6 when classified by rank and file.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The players begin by setting up their pieces however they choose on their home rows. Highlighted are the four trap squares.
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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The game begins with an empty board. Gold arranges his pieces on the first and second ranks, in whatever configuration he chooses—Arimaa pieces do not have fixed starting positions. Silver then arranges his pieces on the seventh and eighth ranks. The diagram at right shows opening setups which are fairly common.

Once the pieces are on the board, Gold and Silver alternate turns, starting with Gold. A turn consists of one to four steps—each player is allowed four steps per turn and must take at least one, but can choose to pass on any of the rest. The steps of a turn can be distributed however the player chooses—four different pieces can each take one step, one piece can take four steps, two pieces can each take two steps, etc. A turn must make a net change to the position—one may not, for example, simply take one step left and one step right with the same piece, effectively passing the turn. A player who has ended two different turns on an identical position may not end another turn with that same position, no matter how far apart the turns are. As a further safeguard against indefinite loops of pieces moving back and forth, rabbits are prohibited from retreating homeward. Elephants, camels, horses, dogs, and cats may each step left, right, forward, or backward; rabbits may only step left, right, or forward.

The objective of the game is to move any 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. It is no easy task, however, to get a rabbit past an opposing army of stronger pieces—a player must out-maneuver the other army while at the same time keeping all enemy rabbits at bay.

The second diagram, portraying a position which could occur later in a game, helps illustrate the remaining rules of movement.

A friendly piece can pull or push a weaker enemy piece which is next to it, provided there is an empty square allowing for the necessary movement. To pull, a piece steps onto an adjacent empty square and drags the weaker enemy piece onto the square from which it came. The silver elephant on d5 could step to d4 (or c5 or e5, since the pulling piece and the pulled piece don't have to move in the same direction) and pull the gold horse from d6 to d5. To push, the weaker enemy piece is moved onto an adjacent empty square, and the piece which pushed it moves onto the square it had occupied. 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 already occupied. Remember also that a piece can only push or pull a weaker enemy piece—for example, a dog may dislodge an enemy rabbit or cat, but not a dog, horse, camel, or elephant. A push or pull uses two steps, since two pieces are moved. Although a rabbit may not retreat toward its home row, it can nonetheless be pushed or pulled in that direction. For example, the gold rabbit on g3 could not itself step back to g2, but the silver horse could take a step right and then pull it down.

A piece may not push and pull simultaneously—for example, the gold elephant on d3 could not push the silver rabbit from d2 to e2 and at the same time pull the other silver rabbit from c3 to d3.

A piece which is adjacent (in any cardinal direction) to a stronger enemy piece is frozen, unless it is also adjacent to a friendly piece. A piece which is frozen may not be moved by its owner, but may still be pushed or pulled by the opponent. The silver rabbit on a7 is frozen, but the one on d2 is able to move because it is next to another 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, as they are of equal strength. An elephant cannot be frozen per se, just as it cannot be pushed or pulled since there is nothing stronger. However, if enough enemy pieces are utilized against it, an elephant can be blockaded so that it has nowhere to go—this is possible because only one piece can be pushed at a time, regardless of relative strength.

A frozen piece can freeze another still weaker piece just as it would otherwise. The silver elephant on d5 freezes the gold horse on d6, which itself freezes the silver rabbit on e6.

A piece which enters a trap square is removed from the game unless there is a friendly piece adjacent to the square. The silver elephant could capture the gold horse by pushing it from d6 to c6. The silver rabbit on c3 is still alive because of the silver rabbit on c4 as well as the silver horse on c2—however, if the silver pieces protecting the c3 trap were to leave their positions, either voluntarily or by being dislodged, the silver rabbit on c3 would be lost.

Note that a piece may voluntarily step into a trap square, even if it is lost thereby. 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 horse from f2 to f3, which would be suicide for 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 enemy rabbit from a7 to a8, and when the dog is on a7, it unfreezes its own rabbit on b7, which can then step to b8 for the victory.

Note that an adjacent friendly piece does not protect against pulling or pushing. For example, on Silver's turn, its horse on c2 could push the gold cat from c1 to b1, or pull it to c2, even though the gold cat has a friendly piece next to it on d1.

Although most games conclude with a rabbit reaching goal, there are two other circumstances which can bring about an end.

  • Complete immobilization. If upon his turn a player has no legal move available, that player loses.
  • Loss of all rabbits. A player can win by capturing the last remaining enemy rabbit, even if he sacrifices his own last rabbit in the same turn.

Finally, if an enemy piece dislodges a rabbit onto its goal line and dislodges it off within the same turn, the game continues.

Setting up

As mentioned previously, Arimaa players decide how to arrange their pieces at the beginning of a game, with the only restriction being that they must be placed on the player's first two rows (i.e. the first and second ranks for Gold, and the seventh and eighth ranks for Silver.) The finer points of initial piece placement will be explored later, but for now you just want a setup that gives you flexibility and denies the opponent an easy attack plan. Most or all rabbits should start in the back row—it might seem logical to put the rabbits closer to their goal, but you will not achieve a goal without first clearing a path for it, and the strong pieces must take the lead in accomplishing this. Strong pieces should not begin stuck behind weaker pieces, and rabbits should not begin exposed to enemy forces, which could quickly pull them up and capture them (remember that unlike other pieces, rabbits cannot retreat homeward.) At first, think of your rabbits as forming a final barrier against an enemy goal.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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Gold has used the 99of9 setup; Silver has switched the dogs and cats.

The 99of9 setup is optimal for beginners. In the diagram at right, Gold has chosen the classic 99of9 setup. The elephant is placed in the center so that it can quickly attack any part of the board. The camel is placed in the center as well, but for defensive purposes. A centralized camel will make it harder for an enemy horse to attack on either wing. Furthermore, a centralized camel can't be pinned against the edge of the board by the enemy elephant as the first stage in taking a camel hostage.

The gold horses are set to move up to b3 and g3, to assert control of the c3 and f3 traps. A rabbit is placed forward on each flank, because a2 and h2 are likely to be safer places for rabbits than the centralized d1 and e1 would be.

Beginning back and center, the gold dogs will soon move up. The gold cats will most likely remain behind the traps for a while, ensuring that Gold will not lose a camel, horse, or dog in his own home trap.

Since Silver goes second, he can take Gold's setup into account when deciding on his own. At right, note that the silver elephant is not placed directly opposite the gold elephant. If the elephants directly face each other to begin, Gold can put Silver in an awkward spot by advancing the gold elephant four squares to fence in the silver elephant. Silver has used a variation of the 99of9 setup, switching the dogs and cats. Gold has chosen "cats behind the traps, dogs back", while Silver has used "dogs behind the traps, cats back". The strategic difference is subtle. Silver's dogs are guarding his home traps, and thus are potentially vulnerable to capture themselves—Silver stands to lose a dog in the opening if he's not careful. Silver is, however, compensated for this disadvantage in that his home traps are less vulnerable to takeover as long as they are guarded by multiple pieces, since the dogs could not be pushed aside by enemy dogs. Cats, however, are most commonly placed behind one's home traps, as most players prefer to have their dogs readily available for other purposes.

You may wonder why rabbits do not begin behind the home traps; while some players do in fact go this route, the downside is that a rabbit could be pushed or pulled into a protected trap and then wouldn't be able to step back, creating a mess for the home defender. Unless you are willing to quickly abandon a home trap, you need something stronger than a rabbit serving as its final defender. Typically, those who begin with rabbits behind their home traps are planning to play very aggressively, and thus don't mind losing control of their home territory.

Introduction to Tactics

In Arimaa, a tactic is usually a one- or two-turn plan which can be calculated precisely. Since there are up to four steps per turn, plans which aim farther than two turns ahead are usually impossible to calculate exactly, 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.

The most basic tactics to master are those that bring a friendly rabbit to goal or capture an enemy piece within four steps. If a player can't achieve victory or capture in a single turn, the opponent may bring in reinforcements which change the situation dramatically. Such contingencies require strategic rather than tactical planning.

One-move goal

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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Most simply, a rabbit may be able to move to the goal within one turn, even if the path 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 to goal. The rabbit will theoretically 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, if the gold cat on h7 moves to h6 while pulling the silver rabbit from g7 to h7, the gold rabbit on g6 can then reach the goal in the turn's final two steps. Note that the gold camel on f6 is lost when the rabbit advances, but reaching the goal is worth any sacrifice. Many beginners seem to prefer pushing over pulling, but note that a push is ineffectual in this case. If the gold 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 silver camel moving from b4 to b3. 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 sliding from g3 to f3, unfreezing the silver rabbit on f2, which could then step to the vacated g2 and then to g1, winning the game. The silver elephant would be lost as soon as the silver rabbit stepped away from the trap, but since Silver wins immediately, the elephant loss is of no consequence. With the silver rabbit beginning the turn blocked in all directions and also frozen, this one-turn goal is very easy to overlook.

Trap control

See also: Introduction to Strategy: Trap Control

The rules of Arimaa dictate that a piece is captured if it is located on a trap square and not next to any friendly pieces. In the case of the c6 trap, the relevant squares are b6, c5, c7, and d6. As long as Gold is able to maintain at least one piece on one of those four squares, Silver will never be able to capture any of Gold's pieces in that trap. If both players are able to maintain at least one piece next to the trap, neither player will be able to make a capture.

The decision of which traps are worth fighting for and how to allocate forces is strategic, but the mechanics of fighting for control of a trap are in the realm of tactics. In particular, at any given point in time a trap is either owned by Gold, owned by Silver, or contested.

One-move capture

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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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 a player 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

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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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. However, mutual protection cannot defend against three attackers because they 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 players may wish to defend a trap square without bringing a second piece adjacent to it. In this case a player 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).

False protection

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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After one has learned the basic ways of defending a trap square, one may be lured into a false sense of security by an illusion of defense. At right, Silver has two pieces next to the northwest trap square, but since both pieces can be dislodged, Gold has a one-move capture. First the gold elephant on b5 can push the silver dog from b6 to c6, and then the newly unfrozen gold cat on b7 can push the silver rabbit from c7 to c8, capturing the dog. Alternatively Gold can capture the rabbit by pushing or pulling the dog away from b6, then pushing the rabbit from c7 into c6.

Around the northeast trap square, the gold camel is blocked from pushing the silver horse into the trap square in one move, but not from pulling it there. Gold could first step the gold rabbit on g5 to f5, making it safe for the gold camel to step to f6, after which the gold camel could step either east or west while pulling the silver horse south and capturing it.

In the southwest corner Gold has calculated that even if the silver elephant pushes the gold rabbit from b4 to a4, the gold dog on b3 is safe from a push. This is correct but insufficient, because the silver elephant may still pull the gold dog from b3 to b4, capturing the gold horse on c3.

The situation around the southeast trap square demonstrates that an enemy piece in a trap square provides less protection than a friendly one. Silver may voluntarily step east or west with the dog on f3, and still have three steps left to enter the trap square with the camel on f4, then pull the gold dog from f2 to f3, capturing it.

The fork

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The gold elephant has forked the silver dog between the c3 and f3 traps

The fork (or hostage fork) threatens a single frozen piece with capture in two different traps. It is the most basic of the two-move tactics: On the first move an enemy piece is threatened, and on the second move it is inevitably captured. In the diagram at left, from this game, Gold has threatened to capture the silver dog in either the c3 or the f3 trap square. Silver to move would need one step to defend the c3 trap square with the elephant, and three steps to defend the f3 trap square with the cat. Unfortunately for Silver, the gold rabbit on e4 is threatening to reach goal next move, so Silver must spend at least one step defending the goal, and the silver dog is lost.

Forks resulting in immediate material gain are more common later in the game when fewer defenders are available, but even if both relevant traps can be defended, holding a hostage near one or two traps can be an effective strategy to commit enemy pieces to defensive positions.

The fence

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The silver horse on c4 has been fenced in; Silver to move can't save it.

The fence (or hostage fence) is a two-move tactic somewhat less common than the fork. On the first move a hostage is brought next to a trap, and hemmed in on two sides. The piece may be unfrozen from a third side, but then its only direction of retreat would be into the trap. In the diagram at right, from this game, Gold has fenced the silver horse next to the c3 trap square. Silver is on move, but can't prevent the capture of the horse. The silver elephant could indeed unfreeze the fenced horse by moving to c5 (incidentally capturing the gold rabbit in the c6 trap square), but the technically unfrozen silver horse can't then escape, as its only available move would be suicide. Furthermore, the silver elephant can't get adjacent to the c3 trap in four steps, a blocking motif common in the opening when the board is still crowded.

Limitations of tactics

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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There are no forced captures even in this sharp opening.

Beyond the above-listed tactics, there are not many ways to achieve immediate material gain in Arimaa. Particularly at the beginning of the game, defenders are plentiful, and captures can be postponed. The position at left arose from a relatively sharp opening in this game. Gold advanced all the heavy pieces, and Silver on the last move froze three of them. The horse on f4 and camel on e5 are both threatened with capture in the northeast trap square. Also the gold horse on d4 can be pushed to d6 with a hostage fork, if the gold elephant evacuates d6. Yet in spite of these multiple threats, Gold can move to prevent any captures in at least the next three moves of Silver, certainly beyond the ability of humans or computers to analyze exhaustively.

In fact, it may not be obvious to inexperienced Arimaa players that Silver is winning. Material is even, and Gold, with more pieces activated, may appear more able to make the first capture. Gold could press the attack by stepping cat to g4, horse to f5, and camel to e6, with a step to spare. (Perhaps dog to h3 to prepare to defend the southeast trap square again.) It is entirely a strategic judgement that the vulnerability of the advanced gold pieces is, in this particular position, a greater liability than their attacking power is an asset.

After becoming acquainted with the most basic Arimaa tactics, you should turn your attention to long-term strategy, which will better enable you to get into positions where tactics are of use.

Introduction to Strategy

Arimaa strategy involves straightforward concepts which overlap in complex ways. Very different styles of play remain popular; some players are very aggressive, quickly attacking enemy territory with the aim of taking over one or both of the enemy's home traps. Others prefer to play a "home game", and try to capture enemy pieces mainly by pulling them down. It is not yet clear what is the optimal balance between offense and defense.

In such an environment, it pays to become somewhat familiar with all of the basic concepts, before focusing on fine detail. After gaining some experience and seeing different strategies in action, you can revisit this material for a closer reading, and perhaps even bring it up to date.

Initial Setup

Understanding gameplay

Arimaa, like chess, is played on an 8×8 square board. The two players, Gold and Silver, control sixteen pieces each: one elephant (Elephant), one camel (Camel), two horses (Horse), two dogs (Dog), two cats (Cat), and eight rabbits (Rabbit), in descending order of strength. If Arimaa is played using a chess set, the pieces may be represented by the king, queen, knights, bishops, rooks, and pawns respectively. Unlike their chess counterparts, however, Arimaa pieces move only in cardinal directions, and must all step from square to square. The relative strength of each piece lies in its power to push, pull, or immobilize weaker enemy pieces. Four squares of the game-board are distinguished as trap squares, on which a piece can potentially be lost—these are the squares designated as c3, f3, c6, and f6 when classified by rank and file.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The players begin by setting up their pieces however they choose on their home rows. Highlighted are the four trap squares.
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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The game begins with an empty board. Gold arranges his pieces on the first and second ranks, in whatever configuration he chooses—Arimaa pieces do not have fixed starting positions. Silver then arranges his pieces on the seventh and eighth ranks. The diagram at right shows opening setups which are fairly common.

Once the pieces are on the board, Gold and Silver alternate turns, starting with Gold. A turn consists of one to four steps—each player is allowed four steps per turn and must take at least one, but can choose to pass on any of the rest. The steps of a turn can be distributed however the player chooses—four different pieces can each take one step, one piece can take four steps, two pieces can each take two steps, etc. A turn must make a net change to the position—one may not, for example, simply take one step left and one step right with the same piece, effectively passing the turn. A player who has ended two different turns on an identical position may not end another turn with that same position, no matter how far apart the turns are. As a further safeguard against indefinite loops of pieces moving back and forth, rabbits are prohibited from retreating homeward. Elephants, camels, horses, dogs, and cats may each step left, right, forward, or backward; rabbits may only step left, right, or forward.

The objective of the game is to move any 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. It is no easy task, however, to get a rabbit past an opposing army of stronger pieces—a player must out-maneuver the other army while at the same time keeping all enemy rabbits at bay.

The second diagram, portraying a position which could occur later in a game, helps illustrate the remaining rules of movement.

A friendly piece can pull or push a weaker enemy piece which is next to it, provided there is an empty square allowing for the necessary movement. To pull, a piece steps onto an adjacent empty square and drags the weaker enemy piece onto the square from which it came. The silver elephant on d5 could step to d4 (or c5 or e5, since the pulling piece and the pulled piece don't have to move in the same direction) and pull the gold horse from d6 to d5. To push, the weaker enemy piece is moved onto an adjacent empty square, and the piece which pushed it moves onto the square it had occupied. 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 already occupied. Remember also that a piece can only push or pull a weaker enemy piece—for example, a dog may dislodge an enemy rabbit or cat, but not a dog, horse, camel, or elephant. A push or pull uses two steps, since two pieces are moved. Although a rabbit may not retreat toward its home row, it can nonetheless be pushed or pulled in that direction. For example, the gold rabbit on g3 could not itself step back to g2, but the silver horse could take a step right and then pull it down.

A piece may not push and pull simultaneously—for example, the gold elephant on d3 could not push the silver rabbit from d2 to e2 and at the same time pull the other silver rabbit from c3 to d3.

A piece which is adjacent (in any cardinal direction) to a stronger enemy piece is frozen, unless it is also adjacent to a friendly piece. A piece which is frozen may not be moved by its owner, but may still be pushed or pulled by the opponent. The silver rabbit on a7 is frozen, but the one on d2 is able to move because it is next to another 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, as they are of equal strength. An elephant cannot be frozen per se, just as it cannot be pushed or pulled since there is nothing stronger. However, if enough enemy pieces are utilized against it, an elephant can be blockaded so that it has nowhere to go—this is possible because only one piece can be pushed at a time, regardless of relative strength.

A frozen piece can freeze another still weaker piece just as it would otherwise. The silver elephant on d5 freezes the gold horse on d6, which itself freezes the silver rabbit on e6.

A piece which enters a trap square is removed from the game unless there is a friendly piece adjacent to the square. The silver elephant could capture the gold horse by pushing it from d6 to c6. The silver rabbit on c3 is still alive because of the silver rabbit on c4 as well as the silver horse on c2—however, if the silver pieces protecting the c3 trap were to leave their positions, either voluntarily or by being dislodged, the silver rabbit on c3 would be lost.

Note that a piece may voluntarily step into a trap square, even if it is lost thereby. 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 horse from f2 to f3, which would be suicide for 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 enemy rabbit from a7 to a8, and when the dog is on a7, it unfreezes its own rabbit on b7, which can then step to b8 for the victory.

Note that an adjacent friendly piece does not protect against pulling or pushing. For example, on Silver's turn, its horse on c2 could push the gold cat from c1 to b1, or pull it to c2, even though the gold cat has a friendly piece next to it on d1.

Although most games conclude with a rabbit reaching goal, there are two other circumstances which can bring about an end.

  • Complete immobilization. If upon his turn a player has no legal move available, that player loses.
  • Loss of all rabbits. A player can win by capturing the last remaining enemy rabbit, even if he sacrifices his own last rabbit in the same turn.

Finally, if an enemy piece dislodges a rabbit onto its goal line and dislodges it off within the same turn, the game continues.

Setting up

As mentioned previously, Arimaa players decide how to arrange their pieces at the beginning of a game, with the only restriction being that they must be placed on the player's first two rows (i.e. the first and second ranks for Gold, and the seventh and eighth ranks for Silver.) The finer points of initial piece placement will be explored later, but for now you just want a setup that gives you flexibility and denies the opponent an easy attack plan. Most or all rabbits should start in the back row—it might seem logical to put the rabbits closer to their goal, but you will not achieve a goal without first clearing a path for it, and the strong pieces must take the lead in accomplishing this. Strong pieces should not begin stuck behind weaker pieces, and rabbits should not begin exposed to enemy forces, which could quickly pull them up and capture them (remember that unlike other pieces, rabbits cannot retreat homeward.) At first, think of your rabbits as forming a final barrier against an enemy goal.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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Gold has used the 99of9 setup; Silver has switched the dogs and cats.

The 99of9 setup is optimal for beginners. In the diagram at right, Gold has chosen the classic 99of9 setup. The elephant is placed in the center so that it can quickly attack any part of the board. The camel is placed in the center as well, but for defensive purposes. A centralized camel will make it harder for an enemy horse to attack on either wing. Furthermore, a centralized camel can't be pinned against the edge of the board by the enemy elephant as the first stage in taking a camel hostage.

The gold horses are set to move up to b3 and g3, to assert control of the c3 and f3 traps. A rabbit is placed forward on each flank, because a2 and h2 are likely to be safer places for rabbits than the centralized d1 and e1 would be.

Beginning back and center, the gold dogs will soon move up. The gold cats will most likely remain behind the traps for a while, ensuring that Gold will not lose a camel, horse, or dog in his own home trap.

Since Silver goes second, he can take Gold's setup into account when deciding on his own. At right, note that the silver elephant is not placed directly opposite the gold elephant. If the elephants directly face each other to begin, Gold can put Silver in an awkward spot by advancing the gold elephant four squares to fence in the silver elephant. Silver has used a variation of the 99of9 setup, switching the dogs and cats. Gold has chosen "cats behind the traps, dogs back", while Silver has used "dogs behind the traps, cats back". The strategic difference is subtle. Silver's dogs are guarding his home traps, and thus are potentially vulnerable to capture themselves—Silver stands to lose a dog in the opening if he's not careful. Silver is, however, compensated for this disadvantage in that his home traps are less vulnerable to takeover as long as they are guarded by multiple pieces, since the dogs could not be pushed aside by enemy dogs. Cats, however, are most commonly placed behind one's home traps, as most players prefer to have their dogs readily available for other purposes.

You may wonder why rabbits do not begin behind the home traps; while some players do in fact go this route, the downside is that a rabbit could be pushed or pulled into a protected trap and then wouldn't be able to step back, creating a mess for the home defender. Unless you are willing to quickly abandon a home trap, you need something stronger than a rabbit serving as its final defender. Typically, those who begin with rabbits behind their home traps are planning to play very aggressively, and thus don't mind losing control of their home territory.

Early Ideas

Direct Goal Is Impossible

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
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Gold has no way to make progress with the h6 rabbit

The most straightforward strategy in Arimaa is to advance a rabbit and some strong pieces, attempting to rip a hole in the opponent's defenses through which the rabbit can score a goal. If both players try to do this, the game turns into a race. Each player in a race must judge how many steps to spend delaying the opposing rabbit, and how many steps to use furthering their own rabbit.

Before long, however, the Arimaa community discovered that if one player tries to force goal in the opening while the other player defends, advantage accrues to the defender. The board is too crowded for rabbits to make headway until some defenders have been captured, but the defender can easily protect his troops from capture while they are all still at home. Meanwhile the attacking rabbit is vulnerable to capture in the defender's home traps. In the diagram at right, the h6-rabbit is no threat to reach the goal, but the rabbit itself is in danger of capture in f6. Furthermore, rabbits can't retreat, so the h6-rabbit can't save itself. Finally, if the gold elephant advances to g6 to protect the h6-rabbit, the elephant will be blockaded.

Capturing Is Impossible Too?

In light of the futility of forcing an early goal, top Arimaa players next turned to capturing pieces as the most plausible strategy. Each player can easily defend his home traps, so the main offensive strategy was to drag an opposing piece to one's own home traps for capture. (An elephant can safely go hunting for a piece to pull home, because elephants aren't vulnerable to capture.) Unfortunately, this strategy also initially appears futile, because the opposing elephant can camp out by the trap in which the dragged piece has been threatened, and permanently make that trap safe.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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Silver has defended against an elephant with an elephant, and against a camel with a camel

Since a defensive elephant can make any one trap safe forever, attackers resort to overloading the defending elephant by simultaneously threatening to capture two different pieces in two different traps. Yet even this may lead to one more round of frustration. If you make a capture threat with your elephant, the opposing elephant will block it, and if you make a second capture threat with your camel, the opposing camel will block it. In the diagram at left, Gold is apparently stymied despite having generated a capture threat in each home trap.

Indeed, one might wonder why pieces of equal strength don't stalemate each other on down the line, resulting in a quagmire where neither side can ever make progress. Some Arimaa players feared that the game was by nature too defensive for attacking play to ever pay off.

The strategic answer to this conundrum, namely taking a camel hostage, was the first deep strategy ever discovered in Arimaa. It gave humans their first advantage over computer programs in what had been, before then, a nearly equal contest. Indeed, to this day the fact that neither player can afford to give up a camel hostage informs every opening strategy. It makes the lone elephant attack ultimately effective, and discourages counter-attack when one launches an elephant-horse attack.

Throughout the opening and midgame, a player who can't punish an opposing camel advance will likely be unable to make any progress whatsoever, because most trouble apart from a camel hostage can be relieved by the opposing elephant and/or camel. Applying secondary strategies first, even successfully, will only lead to frustration when the opposing heavy pieces sail to the rescue. Gaining a camel hostage, in contrast to subtler plans, not only gives one an advantage, it also makes the resulting position simpler and easier to play. Therefore the camel hostage should be every beginner's first study after the basic tactics.

Camel Hostage

Strongest Free Piece

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
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a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The gold elephant is holding the silver camel hostage.

In the position at right, the gold elephant is holding the silver camel hostage. The camel is frozen, so it can't escape, and on any turn Gold could flip or pull it into the c3 trap. To preserve the camel, Silver must station his elephant next to the c3 trap. Silver can't defend with less than the elephant, because any other silver defender could itself be captured in c3, unless there were two silver defenders securely in place, which would be hard to maintain if the silver elephant were elsewhere.

Herein lies the answer to the defensive conundrum of equal forces stalemating each other. Gold's elephant is tying up Silver's elephant and camel both, leaving Gold's camel unopposed as the strongest free piece. As long as the camel hostage remains in place, Gold can dominate the other side of the board. The silver camel obviously can't defend f3, and at present the silver elephant can't defend f3 without losing the silver camel.

Horses Advance

Gold might be inclined to immediately advance the free camel, but advancing horses is much safer. Imagine in this diagram that instead of the horse, the gold camel were on a6. Silver to move could send his elephant to b6, setting the stage for an even camel trade, which would be a disappointing outcome for Gold. Since the camel captures would each use all four steps of a turn, neither elephant could come back and save its own camel.

The gold horse on the west wing is actually not attacking so much as it is defending the c6 trap. The hostage camel keeps Silver's elephant next to the c3 trap, but it can still dart around as long as it is on c4, d3, or even c2 at the end of a turn. This mobility could allow the silver elephant to dislodge a small gold piece and gradually move it up. If the silver elephant could get a weak gold piece into silver territory, and Gold had nothing there to protect it, that gold piece would soon be lost while the silver camel would remain safe. The gold horses are prepared to defend the c6 and f6 traps, short-circuiting any such shenanigans by Silver. The gold horse in the west must keep some distance from the silver elephant, to avoid being frozen on c5 and then captured in two steps.

As long as the camel hostage is intact, Gold's best bet is to use his horse on the east wing to drag down one little piece at a time for capture in f3. This seemingly tedious strategy can ultimately decimate Silver's forces, if the silver camel and elephant both remain away from the action. Silver rabbits in the northeast are especially vulnerable, since Silver cannot hope to shield them all, and a rabbit cannot retreat backward. With the silver camel stuck in the west and the silver elephant committed to defending it, any available silver piece which tries to protect the f3 trap will itself be captured by the gold camel. On g3, the gold camel can deter silver horses from advancing, or capture them if they are reckless enough. The gold horses are the pieces which should operate away from home territory for the time being. Even if a smaller gold piece were to get pulled up, the gold horses could defend the c6 and f6 traps, and still deliver smaller silver pieces to the f3 trap. Depending on what Silver does, Gold may later want to advance his camel, but should keep f3 protected, and avoid offering Silver a camel trade.

Realizing that Gold is poised to take control, what should Silver do? Silver needs to free his elephant. It will be hard to save the camel, but in this case Silver can contain the damage. If Silver moves his elephant to b6, and the silver camel is then captured, Silver can pull the gold horse from a6 into c6, capturing it in return. Alternatively, if Silver still wanted to try to preserve his camel, he could unfreeze it with his horse and push the gold dog away, burrowing the camel so that it could not be captured in one turn—this is an option only because Gold has neglected to put a piece on b2, which would have blockaded any a3 hostage in place. Even as things are, however, burrowing the camel would only be a delaying tactic by Silver, since the camel would hardly be free, and could still be captured if the silver elephant left.

Active Defense

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The silver elephant has regained mobility, so Gold has no advantage from holding a camel hostage

Whether or not one can get immediate compensation for a camel loss, it is crucial that one's elephant be freed from a hostage which gives the enemy the strongest free piece. The best answer to a camel hostage is to rotate out one's elephant. Two other pieces can defend the trap if they have support behind them. In this game, diagrammed at right, Gold is strategically lost because the arrival of the silver dogs has freed the silver elephant to roam, whereas the gold elephant must stay put to prevent an immediate goal. Suddenly the tables have turned and Silver's elephant is the strongest free piece.

This is the last and most subtle reason for the gold camel to stay home on defense when Gold has the silver camel hostage. Imagine that the gold camel were at home in the diagram at right, lurking on d2 for example. Once the silver elephant left the f3 trap, the gold camel could confront the new silver defenders. Silver would then have to scramble to prevent multiple captures.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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Gold must now attack Silver's home traps

The most active defense against a camel hostage is to bring up pieces either numerous enough or strong enough to contest both enemy traps. In this game, diagrammed at left, Gold can't make a capture in either c3 or f3, despite holding the silver camel hostage. The gold camel would like to help regain control of c3, but the silver elephant is well-positioned in the way.

Who has the strongest free piece now? The silver elephant is more free than the gold elephant, but it isn't totally free, because if it crosses back to the silver side of the board, the gold camel will smash across into the silver horses around c3. Likewise, the gold camel isn't really a free piece, because it needs to lurk around in order to keep the silver elephant committed to defending c3.

As it turns out, the gold horses are suddenly the strongest free pieces. Gold can't yet force a capture at home, but he can use both horses to take over an enemy trap (most likely f6), threatening captures and ultimately goals. Silver might then move his elephant to the defense of that trap, and further threaten one or both of the gold horses with capture, but in that case Gold will stand to gain even more than that in c3 and f3. Gold's upper hand traces back to the camel hostage.

These few positions are only scratching the surface of possible play when a camel is held hostage, but they should give an inkling both of how a camel hostage gives one an advantage, and of how it is the foundation of much deeper strategies.

Elephant Blockade

Strongest Free Piece Revisited

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
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The silver elephant has no legal move, so the gold elephant is the strongest free piece.

Not long after discovering the camel hostage strategy, human players accidentally discovered that some computer programs could be lured into an elephant blockade by the offer of a free piece for capture. Much later it was further discovered that the program may squirm to try to free its elephant, and in the process get jammed all the way against an edge of the board.

In this game, diagrammed at right, Gold has lost a cat while Silver has a full army, but Silver is nonetheless completely lost. Not only is there no empty square for the silver elephant to step into, there is no empty square into which it can push its tormentors. The gold elephant and camel will keep an eye on the silver camel and horses respectively. Gold can ward off any attempts to break the silver elephant out of its prison, while herding silver pieces at will into the f3 trap. This is a much more decisive advantage than a camel hostage. Silver has no plausible lines of play in this case.

Rotation

Most elephant blockades, however, are not as hugely advantageous as that one. It is a rare opponent who will voluntarily move his elephant to the edge of the board when a blockade is looming. The diagram below left, from this game, shows a slightly less advantageous situation with the blockaded elephant one square away from the edge of the board.


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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
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The gold elephant is blockaded, but the game is not over.

Here the gold elephant can't move, true, but nine silver pieces are required to maintain the blockade, including both the silver elephant and camel. Indeed, if all the pieces involved in the blockade stayed put, the strongest free piece would actually be the gold camel.

As it happens, however, Silver can undertake a rotation (or replacement) of the pieces participating in the blockade. When it comes to being in the way, a weak piece serves just as well as a strong one. Silver to move can free his camel for duty in only four steps, while maintaining the blockade: camel h6 south, rabbit h7 south, rabbit h8 south, and rabbit g8 east. The gold elephant can't make use of g8 to dig its way to freedom without getting smothered against the edge, so Silver can fill in that hole next move. Thus Silver needs only one turn to equal Gold for having the strongest free piece.

Furthermore, if Gold plays passively, Silver can continue to rotate pieces, freeing his elephant as well in two or three more turns. Because of this threat, it is very important that Gold not remain passive. Gold must immediately begin preparing a rescue mission to erode the blockade from the side, or even from the front if the silver elephant tries to leave. This will necessarily expose gold pieces to danger, but at least it puts some play into the position. For Gold to hang back is to await execution.

Note that even if Silver manages to rotate the elephant out of the blockade, it will require a few more pieces to maintain than a blockade on the edge. Furthermore those pieces will protrude one square further, making them slightly easier targets for would-be blockade busters. Still, the blockade is quite advantageous to Silver.

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a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
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Silver has blockaded the gold elephant, but would gladly give up the blockade to take the Gold camel hostage.

When Rotation Is Impossible

The diagram at right, from this game in the 2006 Arimaa World Championship, features an elephant blockaded one step further from the edge of the board, which is correspondingly less advantageous. Indeed, it is no longer realistic for Silver to expect to be able to free his elephant by rotating blockaders appropriately. True, the f6 trap is participating in the blockade at the moment, but Gold might bring a piece to f5 or e6, allowing the gold elephant to step to freedom, so Silver must soon occupy at least the latter squares.

An elephantless blockade would require silver pieces on g8, f7, g7, h7, e6, h6, f5, g5, h5, and g4. Not only are ten pieces necessary, but the bubble of blockaders also presents a large surface area for Gold to assail, extending within two steps of Gold's home trap at f3. Silver would be too busy warding off threats to the blockade to ever start capturing pieces with his freed elephant.

Since Silver can't rotate his elephant out of this blockade in practice, it does not give him the strongest free piece like an ideal blockade does. Yet the blockade is not worthless. Its value is that, although both Gold and Silver have a free camel, the silver camel is more free.

Suppose that Silver, while maintaining the blockade, were to use his camel to attack the c3 trap. Gold could defend c3 with his own camel, but couldn't endanger the attacking silver camel. In contrast, if Gold were to attack c6 with his camel, Silver would have the option of giving up the blockade to cross wings and take the gold camel hostage. (This principle recurs again and again in the study of elephant mobility.)

This difference of freedom pegs the value of the blockade to Silver at somewhat less than a camel hostage. Silver certainly can't expect to get more out of the position, because if Gold is willing to give up his camel as a hostage, he can frustrate anything else Silver might undertake. Indeed, the gold camel can probably break the blockade at any time if it is willing to expose itself. On the other hand, Silver can't necessarily force Gold to expose his camel. Gold can play in the west as well as hovering in the east making threats to break the blockade. If Silver has trouble generating a threat in the west while maintaining the blockade in the east, he may be forced to give up the blockade for an advantage smaller than a camel hostage.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The silver elephant is too close to the center for a blockade to be reasonable.

Worthless Elephant Blockades

All the top Arimaa computer programs are now aware of elephant blockades, but most are still confused by the huge disparity in value from one blockade to the next. In the diagram at left, from this game, Gold's blockade of the silver elephant actually has negative value, for several reasons.

  • Gold will never be able to rotate the gold elephant out of d4.
  • The gold camel is so buried on c2 that it has less freedom of movement than the silver camel, despite the theoretical difference in elephant mobility.
  • The gold pieces on e4 and f3 (at the edge of the blockade bubble) are in jeopardy.
  • The blockade is unmaintainable. Silver doesn't actually need to break the blockade in order to make progress, but could break it if necessary.

Elephant Mobility

Partial Freedom

As long as an elephant is not blockaded, it retains a theoretical freedom to roam the board. There are, however, fine distinctions in exactly how free a free elephant is. A whole spectrum of positional features may limit an elephant's ability to switch its field of operation. Extreme contraints include guarding against an imminent goal, or guarding against the capture of a camel hostage. The range runs all the way down to minor constraints such as guarding against a rabbit being captured, or even guarding against a small positional disadvantage the opponent might create in the elephant's absence.

At the start of each game, both elephants are free, but differences in elephant mobility soon emerge. Some observers are astonished that tiny advantages in Arimaa seem to snowball; it is easy to underestimate how a more mobile elephant can pose increasingly tricky defensive puzzles for a less mobile elephant to solve. As the game progresses, one or both elephants tend to become more and more committed to the defense of a particular trap. When an elephant finally can't leave a trap without its army suffering captures in that trap, that elephant has lost mobility in a clearly identifiable sense.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The gold elephant is pinned to the defense of the horse on c6, which is framed.

Frames

A piece which is on a trap square, surrounded on three sides by opposing pieces which prevent it from pushing its way off the trap square, has been framed. The lone friendly piece providing support is pinned, because as soon as it moves, the piece on the trap square will disappear. In the position at right, from this game, the gold elephant is pinned to the defense of the gold horse which has been framed on c6.

Pins are most effective in the opening when the board is crowded, so that strong pieces participating in the frame can be rotated out and replaced with the plentiful weaker pieces. At right Silver can rotate the a5-horse to c5, making the silver elephant the strongest free piece. If not enough weak pieces are available for rotation, though, the side maintaining the frame will have to commit three pieces at least as strong as the piece being framed, which may not be worthwhile on a relatively empty board.

At right the silver camel has a freedom to attack that the gold camel does not. The silver elephant can switch wings at any time to threaten the gold camel at no cost other than giving up the frame. In contrast, the gold elephant can't threaten the silver camel without abandoning the framed gold horse to instant capture.

A pin is not as absolute as a blockade, because the pinned piece may choose to leave at any time. If the silver camel becomes too exposed, the gold elephant may cross over in a single move, and possibly even force the capture of the silver camel. More subtly, if Silver rotates the a5-horse to c5 and then attacks the east wing with the silver elephant, the gold elephant may be able to abandon the framed horse and in the same move push the silver horse from c5 to c4, threatening to capture it in the following turn. It would be a huge relief for Gold to abandon the pin in a way that resulted in only an exchange of horses rather than a loss.

A frame may be broken by the arrival of a piece strong enough to dislodge the framing piece or pieces on one side of the trap. If a piece on a trap gains a second supporter, it is no longer framed. At right only the gold camel would be strong enough to break the frame. Gold must judge whether breaking the frame is worth the danger of the gold camel being taken hostage, as well as considering the damage that the silver camel will do on the east wing in the mean time.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The gold elephant is nearer the center than the silver elephant.

Horse Hostages

An elephant holding a camel hostage is only one example of a whole class of positions where a frozen piece is held in danger of capture, committing the opponent to defense. It is, however, the most effective. Suppose, for example, that an elephant holds a horse hostage instead. This commits the opposing elephant to defense, true, but who has the strongest free piece? The defending elephant can't leave without losing a horse, but the hostage-holding elephant often can't leave either, because the newly-freed horse would cooperate with the formerly-defending elephant to wreak havoc at that trap.

In the diagram at left, from this game, Silver is holding a hostage, but neither elephant wishes to leave the northeast trap square. Indeed, it is the gold elephant which has the greater mobility because it is two steps closer to the center, and is thus a greater threat to the silver camel than the silver elephant is a threat to the gold camel. Since the gold camel has more freedom of movement, Gold may be able to make progress in this position despite having a horse held hostage.

Silver's hope in this position must be either to frame the hostage horse and rotate out the silver elephant (see above), or to pass the hostage horse off to the silver camel (see below), which would also free the silver elephant.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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It is difficult to maintain a horse hostage with a camel on the side of a trap if the opposing elephant is nearby.

After an elephant holding a camel hostage, the second-most effective hostage situation is a camel holding a horse hostage. Ideally this ties the opposing elephant to defense, while the friendly elephant is free to roam. However, the situation is often unstable because the "defending" elephant can attack the hostage-holding camel, freeing the hostage horse, and possibly creating offensive threats. This means that the "free" elephant must work as quickly as possible to score material gain, before the hostage situation collapses.

When a horse is held hostage by a camel, the deeper the hostage is held, the greater the threat. In the diagram at right, the gold elephant has no easy way to approach the silver camel and switch to offense. In contrast, if a hostage-holding camel has no friendly pieces in front of it, the opposing elephant can more easily free the horse. At right, the silver elephant can pull the gold camel to b4 in three steps (returning to c4 after the pull) and the final step can be used to move the silver horse to b3. This would threaten the gold cat with capture on the following turn. Also the silver elephant has the option of flipping the gold camel to c5 on the following turn, threatening another capture.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The gold dog would be most valuable as a hostage if held by a horse.

Cat and Dog Hostages

A smaller piece may also make a valuable hostage. In the diagram at left, from this game, Silver has an advantage due to the gold dog held hostage by the silver camel. The silver elephant is a free piece, which will try to pull something to capture in the f6 trap. The Gold elephant will be overloaded if it must simultaneously defend c6 and f6.

However, the situation is less than ideal for Silver, for two reasons. First, the silver cat on c6 means that the hostage dog isn't threatened with one-turn capture. If the gold elephant leaves, it will take five steps for Silver to capture the hostage dog, giving the gold elephant that much more mobility. Second, the gold rabbit on h5 gives the gold camel more freedom to advance without fear of being taken hostage by the silver elephant. Silver would like to send his elephant after the gold camel, but in practice will probably have to try to capture the h5-rabbit before this is feasible.

In general, the advantage of a small piece hostage over a horse hostage is twofold. First, the smaller piece can be held hostage by a horse, sparing both the friendly elephant and the friendly camel for other duty. Second, it is easier to frame a small piece than to frame a horse. In the diagrammed position, Silver would have a greater advantage if the c7-camel were swapped with the g6-horse. Then the silver camel could defend the f6 trap while the silver elephant either hunted for a second piece to drag for capture, or assisted in framing the hostage dog. As it stands, the silver elephant can't help frame the c8-dog, because of the damage the gold camel would do in the mean time.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The gold elephant is cut off from the center.

The Center

A more subtle, positional aspect to elephant mobility is the ability of the elephant to access the areas of most importance at the current stage of the game. From the centre four squares of the board the elephant has access to all four traps, so as a general rule it is better for the elephant to be centrally located. Also, a centralized elephant is almost impossible to blockade. On the other hand, in order to attack or defend a trap, an elephant must decentralize itself at least to one of the eight squares in the ring around the center, and perhaps further if it is hunting down a fleeing piece. The possibilities available to an elephant in the center usually remain mere possibilities until the elephant leaves the center to pursue one of them.

It is occasionally possible to station a clump of friendly pieces in the center of the board, which can cut off the opposing elephant's ability to switch wings at will. In the diagram at right, from this game, Gold is losing because the gold elephant can't access the f3 trap.

This strategy must be used with great caution, however, because if the opponent is able to erode the dividing wall, there will probably not be time to retreat all of the participating pieces to safety. At right, if the gold elephant could break through the wall, some silver piece would likely perish in f3. In general, pieces other than the elephant should stay out of the center. It is asking for trouble to give the opposing elephant a target in the center where that elephant wants to be anyway.

Race Positions

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The players are racing to capture pieces as quickly as possible

Race positions occur when the two elephants each take control of an opposing trap as in this game diagrammed at right. In a race position, both elephants are clearly free, so elephant mobility is of secondary importance. The critical issue is which elephant can do more damage more quickly.

Race positions are rare because they typically require one player or the other to have misjudged the position in a costly way. The player who is going to lose the race shouldn't have raced in the first place. The ultimate loser will wish he could rewind the game to the move on which he used his elephant to attack rather than defend.

By the same token, however, it is important to know how to play in a race position, so as to be able to judge beforehand whether or not to race. One strategic point is that an elephant assisted by other pieces usually makes progress faster than an unaided elephant. Another point is that an advanced rabbit makes an attack much more potent by threatening to make goal, and may force the opposing elephant to break off its attack, wasting a couple of moves to cross over and defend.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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Goal threats trump capture threats.

In the position at left, from this game in the 2006 Arimaa World Championship, Silver is on move. Each player is threatening goal. As soon as one player is forced to defend with his elephant, the game will essentially be over, because the other player will capture the little attacking pieces near the trap where the attack has just been abandoned. In other words, both players have staked the game on the outcome of this race.

Silver should consider using his camel to push the d6-rabbit back to d5. It may endanger the silver camel to bring it nearer to the gold elephant, but in a goal race it may be worth sacrificing a camel to buy time. Meanwhile if Gold ends Silver's goal threat by capturing the g3-rabbit, the silver elephant will respond by heading home for defense and capturing the gold f4-camel in f6. Therefore Gold would have to keep racing instead.

Because the cost of losing a race is so high, and because the side with more pieces committed usually makes progress faster, counter-attacks in Arimaa are rare. Usually as soon as one side commits multiple pieces to an attack, the other side defends, hoping to gain a useful hostage or frame, and eventually win slowly due to superior elephant mobility.

Trap Control

Trap Ownership

To own a trap square is to be safe from capture in it, and further to be able to capture opposing pieces in it. At the beginning of the game each player owns the two nearest trap squares, called home traps. The most basic offensive strategy is to go hunting with the lone elephant to dislodge an opposing piece towards one's home traps, in hopes of capturing that piece.

A more ambitious offensive strategy is to try to gain ownership of one of the enemy's home traps, most often with an elephant and horse. It raises the stakes to try to invade enemy territory in this fashion; if the opposing elephant doesn't defend (making the position a contest in elephant mobility), the result will be a race position.

An enemy trap usually has plenty of enemy pieces nearby, which can be captured in quick succession, as opposed to the laborious process of repeatedly prying loose a single enemy piece and dragging it all the way to a home trap for capture. An equally important benefit of owning an enemy trap is that if the opposing pieces scatter, it leaves a hole through which a friendly rabbit can march to the goal. Threatening to own an opposing trap is quite forcing; it limits the opponent's options to immediate defense or sharp counter-attack.


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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The traps c3, f3, and f6 are all deadlocked.

Deadlocked Traps

A trap is deadlocked when the strongest adjacent gold piece is equal to the strongest adjacent silver piece. At a deadlocked trap, neither player can hope to capture anything, ever, because the other player's strong piece keeps the trap safe. In the diagram at right, from this game, each player has an elephant next to the f6 trap, and both players have a camel near c3 and a horse near f3. A tie for strongest will never force a capture; you have to have the strongest piece next to a trap to own it.

Nothing can break an elephant deadlock except the voluntary departure of one of the elephants. A camel deadlock, in contrast, can be broken by the arrival of an elephant, and a horse deadlock can be broken by the arrival of an elephant or camel. At right Gold immediately moved his camel to e4 in order to re-establish ownership of the f3 trap.

Each trap that becomes deadlocked beyond the first trap adds a layer of strategic decision. In addition to maneuvering locally for mobility around each trap, there is a global question of which strong pieces to allocate to which quadrant. Crossing from one quadrant to another creates critical tactical timing questions of the form, "If I take ownership of that trap, will I gain enough to compensate for losing ownership of the trap I am leaving?"

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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Gold to move can profit by abandoning the f6 trap


If at any point an elephant can gain more on a different front than it stands to lose by abandoning its current front, then it should switch quadrants. For example, in the position at left (from this game) the elephants are deadlocked around the f6 trap, and the camels are deadlocked around the c6 trap. Gold faces the loss of a horse if he moves his elephant west, but if the elephant moves to c5 pushing the silver camel to c4, and Silver uses two steps of its next turn to capture the gold horse, Gold can then immediately take the silver camel in c3, and return his elephant to c5, having gained a camel and temporary control of the c6 trap in exchange for a horse.

When a deadlock is broken by the arrival of a stronger piece, the stronger side will almost always be able to own the trap. If there had been a mobility fight at that trap before the stronger piece arrived, that will only affect the timing of the captures. For example, if a camel is pinned in a camel deadlock, and the opposing elephant arrives, it will capture the framed piece on the first turn and the pinned camel the second turn. In contrast, if a camel is pinned in a camel deadlock and a friendly elephant arrives, it will first relieve the pin then extract the framed piece from the trap, then gain ownership of the trap, and then start capturing. The end result is the same, but the amount of time taken can be important.

Contested Traps

A trap is contested when one player has the strongest adjacent piece but the other has several weaker pieces to protect one another from capture and unfreeze one another when taken hostage. Momentarily neither player can make a capture, but the player with the strongest local piece is guaranteed safety, and may be able to eventually own the trap if the attack is supported with at least one additional piece.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
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The players are contesting all four traps


In this game, diagrammed at right, the players are contesting all four traps. Until one player can establish total control of some trap, no captures will be possible by either player. Nevertheless, this is a somewhat less stable situation than when multiple traps are deadlocked, because at each trap the player with the strongest piece can fight to own it. Whoever gets ownership of a trap first will start capturing pieces, quickly gaining a huge advantage.

This is a positional race, as opposed to a capturing race or goal race. Whichever player is going to lose the race to own a trap will be obliged to globally re-allocate forces before disaster strikes. Yet each re-allocation presents new opportunities to the opponent. For example, if the silver elephant took ownership of f6 at right, the gold camel would cross and take ownership of c3, a poor exchange for Silver. It would perhaps be wiser for a silver dog to assist the silver elephant in owning f3, although that would make the gold horse near f6 all the more potent a threat.

Losing control

To attack opposing traps entails the risk of exposing pieces which can be taken hostage or framed. On the other hand, it is a significant advantage to own at least one trap when the opponent owns none. A player who owns no traps can't make any captures at all. In particular there is not even the strategic threat of dragging a piece across the board to a place it can be captured, because there is no such place.

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a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6