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 to win matches against top-ranked human players. David Wu's bot Sharp accomplished this in 2015.

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 online 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 online every week. Omar Syed has hosted various events in the gameroom:

  • The World Championship is an annual tournament for human players. Jean Daligault of France has won the championship six times, in 2007, 2009, 2010, 2011, 2013, and 2014. Matthew Brown won it in 2015, 2016, and 2017.
  • From 2004 to 2015, the Computer Championship matched the top Arimaa bots in an elimination tournament. David Fotland's program Bomb placed first from 2004 to 2008. Jeff Bacher's Clueless prevailed in 2009. Mattias Hultgren's Marwin won in 2010 and 2012. David Wu's Sharp triumphed in 2011, 2014, and 2015. Ricardo Barreira's Ziltoid was victorious in 2013.
  • The Arimaa Challenge took place following the Computer Championship. The top two bots were available to play during a "screening" period. A human could play two games against each of these two bots; the bot with the better record in those matches advanced to the Challenge. That bot faced three ultimate human defenders, who were selected beforehand and hadn't played in the screening. If this bot could win three-game matches against all three human defenders, its developer would win the $10,000 prize. Humans dominated until 2015, when Sharp defeated Jean Daligault, Lev Ruchka, and Matthew Brown.

In addition to these events, the 1st Arimaa Online Festival was 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.

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png
The players begin by setting up their pieces however they choose on their home rows. Highlighted are the four trap squares.


37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png


The game begins with an empty board. Gold arranges their pieces on the first and second ranks, in whatever configuration they choose—Arimaa pieces do not have fixed starting positions. Silver then arranges their 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 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 east, west, or south 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 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. Now on a7, the gold dog 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, their 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 their 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 they sacrifice their 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

Gold arranges their pieces on the first and second ranks, and then Silver arranges their pieces on the seventh and eighth ranks. 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— the strong pieces must take the lead in clearing a path for a rabbit to goal. Moreover, rabbits cannot retreat homeward, and thus could quickly be pulled up and captured if they began exposed.

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png
Gold has used the 99of9 setup; Silver has switched the dogs and cats.


The 99of9 setup (named for the player who popularized it) 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, so that it can easily go to either wing depending on Silver's setup.

The gold horses will quickly move up, to assert control of Gold's home traps and then perhaps to attack Silver's traps. A rabbit is placed forward on each flank, because a2 and h2 tend 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. As long as the gold cats remain behind the traps, they make it hard for Silver to capture a gold camel, horse, or dog in Gold's own home trap.

Since Silver goes second, they can take Gold's setup into account when deciding on their own. At right, note that the silver elephant is not placed directly opposite the gold elephant. If the silver elephant directly faced its gold counterpart, Gold could advance their elephant four squares and temporarily 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 their home traps, and thus are potentially vulnerable to capture themselves—Silver stands to lose a dog in the opening if he's not careful. As long as another silver piece is next to the trap, however, a dog behind that trap strengthens Silver's control of it, since dogs cannot be pushed aside by enemy dogs.

You may wonder why rabbits do not begin behind the home traps. Experienced players will sometimes do this, but the downside is that a rabbit could get pushed into a protected trap and then could never step back, creating a mess for the home defender.

Notation

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png

You can become a decent Arimaa player without learning the standard notation for moves. For brevity's sake, however, examples will sometimes use the notation rather than spell out each individual step. You can learn the notation now or later.

A move is a sequence of steps, where each step is notated as ⟨piece⟩⟨initial square⟩⟨direction⟩.

  • ⟨piece⟩ is a single letter:
    • E for Elephant
    • M for caMel
    • H for Horse
    • D for Dog
    • C for Cat
    • R for Rabbit
    Upper case is used for the gold pieces, and lower case for the silver pieces.
  • ⟨initial square⟩ is a two-character square name as in chess notation (e.g. e4).
  • ⟨direction⟩ can be n, e, s, or w for north, east, south, or west. This is always from Gold's point of view. In addition the special direction x indicates that the piece is captured.

Consecutive steps by the same piece may be condensed, with the piece letter and initial square given just once.

Suppose Gold is to move in the diagram. The capture of the silver elephant is described as Mb4e db3n cd3e Hd4s ec3x. The goal in the north-east is Eg7s rh7w Rh6n Rh7n, or, equivalently, Eg7s rh7w Rh6nn.

Although this notation is rather verbose, proposed replacements have not become standard.

Introduction to Tactics

In Arimaa, a tactic is usually a one- or two-turn plan which can be calculated precisely. With up to four steps per turn, plans which aim more than a turn ahead are usually impossible to calculate exactly, and therefore generally belong to the realm of strategy. This is in contrast to chess, where one can often limit the opponent's options enough to have a precise three-turn plan.

The essential tactics are those that bring a friendly rabbit to goal, capture an enemy piece, or thwart the opponent's plan for a goal or capture. If neither side can achieve victory or capture in a single turn, one might work toward doing so in the next turn; the opponent could then counter, perhaps bringing in reinforcements to strengthen a weak area. This quickly gets out of tactics and into strategy.

One-turn goal

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png
Each side to move has a one-turn goal on each wing.


A rabbit might reach goal within one turn, even if the path appears blocked by a trap square and/or opposing pieces. 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. 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 goal in the turn's final two steps. The gold camel on f6 is lost when the rabbit advances, but reaching goal is worth any sacrifice. Beginners often incline more toward pushing than pulling, but note that a push is ineffectual here. If the gold cat pushes the silver rabbit, the cat will itself block the friendly rabbit, which can't then reach goal in the turn's final two steps.

Near a depleted goal line, an enemy rabbit 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 can be unblocked and unfrozen at the same time. Silver to move could slide the silver elephant from g3 to f3, pulling the gold rabbit from g2 to g3; the newly unfrozen f2 rabbit could then step to the vacated g2 and then to g1, winning the game. The silver elephant would be lost on the third step, 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.

Goal defense

In the above position, each side to move can win the game, but pretend that Silver does not have a one-turn goal. Silver would then have to stop both Gold goal threats. Fortunately, this can be done in four steps. To block the eastern goal, Silver must occupy g8 and still also occupy f8, since moving the f8 rabbit to g8 would open a different goal path. To stop the western goal, Silver can likewise block it, or can use the b4 camel to push the b5 rabbit to a5.

Suppose Gold had to stop Silver's goals. The f1 dog could push the rabbit into the trap, which it couldn't easily get out of (per the rules it couldn't step itself to f4, and if the silver elephant stepped away, a silver piece in the trap would be lost.) To stop the western goal in two steps, the c2 horse must push the rabbit to a2; any other two-step option would still allow the silver camel to unfreeze the rabbit and get it to goal within four steps. If the b2 rabbit were the only silver goal threat, Gold could capture it in four steps, since Silver is currently not defending c3.

One-turn capture

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png
Gold to move has one-turn captures in c6 and f3; Silver to move has one-turn captures in c3 and f6.


A non-elephant piece next to a trap is vulnerable. With Gold to move in the position at right, the gold elephant can step from d5 to c5 to b5, and then push the b6 dog into the c6 trap. Since there is no other silver piece next to the trap, this captures the silver dog, in one turn.

Sometimes one may need to push aside an obstructing piece to get at a vulnerable piece. With Gold to move, the g6 camel can push the g7 cat to h7; now on g7, the gold camel can push the f7 horse to f6, capturing it in a turn where one friendly piece pushed two different enemy pieces.

A piece on a trap square will be captured if its sole defender is dislodged. Silver to move could step the silver elephant from d4 to c4, then to b4, and then dislodge the b3 dog, capturing the c3 horse.

Since it is two squares away from an unprotected trap, the g2 cat can be captured in four steps. The dog could end the capture on any of three squares: f4 (after two pulls via f3), f2 (after two pushes via g2 or two pulls via f3) or g3 (via g4, with a pull followed by a push). The latter maneuver, where a piece returns to its initial square after moving an opposing piece two squares, is known as a flip.

Capture defense

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png


Often, the best defense against capture is to move away the threatened piece. By defending the c3 trap, the silver elephant is protecting the b3 cat, but there is currently a better solution that does not tie down either piece. The silver elephant could step to b4, unfreezing the b3 cat, which could then step to a3 and then to a4.

Other than that, the simplest capture defense is to station one's elephant next to a trap square. Since an elephant can't be dislodged, no friendly piece can be captured in that trap until the elephant chooses to leave. At left, no gold piece can be captured in c6, and no silver piece can be captured in c3, unless the respective elephant moves away.

A non-elephant can sometimes defend a trap alone, but that is usually precarious. The enemy elephant may lose something if it moves away from a different trap, but if it captures a stronger piece in return it was worth it. If any non-elephant is the sole defender of a particular trap, it must keep a constant eye on all enemy pieces stronger than itself.

Two non-elephant pieces can defend a trap together; this is called mutual protection. Since a trap's four key squares are not right next to each other, four steps are not enough for any single attacker to dislodge one defender and capture the other. At left, Silver has defended the c6 trap with a dog on b6 and a rabbit on c7. Even if the gold elephant started on b5, it could not immediately capture anything in c6. Gold could at most dislodge the dog to set up a capture threat for the following turn, but then Silver could add another defender, such as the d7 horse.

Instead of adding a second defender to a trap, one can sometimes obstruct the path of an attacking piece. The gold camel on g6 can't capture the 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. The silver pieces on f7, g7, h7 and g8 form a phalanx blocking the gold camel.

One counter-intuitive way to block an enemy's path is to station a friendly piece on a trap square in front of another friendly piece. The silver elephant could push the gold cat to d3, but that would temporarily give the c3 trap two gold defenders, making it impossible for Silver to capture anything in the final two steps. This only works because there is nothing else to immediately threaten the c2 dog; if it were dislodged with the cat still on c3, the cat would be lost.

In the southeast, Silver to move can step the g3 camel into the trap, where it is safe due to the silver dog on f2, and then capture the gold horse by pulling it into the trap. Gold to move could prevent this: if the horse steps to e2 and then back to e3 while pulling the f2 dog to e2, and then the f1 dog steps to f2, the gold horse and dog mutually protect the trap. This four-step maneuver is called a pull and replace.

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 may only delay captures, if the weak pieces have no safe place to go. Furthermore, scattering away from a home trap may leave a hole through which an opposing rabbit can march to goal. On the other hand, delaying captures may buy time to make progress elsewhere on the board. Scattering is most effective when one has strong threats of their own in enemy territory, and could reach goal first.

More capture patterns

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png
Although not immediately obvious, a one-turn capture is possible in each trap.


It is easy to become too confident about one's defense. At right, Silver has two pieces next to the northwest trap, but Gold has a stronger piece next to each, and thus has a one-turn capture. First the gold elephant can push the silver dog from b6 to c6, and then the newly unfrozen gold cat can push the silver rabbit from c7 to c8, capturing the dog. If Gold prefers to capture the rabbit instead, he can pull the dog away, and then push the rabbit into c6. This situation, where one defender can be dislodged and the other captured, is known as false protection. Such a capture is only possible if there are two attackers, each adjacent to a weaker defender.

Around the northeast trap, the silver camel cannot push the gold dog into the trap, since the horse is in the way. However, Silver can play a split capture: the camel pushes the dog to g6, then the horse pulls the dog into the trap.

In the southwest corner, Gold has calculated that even if the silver elephant pushes the gold rabbit from b4 to a4, the b3 horse cannot be pushed. This is correct but insufficient, because the silver elephant may still pull the gold horse from b3 to b4, capturing the gold camel on c3.

The southeast shows that a weak enemy piece in a trap square may not prevent a one-turn capture. The f3 dog can simply step east or west, leaving three steps for the f4 horse to enter the trap square and pull the gold dog from f2 to f3, capturing it. If instead a gold piece were on f3, Silver wouldn't have a one-turn capture.

The fork

37px-Arimaa board.jpg
Arimaa-border.png
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
Arimaa-border.png
Arimaa-border.png
The gold elephant has forked the silver dog between the c3 and f3 traps.


The fork threatens a single frozen piece with capture in two different traps. It is a basic two-turn tactic: a piece is pushed onto a square between traps, and is then captured in one or the other, unless the defender can do some heavy-lifting in between. In the diagram at left, from this game, Gold has threatened to capture the silver dog in either c3 or f3. Silver to move would need one step to defend the c3 trap with the elephant, and three steps to defend the f3 trap with the cat. Unfortunately for Silver, the gold rabbit on e4 is threatening to reach goal next turn, and cannot be frozen in place since it is next to the gold elephant; Silver must spend at least one step defending goal, and the silver dog is lost.

Against a weak opponent, a fork can be a way to make a quick capture. If all non-elephant pieces are still in their home territory, and then the gold elephant flips a silver piece into Gold territory, that piece is well on its way to being hopelessly forked. A weak opponent may even lose more pieces while trying to defend the forked piece. A skilled opponent, however, can usually prevent a fork, although that requires a bit more foresight than other tactical defenses; the key is to not leave a piece where it could be flipped directly to a place where it couldn't be rescued from a fork threat. Between skilled opponents, capture via a fork may happen later in the game, when pieces are more scattered and therefore vulnerable.

The fence

37px-Arimaa board.jpg
Arimaa-border.png
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 is a less common two-turn capture tactic. A piece is brought next to an unprotected trap, and hemmed in on two sides. The piece may be unfrozen from a third side, but would have nowhere to go, as the trap is on the fourth side. In the diagram at right, from this game, Gold has fenced the silver horse next to the c3 trap. Silver is to move, but can't prevent the capture of the horse. The silver elephant could unfreeze the fenced horse by moving to c5 (incidentally capturing the gold rabbit in c6), but the unfrozen silver horse can't then escape, as its only available move would be suicide. Furthermore, the gold pieces prevent the silver elephant from reaching d3, c2, or b3 in four steps, and no other silver piece is close enough to defend the c3 trap.

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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With Gold to move, there are no forced captures, despite tactical threats.


When both players understand these tactics, it will be difficult for either to achieve immediate material gain, since it is easy enough to prevent a one- or two-turn capture if you recognize the threat. In this game, shown at left, Silver has just flipped the gold camel from e3 to e5, and is threatening to capture it in f6. Besides defending the camel immediately, Gold must not let it get forked between c6 and f6. Even if the camel escapes, the b5 horse could be threatened. Despite these tactical ideas, the game ended without a single capture. Although this is unusual, captures in Arimaa are much less common than in chess.

After becoming acquainted with the most basic Arimaa tactics, you should turn your attention to long-term strategy, which will in turn open up more tactical possibilities.

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 advancing piece after piece with the aim of controlling the entire board. Others tend to play a "home game", initially sending few pieces to enemy territory. However you prefer to play, it is best to understand any strategy which an opponent might use against you.

If you are playing Arimaa bots, you may have already defeated the weakest ones just by using basic tactics. Each strategy you learn will get you further up the bot ladder. There's no need to study all the subsections right away; just understanding basic concepts could take you far. Later, you could revisit this material to learn how to build on an advantage or turn around a disadvantage. As you progress up the bot ladder, you'll need deeper and deeper strategies.

A recurring theme in Arimaa strategy is elephant mobility. Sometimes an elephant gets blockaded and can't move, but more often it would be too costly for an elephant to leave a particular area. Since the elephants are key to everything, even a slight elephant advantage can snowball.

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 their pieces on the first and second ranks, in whatever configuration they choose—Arimaa pieces do not have fixed starting positions. Silver then arranges their 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 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 east, west, or south 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 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. Now on a7, the gold dog 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, their 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 their 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 they sacrifice their 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

Gold arranges their pieces on the first and second ranks, and then Silver arranges their pieces on the seventh and eighth ranks. 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— the strong pieces must take the lead in clearing a path for a rabbit to goal. Moreover, rabbits cannot retreat homeward, and thus could quickly be pulled up and captured if they began exposed.

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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 (named for the player who popularized it) 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, so that it can easily go to either wing depending on Silver's setup.

The gold horses will quickly move up, to assert control of Gold's home traps and then perhaps to attack Silver's traps. A rabbit is placed forward on each flank, because a2 and h2 tend 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. As long as the gold cats remain behind the traps, they make it hard for Silver to capture a gold camel, horse, or dog in Gold's own home trap.

Since Silver goes second, they can take Gold's setup into account when deciding on their own. At right, note that the silver elephant is not placed directly opposite the gold elephant. If the silver elephant directly faced its gold counterpart, Gold could advance their elephant four squares and temporarily 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 their home traps, and thus are potentially vulnerable to capture themselves—Silver stands to lose a dog in the opening if he's not careful. As long as another silver piece is next to the trap, however, a dog behind that trap strengthens Silver's control of it, since dogs cannot be pushed aside by enemy dogs.

You may wonder why rabbits do not begin behind the home traps. Experienced players will sometimes do this, but the downside is that a rabbit could get pushed into a protected trap and then could never step back, creating a mess for the home defender.

Notation

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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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You can become a decent Arimaa player without learning the standard notation for moves. For brevity's sake, however, examples will sometimes use the notation rather than spell out each individual step. You can learn the notation now or later.

A move is a sequence of steps, where each step is notated as ⟨piece⟩⟨initial square⟩⟨direction⟩.

  • ⟨piece⟩ is a single letter:
    • E for Elephant
    • M for caMel
    • H for Horse
    • D for Dog
    • C for Cat
    • R for Rabbit
    Upper case is used for the gold pieces, and lower case for the silver pieces.
  • ⟨initial square⟩ is a two-character square name as in chess notation (e.g. e4).
  • ⟨direction⟩ can be n, e, s, or w for north, east, south, or west. This is always from Gold's point of view. In addition the special direction x indicates that the piece is captured.

Consecutive steps by the same piece may be condensed, with the piece letter and initial square given just once.

Suppose Gold is to move in the diagram. The capture of the silver elephant is described as Mb4e db3n cd3e Hd4s ec3x. The goal in the north-east is Eg7s rh7w Rh6n Rh7n, or, equivalently, Eg7s rh7w Rh6nn.

Although this notation is rather verbose, proposed replacements have not become standard.

Early Ideas

Early Arimaa players had no blueprint, and could only experiment. The object was to get a rabbit to goal first, but was it better to aggressively go for it, or bide one's time and focus mainly on home defense? If defense proved most important, would Arimaa become a stalemate?

Direct Goal

The most direct strategy in Arimaa is to try to rip a hole in the goal line and then march a rabbit through. If both players go this route, the game turns into a race.

However, if one player tries to force a quick goal while the other player defends, advantage accrues to the defender. With the entire defending army still at home, the pieces support each other, and thus can prevent attackers from simply tearing through. Other than the elephant, attacking pieces risk capture in the defender's home traps. The attacking elephant can protect one trap, or perhaps help threatened pieces retreat, but having one's elephant on defense is not a promising way to force a goal.

Capture

Nevertheless, a goal path must eventually be cleared somehow. The next strategy to consider is that of capturing pieces, though that is not straightforward either. An opponent committed to defense will have two or three pieces next to each home trap. Even if attackers try to overwhelm a particular trap, the defending elephant can protect it and likely turn the tables on an attacker.

A safer strategy would be to drag an enemy piece down to one's own home trap for capture, but even then the enemy elephant can defend. Suppose Gold threatens a silver piece with capture in f3. The silver elephant can camp out by that trap, and Gold can never capture anything there: an elephant can't be displaced, and protects any friendly piece which might end up on an adjacent trap square. In order to make progress, Gold must create threats around two different traps.

Two threats

For this reason, each player wants to be the first to make two strong threats. This could be as simple as forking one enemy piece between two traps; any non-elephant must be careful about advancing in the center. It is tempting to send one's elephant hunting and just keep everything else at home, but a skilled opponent could make that very slow, while developing more sophisticated threats of their own.

When a threat is in place, an important question is who has the strongest free piece. For example, if the gold elephant corners the silver camel, forcing the silver elephant to stay next to a trap, the gold camel is the strongest free piece, if in fact that camel is still on the board and able to move around. The gold camel could then create a second threat, which only the tied-up silver elephant or camel could defend against.

Hostage

That configuration became known as a camel hostage. Once players realized the problems with direct goal or capture, camel hostages became quite popular. Once again, however, it is hard to directly force this, since a camel can stay at home where it has the support of friendly pieces. A camel at home might still be forced into a corner, or at least cut off from one of its home traps. This could allow a horse to safely advance on the other wing; even if an elephant takes a horse hostage under such circumstances, that is not really an advantage. A horse-by-elephant hostage does not give one the strongest free piece, and with the hostage-holder's camel stuck on the other wing, the hostage defender has a better alignment. There is, however, a risk of a horse frame, which could be very effective.

Frame

A frame occurs when a piece is stuck on a trap square, unable to push its way off. If a horse hostage became a frame, and the defending elephant then stepped away, the horse would be lost instantly. By contrast, if an elephant abandons a hostage, the opponent must still use up a turn capturing it, letting the former defender get a head-start elsewhere. Thus a player holding a horse hostage might be hesitant to advance their camel, fearing that the defender could trade his horse for it. A player holding a frame has more time for advances on the other wing.

Blockade

A frame is one type of blockade, in which a particular piece is surrounded and has nowhere to go. Blockades are often not worth all the pieces they use, but if a blockade can immobilize the enemy elephant, it may be worth it. Sometimes the elephant itself can get blockaded, usually near a side of the board. If a blockaded elephant can't be freed, and the other elephant is not itself a direct part of the blockade, the game is likely over; the one free elephant will reign supreme.

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. For now, Silver can't defend with less than the elephant, because any other silver defender could itself be captured in c3, unless there were two weaker silver defenders securely in place, which would require some work.

The gold elephant must be next to the trap, to prevent the silver elephant from capturing a gold piece in that trap. A hostage is usually not held right next to a trap, as that would prevent the hostage-holder from protecting the trap.

If Silver ever abandons his camel, the gold elephant should then pull it into the trap, finishing on c4 or d3, near the action. It is thus important that the gold cat remain on c2.

For now, Gold's elephant is tying up Silver's elephant and camel both, leaving Gold's camel unopposed as the strongest free piece. The silver camel obviously can't defend any trap, and the silver elephant can only defend one trap at a time. This makes the gold camel a grave threat in the east.

Note:
In Arimaa, the strength and function of a piece depends on what pieces remain on the board. For instance, if each side has lost a camel and a horse, the remaining horses are the unique second strongest pieces on each side, and so have essentially the role that the camels had initially. When this book discusses strategic features such as camel hostages, it should be borne in mind that the actual pieces involved may differ if exchanges have taken 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 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 holding the silver camel hostage.

Use of Free Pieces

As the strongest free piece, Gold's own camel is key to his advantage. However, that does not mean the camel should simply march forward. Imagine in the above diagram that instead of the horse, the gold camel were on a6. Silver to move could send her 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.

Even if the gold camel advanced on the other wing, it would have to be careful. Gold might advance his camel expecting that, if the silver elephant went east, the gold elephant could take care of everything. However, the silver elephant would have a head start while the gold elephant captured the silver camel, and other silver pieces could temporarily block the gold elephant's path to e6 or f5; Gold can't just assume his elephant could get there in time to save an advanced piece. Remember also that an advanced piece can retreat only if it is not frozen.

Usually, at least one horse should advance first. Besides protecting the friendly camel, advanced horses can also protect weaker pieces. The gold horse on the west wing is 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 Silver 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.

While holding the camel hostage, Gold wants to make a strong second threat and force the silver elephant to choose which to defend against. If Gold wants to threaten captures in f6, his own camel will likely have to advance in the east, since a silver horse could defend a trap from a gold horse. Another option is to keep the gold camel where it is, and use the eastern gold horse to drag weaker pieces down for capture in f3 (remember, rabbits can't retreat homeward). As long as the camel hostage remains in place, the silver horses are the strongest silver pieces which could defend f3, but that would be difficult with the gold camel right there. Indeed, the gold camel could hostage a weak piece much as the gold elephant is hostaging the silver camel, and the silver elephant couldn't defend both. If Silver doesn't abandon her camel, find some other way to defend it, or find such a way to defend f3 while still also defending f6, Gold can do this over and over, decimating Silver's forces.

Realizing that Gold is poised to take control, what should Silver do? Silver needs to free her elephant. In this case, Silver could give up her camel in exchange for a horse: if the silver elephant moves to b6, and the silver camel is then captured, Silver can pull the gold horse from a6 into c6, capturing it in return. That is a material loss for Silver, but not a bad deal considering the position she was in. Alternatively, Silver could use her horse to unfreeze her camel, which could then push the gold dog aside, burrowing the camel so that it could not be captured in one turn. This is an option only because Gold has left b2 empty; gold pieces on a1, a2, and b2 would form a phalanx, fencing in any a3 hostage. Burrowing the camel would only be a delaying tactic by Silver, since the camel would hardly be free, but it would allow the silver elephant to leave c3 temporarily.

Gold to move can prevent Silver from doing either of those things. To fence in the silver camel, Gold could simply advance the b1 rabbit to b2, but a better option would be to slide the a2 dog to b2, the a1 rabbit to a2, and the b1 rabbit to a1; that way, the dog isn't blocked in. On the fourth step, the a6 horse could move to a7, precluding an immediate camel-for-horse trade.

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


When the enemy has the strongest free piece, it is crucial to free one's own elephant within a few turns. Depending on the balance of forces near the trap, this can sometimes be done without giving up the camel; if two weaker pieces can defend the trap, the elephant can leave. 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.

Such a scenario highlights one advantage of keeping the friendly camel home on defense while holding a camel hostage. Imagine that the gold camel were at home in the diagram at right, lurking on d2 for example. The e3 dog, and thus the f4 dog and g2 camel, would not be safe unless the silver elephant stayed nearby.

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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 cannot force a capture in either c3 or f3, despite holding the silver camel hostage and having two strong gold pieces on the other wing. The well-positioned silver elephant prevents any capture in f3, while also protecting the silver horses from the gold camel.

Who has the strongest free piece now? If the gold elephant leaves the c3 trap, Silver will soon wipe out Gold's western forces. As things are, however, the silver elephant isn't totally free either; if it returns to its home territory, the silver horses will be vulnerable to the gold camel. Likewise, the gold camel isn't really free, as it currently can't go anywhere without quickly being confronted by the silver elephant, which could safely leave e3 if the gold camel advanced. The elephants, camels, and silver horses are basically in a stalemate.

As it turns out, the gold horses are 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 then Gold will stand to gain even more than that in c3 and f3. Gold's upper hand traces back to the camel hostage.

Camel hostage value

When Gold is holding a silver camel hostage in Gold's home territory, both sides are usually keen to advance on opposite wings. Gold wants to advance on the non-hostage wing in order to secure material and goal threats, using the strongest free piece as necessary. Meanwhile, Silver wants to advance on the hostage wing in order to rotate out her elephant. The value of the hostage depends on the outcome of this race, and therefore on the initial positions of the pieces. The hostage will tend to be worth more if the gold camel is already on the non-hostage wing, and less if gold pieces (particularly the horse) are stuck behind the hostage.

If Silver is not well-positioned for an elephant rotation, she might abandon her camel and seek compensation elsewhere. For Gold, the hostage value depends on what compensation Silver could get, and what the resulting position would look like. It is typically hard to make use of a camel hostage without exposing a friendly horse to capture. Since a camel is estimated to be worth a horse and cat as an initial trade (see Relative Value of Pieces), it is not usually worth giving up more than a cat to get a camel hostage. If the board is full, and Gold takes a camel hostage but gives up a cat in the process, and then Silver abandons her camel but captures a gold horse in return, material is basically even. If this sequence happens after material has already been exchanged, Silver likely comes out ahead, and thus the hostage was a net loss for Gold even though he captured the silver camel.

If Silver is well-positioned for a quick elephant rotation, a hostage is likely of negative value for Gold. One should not aim to take a camel hostage if the opponent could defend it without their elephant.

Other hostage patterns

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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 double hostage is stronger than a simple camel hostage


At right (game) Gold is holding a double hostage of the silver camel and horse. With three strong silver pieces tied up, Silver's counterplay options are quite limited. Unless Gold is reckless in the west, Silver cannot hope to abandon the hostages and remotely make up for a camel and horse loss. With Gold's large advantage in free pieces, elephant rotation would also be quite difficult for Silver. The silver camel can't escape, due to a phalanx which the frozen g2 horse is part of.

One option for Silver is to occupy g4 with a weak piece. If the silver elephant then leaves, any capture the gold elephant could make would land it on a square other than g3, so the second hostage might escape. If the silver elephant moved west and threatened a gold piece, Gold might only capture one piece in return, despite having held a double-hostage. Realizing that Silver could complicate things by stepping a rabbit or cat to g4, Gold should be careful not to expose his camel to quick capture, since he might only get an even camel trade, and then would have an enemy horse moving through his own back row, now with no camel to threaten it.

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a4 b4 c4 d4 e4 f4 g4 h4
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Gold has an unstable but valuable high hostage

In this game, Gold has a high hostage. This type of pattern is less common than an ordinary camel hostage, since more pieces are required to maintain it. In this case, the hostage is unstable, as the silver horse can erode the blockade on b4 and then unfreeze the camel. If a high hostage can be maintained, it has the advantage of the attacking elephant remaining centralised.

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This central hostage will result in the capture of the camel


A central camel hostage is essentially a camel fork. In the example at right (game), the d2 and c3 phalanxes prevent the camel from escaping, and the silver elephant can't defend both c3 and f3. Such hostages are therefore more in the realm of tactics than strategy. TODO: find better example.

Elephant Blockade

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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 discovered that some bots could be lured into an elephant blockade by the offer of a free piece for capture. From there, the elephant would look for an escape, and thus could be tempted by an empty square on the edge of the board, where the blockade could become stronger.

In this game, diagrammed at right, Gold has lost a cat while Silver has a full army, but Silver is nonetheless completely lost. Due to the b1, d1, and c2 phalanxes, the silver elephant has no move at all. With the only functional elephant, Gold has the most decisive advantage possible. Silver could try to free her elephant, but the gold elephant and camel could ward off (and probably capture) any silver piece which approached the blockade.

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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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.

When rotation is impossible

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Silver has blockaded the gold elephant, but would gladly give up the blockade to take the Gold camel hostage.


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.

Partial blockades

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A partial elephant blockade makes the gold camel the strongest free piece.


Because of the difficulty of blockading an elephant in the centre of the board, complete elephant blockades are uncommon in games between strong players. Partial blockades, keeping the elephant from accessing an important area of the board, are more frequent. In this position (see game), Silver has a horse hostage, which ideally would make the silver elephant the strongest free piece. In this case, however, the elephant is blockaded away from the centre of the board; the strongest free piece is in fact the gold camel. Note that the blockade is not complete: the elephant can escape through the c6 trap. This maneuver would take several turns, allowing the gold camel to capture a piece in the meantime. Furthermore, it would disrupt Silver's defense of c6, probably allowing Gold to share control of the trap.

Swarm

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Silver's many advanced pieces have hopelessly blockaded the gold elephant.


If a defending elephant becomes decentralized while holding a hostage near a home trap, there may be an opportunity for the attacking side to blockade it with a swarm of weak pieces, i.e. rabbits, cats, and dogs. In this 2005 Arimaa Challenge game at left, Silver has exploited the bot's susceptibility to elephant blockades.

Notice that the immobilized gold cat has become part of the blockade holding in the gold elephant. While the gold elephant might try to escape by pushing the a3 horse south and then using its newly mobilized cat to push the b4 rabbit south, that would play further into Silver's hands. The c4 horse could push the cat back to a4, totally blocking in both the cat and the elephant, which couldn't get back to b3 due to the phalanx.

Often in such situations the gold elephant is not absolutely blockaded, because it may still escape to the center of the board via the first two ranks. Even if the gold elephant could so escape, however, the blockading silver pieces would surge forward in its wake, ensuring long-term trap control of c3 for Silver, and consequent indirect goal threats.

The Center

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The gold elephant is cut off from the center.


When an elephant is next to a trap, it is somewhat decentralized, and thus can potentially be cut off from a substantial part of the board. If one has several pieces near the center, and the enemy elephant is not on one of the four central squares, it is occasionally possible to station a clump of friendly pieces on those squares, creating a substantial barrier between that elephant and at least one trap. In the diagram at right, from this game, the six central silver pieces keep Gold's elephant away from his own eastern home trap. Silver could threaten goal in the west while also making a threat in the east.

This strategy must be used with great caution, however; if the dividing wall does not hold, at least one of the pieces will likely be lost. At right, if Gold could break through the wall, some silver piece would likely perish in f3. If an elephant is even mildly mobile, enemy pieces other than its counterpart should be cautious about occupying any central square, let alone all four.

Elephant Mobility

Other than the elephants, any piece can potentially be taken hostage, and often the elephant is the only friendly piece which can feasibly protect it. The value of such a hostage depends largely on which side is left with the strongest free piece.

Not all hostages are good

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The horse hostage does not benefit Silver, since the gold camel is more free than the silver camel (from this game).

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. However, not all such positions are effective. Even with camel hostages, the hostage-holder is often more stuck than the defender, since the hostage could become an attacker if freed. This gets worse if an elephant holds a horse hostage instead, as in this diagram. If the silver elephant left the trap, the gold horse would likely help its elephant make captures, perhaps decimating Silver's eastern forces. Thus both elephants are basically stuck, but consider the overall position. Gold would gladly trade his horse for the silver camel, thus the silver camel must be careful. The gold camel is safer, since the gold elephant could defend it without losing anything should the silver elephant leave the f6 trap, and also because of the current respective distances between elephants and enemy camels. The camels are the strongest free pieces, but Gold's camel is more free, thus the hostage does not give Silver an overall advantage. When the enemy camel is still on the board, an elephant must be careful about what else it commits to.

As long as the gold elephant defends f6, it should stay on e6. If e6 and e7 were clear, the silver elephant could pull the gold horse onto e7 and then fork it between traps. That is a possible advantage of positioning a hostage-holder behind the trap.

In some cases, a horse-by-elephant hostage might be converted to a frame, or passed off to the silver camel, freeing the silver elephant. If Silver were better positioned for that, it would be urgent for Gold to prevent a solid frame or horse-by-camel hostage, which could make the silver elephant the strongest free piece.

Gold can treat this position somewhat like a camel hostage, advancing pieces on the hostage wing to help two weaker defenders replace the gold elephant. This could be relatively easy, since Silver does not have the strongest free piece.

Horse hostages

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The right piece to hold a horse hostage is a camel. Ideally this ties the opposing elephant to defence, while the friendly elephant is free to roam. However, the situation is often unstable because the "defending" elephant can attack the hostage-holding camel, which could quickly turn the tables. Thus the camel needs friendly supporting pieces, to keep the advantage long enough for the friendly elephant to make a strong second threat. A good horse hostage can make one's own elephant the strongest free piece, a greater advantage than could be gained from a camel hostage.

These diagrams illustrate different positions for a horse-by-camel hostage. In the first diagram, the gold piece on a4 is necessary for the hostage to be effective: pulling the gold camel to b4 would be a waste of time, as it could simply return to b3. It may take Silver several turns to break this hostage, which should allow Gold to gain a large advantage elsewhere. Note that the b2 cat allows capture of the hostage in case the silver elephant goes to b4. Also, Gold should leave d3 clear; if d3 were occupied by any gold piece, the silver elephant could then afford to move to b4, since the only one-turn capture would end with the gold camel on c4, where it would be threatened in c6.

Without the gold horse on a4, the hostage would be weak. With ec4w eb4e Mb3n ha3e, Silver could have her elephant on c4 and her horse on b3, with the gold camel frozen on b4. Not only could Silver then threaten captures in c3, but she could also flip the gold camel to c5, with a threat to capture it in c6. Without a gold piece ensuring the friendly camel's mobility, Gold would have much more to lose than to gain here.

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In the second diagram, the silver elephant has no easy way to approach the gold camel. As long as Gold can safely occupy b3, the hostage is fairly secure. This can be the strongest type of horse hostage, provided there are enough pieces to maintain it.

In all of these cases, the side defending the hostage should consider bringing more pieces into the local fight. For instance, in the last example Silver would have a strong position if she could bring a horse to b3, dislodging the gold dog. In the first, the silver camel might be able to attack the a4 horse, which would weaken the hostage pattern.

Cat and Dog Hostages

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In this game, Silver's position would be much stronger if its camel were free and a silver horse held the gold dog hostage.


A smaller piece may also make a valuable hostage. In this example, the silver camel holds a gold dog hostage next to the c6 trap. The silver elephant is free to pull gold pieces toward the f6 trap. The gold elephant can't defend c6 and f6 at the same time.

However, the situation is less than ideal for Silver, for two reasons. First, the silver cat on c6 means that the c8 dog isn't currently 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 probably have to capture the h5 rabbit before making any other threat in f6.

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.

Race Positions

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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 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 go back 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. Silver to move is winning this race because of her still full home rank, as well as the b3 horse which strengthens Silver's control of c3. An elephant by itself is quite limited; Gold to move could capture the silver camel in f6, but that would use all four steps and still leave the gold elephant decentralized. Instead of saving her camel, Silver to move can capture the c2 horse in three steps and advance the a6 rabbit to a5. If Gold does then capture the camel, Silver can continue the race by capturing the b2 dog. If Gold advances his own rabbit and pulls down the f8 rabbit, the e8 and d8 cats can each take one step east, leaving Silver plenty of time to get the now a5 rabbit to goal. If the gold elephant ever abandons the race and comes west, Silver could attack in the east. Gold should have avoided this position.

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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 would like to push the f4 camel to f5 and then move the g3 rabbit to e3, threatening captures and goal. Instead, he must stop the d6 rabbit; merely freezing it would not be enough, since the c7 dog could step to c6 and pull the d7 rabbit to c7, unfreezing and unblocking the d6 rabbit, which could then step to goal. Silver's best option may be for his camel to push the d6 rabbit back to d5; besides setting back the goal threat, this would contest control of c6, where the b6 horse is threatened with capture. To retain control of c6, the gold elephant would likely step to c5; if Gold used his other three steps to capture the g3 rabbit, the silver elephant could then capture the gold camel in f6, and could likely then stop Gold's northwestern threats. Silver would then be poised for a new, stronger attack in the east. If Gold foresees this, he will likely choose not to immediately end Silver's current goal threat, and instead reestablish full control of c6 by pushing the silver camel off of d6. Gold would have capture threats, but not yet a one-turn goal threat, and the race would continue.

Because the cost of losing a race is so high, and because the side with more pieces committed usually makes progress faster, it is rare to respond to a complete loss of trap control by launching an all-out race. Instead, the defender might protect material with the elephant while building towards a slower counter-attack.

Trap Control

Each trap is either owned by one player or shared. To own a trap 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 their two home traps (c3 and f3 for Gold, c6 and f6 for Silver), but this could change quickly as pieces advance. It is very valuable to take complete ownership of an enemy trap. An enemy trap usually has plenty of enemy pieces nearby, which might be captured in quick succession, in contrast to the laborious process of repeatedly prying loose a single enemy piece and dragging it all the way to a home trap for capture. When pieces are threatened in their own home trap, they might delay or avoid capture by scattering, but this is even better for an attacker, since it leaves a hole through which a rabbit might reach 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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a4 b4 c4 d4 e4 f4 g4 h4
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