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. Camel Hostage
  6. Other Hostages
  7. Frames
  8. Trap Control
  9. Distribution of Force
  10. Rabbit Advancement
  11. Elephant Blockade
  12. Race Positions
  13. Relative Value of Pieces
  14. Setup
  15. Attacking
  16. Lone Elephant Attacks
  17. Advanced Tactics
  18. Sample Games
  19. Arimaa Challenge History
  20. Glossary
  21. Resources
  22. GNU Free Documentation License

Overview

Arimaa (pronounced uh-ree-muh) is a two-player board game invented by Omar Syed, a computer engineer trained in artificial intelligence. After Garry Kasparov was defeated by the chess computer Deep Blue, Syed wanted to create a game that could be played on a chessboard with chess pieces, but which could not be won by sheer calculating power. Syed thought that a large branching factor was the key, but soon realized that this would not automatically favor humans, who would need to anticipate a position a few turns ahead. Syed began to see the solution while teaching his young son a simplified version of chess; if movement was limited but each turn allowed for multiple steps, a game could be both high-branching and playable. After much experimenting, Syed came up with a game which, like chess, used a 1-1-2-2-2-8 distribution of pieces. Conceptually, Syed's new game was much simpler than chess, yet the branching factor dwarfed that of chess. For a human or machine, imprecise planning would be key.

Arimaa's first computer test was the Zillions of Games engine, which was designed to competently play any game it was given the rules to. As Syed expected, Zillions was an easy opponent. On November 20, 2002, Syed published the rules for Arimaa and announced a $10,000 prize, available annually through 2020, for the first bot to defeat a top human player in a "Challenge" match. David Wu's bot Sharp claimed the prize in 2015.

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.

Arimaa can be played online at the arimaa.com gameroom. Elsewhere, it can be played using a chess set, since each Arimaa piece corresponds to a chess piece. In 2009, Z-Man Games began producing a commercial Arimaa set. Only one face-to-face tournament has taken place, but various events have been hosted online:

  • The World Championship is an annual tournament for human players. This tournament has taken various formats, and has sometimes lasted several rounds. Jean Daligault of France has been the world champion six times, in 2007, 2009, 2010, 2011, 2013, and 2014. Mathew Brown prevailed in 2015, 2016, and 2017. Karl "Fritz" Juhnke won in 2005 and 2008.
  • 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. At first, the winning bot played an eight-game match against a lone human defender. The format evolved: from 2007 onward, 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, where it faced three ultimate human defenders, who were selected beforehand and hadn't played in the screening. If this bot could win best-of-three 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 Mathew Brown.

An online Arimaa festival was organized in 2010. This event included Arimaa matches, a strategy workshop, and an interview with Omar Syed.

After DeepMind's AlphaZero mastered Go, Chess, and Shogi simply by playing itself, Omar Syed announced a $10,000 prize for the creation of such an Arimaa bot which could win a 10-game match against Sharp. Such a match has not yet occurred, though such bots are in progress.

External links

A collection of articles by the game creator and top bot developers

Playing The Game

Rules

Like classic chess, Arimaa is played on an 8×8 grid. The two players, Gold and Silver, control sixteen pieces each (listed in descending order of strength):

Piece Elephant Camel Horse Dog Cat Rabbit
Number 1 1 2 2 2 8
Symbols Arimaa egb74.png
Arimaa esb74.png
Arimaa mgb74.png
Arimaa msb74.png
Arimaa hgb74.png
Arimaa hsb74.gif
Arimaa dgb74.png
Arimaa dsb74.png
Arimaa cgb74.png
Arimaa csb74.png
Arimaa rgb74.png
Arimaa rsb74.png
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8 Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg 8
7 Arimaa ds.svg Arimaa ms.svg Arimaa cs.svg Arimaa es.svg Arimaa ds.svg Arimaa cs.svg Arimaa hs.svg Arimaa hs.svg 7
6 6
5 5
4 4
3 3
2 Arimaa rg.svg Arimaa hg.svg Arimaa cg.svg Arimaa mg.svg Arimaa eg.svg Arimaa cg.svg Arimaa hg.svg Arimaa rg.svg 2
1 Arimaa rg.svg Arimaa rg.svg Arimaa rg.svg Arimaa dg.svg Arimaa dg.svg Arimaa rg.svg Arimaa rg.svg Arimaa rg.svg 1
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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.

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 step one square at a time. Diagonals have no place in the rules of Arimaa. The relative strength of each piece lies in its power to push, pull, or immobilize weaker enemy pieces. There are four 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.

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 her pieces on the seventh and eighth ranks.

A piece steps from square to square. Rabbits can step left, right, or forward. All other pieces can step left, right, forward, or backward.

A turn (or move) consists of one to four steps. The steps in a turn can be used on four different pieces, all on the same piece, or any combination. After the setup phase is completed, the players alternate turns with Gold going first.

A player may not pass a turn, which must make a net change to the position. Thus one cannot, for example, move the same piece forward and backward and leave it at that. Furthermore, a turn may not result in a position identical to one which the same player has created twice before.

The game's main objective is to get a rabbit across the board; the opponent's home rank is the goal line.

Freezing

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8 8
7 Arimaa rs.svg Arimaa rg.svg Arimaa cs.svg Arimaa ds.svg Arimaa rs.svg 7
6 Arimaa dg.svg Arimaa ds.svg Arimaa hg.svg Arimaa rs.svg 6
5 Arimaa es.svg 5
4 Arimaa rs.svg 4
3 Arimaa dg.svg Arimaa rs.svg Arimaa eg.svg Arimaa rg.svg Arimaa rg.svg 3
2 Arimaa hs.svg Arimaa rs.svg Arimaa hs.svg 2
1 Arimaa cg.svg Arimaa cg.svg Arimaa rg.svg Arimaa rg.svg Arimaa rg.svg Arimaa rg.svg 1
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The second diagram, portraying a position which could occur later in a game, helps illustrate the remaining rules of movement.

A piece which is adjacent to a stronger enemy piece is frozen, unless it is also adjacent to a friendly piece. A frozen piece may not be moved by its owner. The silver rabbit on a7 is frozen, but the one on d2 can move because it is next to another silver piece. 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.

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.

Pushing and pulling

A piece can pull or push a weaker enemy piece which is next to it, provided an empty square allows 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. A push or pull uses two steps, since two pieces are moved.

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. 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 could not dislodge the d2 and c3 rabbits in one move. No push can occur if there is no empty square for a piece to be pushed onto, and no pull can occur if there is no empty square for the pulling piece to step onto. Strong pieces can sometimes be limited by teams of weaker pieces.

Whether a piece is frozen is independent of whether it can be dislodged — an adjacent friendly piece does not protect against pulling or pushing. For example, the d1 cat does not protect the c1 cat from being pushed to b1 or pulled to c2.

Captures

In Arimaa, a capture can only occur on a trap square. A piece which enters a trap square is removed from the board unless there is a friendly piece next to that trap. 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 c4 rabbit and c2 horse — the c3 trap has two silver defenders. If all friendly defenders left or got dislodged, a piece still on a trap square would be lost.

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 pulling piece is lost on the first step. For example, Silver to move could step the silver horse from f2 to f3, losing the horse, and still pull the gold rabbit from f1 to f2 as part of the horse's move.

End of the game

An Arimaa game cannot end in a draw. There are three different ways to win:

  • Goal: The principal object of the game is to move any friendly rabbit all the way across the board. Thus Gold wins by moving a gold rabbit onto the eighth rank, and Silver wins by moving a silver rabbit onto the first rank. This is normally how a game is won.
  • Elimination: One can win by capturing the last remaining enemy rabbit.
  • Immobilization: One can win by creating a position that leaves the opponent with no legal move. This could involve a large blockade.

If one side's move directly brings about a winning condition for the opponent, the opponent wins, except under these rare circumstances:

  • If a move brings about goal or elimination for both sides, the one who made the move wins.
  • Goal or elimination wins the game even if one's pieces are technically left immobilized. Immobilization is only checked at the start of a turn.
  • If a rabbit is dislodged onto its goal line and then dislodged off within the same turn, the game continues.

Setting up

The finer points of initial piece placement will be explored later, but a beginner just needs a flexible setup that denies the opponent an easy attack plan. Strong pieces start in the front row, as they must take the lead in clearing a path for a rabbit to goal.

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8 Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa cs.svg Arimaa cs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg 8
7 Arimaa rs.svg Arimaa hs.svg Arimaa ds.svg Arimaa ms.svg Arimaa es.svg Arimaa ds.svg Arimaa hs.svg Arimaa rs.svg 7
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2 Arimaa rg.svg Arimaa hg.svg Arimaa cg.svg Arimaa eg.svg Arimaa mg.svg Arimaa cg.svg Arimaa hg.svg Arimaa rg.svg 2
1 Arimaa rg.svg Arimaa rg.svg Arimaa rg.svg Arimaa dg.svg Arimaa dg.svg Arimaa rg.svg Arimaa rg.svg Arimaa rg.svg 1
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Gold has used the classic 99of9 setup; Silver has switched the dogs and cats.

In the diagram at right, Gold has chosen a classic symmetric setup. The elephant is placed in the center so that it can quickly reach any part of the board. The camel is placed in the center as well, to deter an enemy horse advance on either wing. Rabbits are kept out of the center; since rabbits can't retreat, a rabbit pulled up in the middle might block its own friendly pieces. For the same reason, rabbits are not placed directly behind the traps.

The gold horses will quickly move up, to assert control of Gold's home traps; one horse or the other might then advance in hopes of creating a threat. Beginning back and center, the gold dogs may soon move up. The gold cats might stay in place for a while, guarding the traps from behind.

Since Silver goes second, she can take Gold's setup into account when deciding on her own. Note that the silver elephant is not placed directly opposite the gold elephant. If both elephants started on the same file, Gold could advance his elephant four squares and temporarily fence in the silver elephant.

Silver has also used a symmetric setup, only 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 pieces guarding their home traps from behind are potentially vulnerable to capture themselves — Silver stands to lose a dog in the opening if she'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.

A player who uses a non-symmetric setup may intend to move quickly on one wing.

Notation

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8 8
7 Arimaa eg.svg Arimaa rs.svg 7
6 Arimaa rg.svg 6
5 Arimaa ms.svg Arimaa cs.svg 5
4 Arimaa mg.svg Arimaa hg.svg Arimaa ds.svg 4
3 Arimaa ds.svg Arimaa es.svg Arimaa cs.svg 3
2 Arimaa rg.svg 2
1 1
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One 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.

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.

Suppose Gold is to move in the diagram. The false protection capture of the silver elephant is described as Mb4e db3n cd3e Hd4s ec3x. The northeastern goal is Eg7s rh7w Rh6n Rh7n, or equivalently, Eg7s rh7w Rh6nn. Consecutive steps by the same piece may be condensed, with only the initial square given.

While this notation may seem verbose, there is a lot to account for in almost any turn.

Introduction to Tactics

A tactic is a narrow plan. A precise multi-turn plan would account for every possible move by the opponent, which is difficult when the opponent has thousands of options. Sometimes, however, one can severely limit the opponent's viable options. If a move creates a one-turn goal threat, the opponent must stop the goal or else lose the game. If a move creates a capture threat that if executed would give one a large material lead, the opponent must either stop the capture or somehow mitigate it, perhaps by capturing a piece in return.

Tactics can be offensive or defensive; a defensive tactic may slow things down considerably. When nothing big is imminent, plans may be more general, with each side aiming for a strong long-term position. This gets out of tactics and into strategy.

Goal

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8 Arimaa rs.svg Arimaa rs.svg Arimaa hs.svg Arimaa rs.svg Arimaa rs.svg Arimaa cs.svg 8
7 Arimaa rs.svg Arimaa cg.svg Arimaa ds.svg Arimaa rs.svg Arimaa cg.svg 7
6 Arimaa ds.svg Arimaa hs.svg Arimaa cs.svg Arimaa mg.svg Arimaa rg.svg 6
5 Arimaa rg.svg Arimaa dg.svg 5
4 Arimaa ms.svg 4
3 Arimaa rg.svg Arimaa hg.svg Arimaa eg.svg Arimaa es.svg 3
2 Arimaa rs.svg Arimaa hg.svg Arimaa rg.svg Arimaa rs.svg Arimaa rg.svg 2
1 Arimaa rg.svg Arimaa rg.svg Arimaa dg.svg Arimaa rg.svg 1
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Each side to move has a one-turn goal on each wing.

At left, the gold rabbit on b5 can step to victory via b6, c6, c7, and c8. The rabbit is never frozen, as it is always next to the b7 cat or c5 dog, which also allows the rabbit to move through the c6 trap. The rabbit will theoretically be frozen once it reaches c8, but Gold will have won regardless.

Even if a rabbit's path is blocked, there may be a four-step goal. If the gold cat on h7 moves to h6 while pulling the silver rabbit from g7 to h7, the gold rabbit on g6 can step to goal. 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 be on g7, blocking its own rabbit with only two steps remaining.

Near a depleted goal line, an enemy rabbit is a constant threat. By pushing the b3 horse to c3, the silver camel would occupy b3 and unfreeze the b2 rabbit, which could then step to victory via a2 and a1. Occasionally a rabbit can be unblocked and unfrozen with one pull. 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 is lost on the third step, but that doesn't matter since Silver has won. With the f2 rabbit beginning the turn blocked and 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 and thus must defend. There are a few ways to stop each gold goal individually, but Silver has only four steps to stop both. To block the eastern goal, Silver must close the path to g8 without opening a different one-turn goal path. This could be accomplished by stepping the h8 cat to g8, leaving three steps for the western defense. In those three steps, the b4 camel or d6 horse could capture the c5 dog and freeze the b5 rabbit. If Silver had only two steps for western defense, the b5 rabbit could be pushed west, or the goal path could be blocked.

Suppose Gold had to stop Silver's goals. The simplest solution is Ee3nww Rh1w, which would freeze and threaten the silver camel while also keeping the f2 rabbit frozen, since unfreezing it would no longer be worth an elephant sacrifice by Silver. If Gold had only two steps for western defense, he couldn't freeze the silver camel, but could impede it.

Goal threat

Even if a goal can be stopped, a one-turn goal threat will force the opponent to use up steps on defense, unless he has a one-turn goal of his own. A goal threat is typically part of a multi-turn plan; the opponent might successfully defend goal, but the necessary moves could be costly for him. One should always be aware of possible goal paths on both sides, so as not to be caught off guard or miss an opportunity.

One-turn capture

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8 Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg Arimaa rs.svg 8
7 Arimaa rs.svg Arimaa hs.svg Arimaa rs.svg Arimaa hs.svg Arimaa cs.svg 7
6 Arimaa ds.svg Arimaa mg.svg Arimaa rg.svg 6
5 Arimaa eg.svg 5
4 Arimaa es.svg 4
3 Arimaa rg.svg Arimaa dg.svg Arimaa hg.svg Arimaa hg.svg Arimaa dg.svg Arimaa rg.svg 3
2 Arimaa rg.svg Arimaa cs.svg 2
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Gold to move has one-turn captures in c6, f6, and f3; Silver to move has a one-turn capture in c3.

A piece adjacent to a trap is said to be a defender of that trap. A non-elephant acting as its side's sole defender of a trap is often vulnerable. With Gold to move in the position at right, the gold elephant can take two steps west 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.

One can sometimes 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 her elephant two squares west and then dislodge the b3 dog, capturing the c3 horse.

The g3 dog can capture the g2 cat in four steps, either by pushing it twice (cg2w Dg3s cf2nx Dg2w), pulling it twice (Dg3w cg2n Df3s cg3wx), or flipping it (Dg3n cg2nwx Dg4s). A flip is a pull and then a push, with the flipped piece moving two squares and the flipping piece returning to its original square.

Capture defense

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6 Arimaa ds.svg Arimaa mg.svg Arimaa rg.svg 6
5 Arimaa eg.svg 5
4 Arimaa es.svg 4
3 Arimaa cs.svg Arimaa cg.svg Arimaa hg.svg Arimaa ms.svg Arimaa rg.svg 3
2 Arimaa hg.svg Arimaa dg.svg Arimaa ds.svg Arimaa cg.svg Arimaa rg.svg 2
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Sometimes, the best way to stop a capture is to move the threatened piece. The silver elephant protects the b3 cat from capture in c3, but neither piece needs to be tied down. The silver elephant could step west and thereby unfreeze the b3 cat, which could then step to a3 and then to a4.

Other than that, stationing one's elephant next to a trap square is the simplest capture defense. 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 this is usually precarious, as the defender itself could be captured. Such a piece must keep a constant eye on all enemy pieces stronger than itself. The opponent may lose something if he moves his elephant, but if he captures more than he lost, it was worth it.

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. A phalanx blocks the gold camel from g7, thus the f7 horse is currently safe even though it is Silver's only direct defender of the f6 trap.

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. Gold has stuffed c3; the silver elephant could push the gold cat to d3, but that would temporarily give the c3 trap two gold defenders, preventing any capture in Silver's final two steps. Occupying c3 only works because no silver piece immediately threatens the c2 dog; if it were dislodged with the cat still on c3, the cat would be lost.

The f2 dog allows the g3 camel to safely step into the trap, and then capture the gold horse with a pull. Gold to move could prevent this with He3sn df2w Df1n, a pull and replace that would give the trap a second gold defender.

If defending a trap is not feasible, threatened pieces might scatter away from it. It may not be possible to get every piece to safety, but even delaying captures may buy time to make progress elsewhere. One scattering pieces away from a home trap should try to block the goal line, as enemy rabbits can easily advance toward an undefended trap.

More capture patterns

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6 Arimaa ds.svg Arimaa hs.svg Arimaa dg.svg 6
5 Arimaa eg.svg Arimaa ms.svg 5
4 Arimaa rg.svg Arimaa es.svg Arimaa hs.svg Arimaa cs.svg 4
3 Arimaa rg.svg Arimaa hg.svg Arimaa mg.svg Arimaa ds.svg 3
2 Arimaa rg.svg Arimaa cg.svg Arimaa dg.svg Arimaa hg.svg 2
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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.

In the northeast, 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, 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.

In the southeast, the f3 dog can 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 there.

The fork

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The gold elephant has forked the silver dog between the c3 and f3 traps. (Game)

The fork threatens a single piece with capture in two different traps. In the diagram, Gold has threatened to capture the silver dog in either c3 or f3. There is currently no chance for the dog to escape, and defending both traps is not a viable option for Silver; while the silver elephant and cat could defend c3 and f3 in four total steps, the e4 gold rabbit could then step to goal (the gold elephant keeps the e4 rabbit unfrozen even if the silver cat comes to f4). Silver must use at least one step to defend goal, and thus concede the dog.

Generally speaking, a solid fork is most likely to occur between the forker's home traps; any other fork would tend to be easier to escape. From the start of the game, each side must be wary of allowing the opposing elephant to flip a piece into the centre, as that can lead to a strong fork. As defenses thin, forks become a greater potential threat in all areas of the board.

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Use this link to make moves on the board.

Problem: Silver to move and win the gold camel

Solution: ed3n Rc4w ed4w cb7s. Silver forks the camel between c3 and c6, blockading b5 so the camel cannot go west. The gold camel has no effective way to escape, and Gold has no effective way to defend both traps at once; Gold could put a rabbit or horse on d3, but the silver elephant could pull that piece into the trap, capturing it while retaining the camel fork.

The fence

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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 capture-forcing 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 save the horse. The silver elephant could move to c5 and unfreeze the fenced horse (while also capturing the gold rabbit in c6), but the unfrozen silver horse couldn't then escape, as its only available move would be suicide. Silver can't defend the trap, as no silver piece can reach d3, c2, or b3 in four steps; gold pieces block the silver elephant from doing so.

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plan window

Problem: Silver to move and win material

Solution: ef4nw Hg5w hg6s blocks the gold elephant out of e6, fences the gold horse on f5, and clears g6 so that the f6 rabbit can step east. The gold elephant can still pull the e6 dog, capturing the f6 rabbit, but in any event Silver can capture the fenced horse.

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A fence may be valuable even if the fenced piece can be defended. At right, the silver elephant can go to d3 to defend the horse, but Gold can then frame the horse with hc4s Ed4w Hb4s. In other cases, Gold might keep the fenced piece on c4 as a hostage, while blockading c3 to stop the piece from advancing. A frame or hostage is worthwhile only if it results in a whole-board advantage, which is usually a strategic rather than tactical issue.

Limitations of tactics

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Gold to move can avoid any capture.

Tactics alone can't accomplish much, if the opponent understands basic threats and defenses. The position at left may look promising for Silver, but Gold went on to win without losing a single piece until the end.

The next thing to learn is basic long-term strategy, which can help one get a whole-board advantage and overwhelm the opponent's defenses.

Introduction to Strategy

Strategy is focused on long-term positions. If no capture or goal can be forced, a tactic may become a strategy. A goal threat's value may lie in the space advantage it creates; enemy pieces forced to defend goal will not be a problem elsewhere. A capture threat may likewise tie enemy pieces to defense, if the opponent is unwilling to give up the threatened piece. By tying up enemy forces, one can build a whole-board advantage.

Direct goal

The ultimate object of Arimaa is to get a friendly rabbit to goal. This requires that space be cleared, but how? Should one immediately try to rip a hole in the goal line and then march a rabbit through? If one side attempts this, how should the other side respond?

It is best not to treat the whole game as a race. Until a fair amount of space has been cleared, forcing a goal is much harder than stopping a goal. Even if a goal path is opened, the defender can often close it easily. One might get a quick goal against a weak opponent, but a skilled opponent is usually attentive to home defense.

As long as there is no real opening in the goal line, it is futile to advance a rabbit for the purpose of reaching goal. A quick rabbit advance may be costly, since an advanced rabbit can never retreat. Defending a rabbit from capture is often untenable, as an elephant has more important things to do, and any other defender could be captured itself.

In short, going for an early goal is a poor strategy which can quickly set one back. Nevertheless, a goal path must eventually be cleared somehow.

Capture

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Because the gold elephant was decentralized, Silver was able to fork a gold dog. (Game)

While a game can finish with minimal captures or even no captures, capture threats are important. A piece is captured only when it is on a trap square with no friendly piece beside it; there are many ways to prevent a capture. If nothing else, the defending elephant can camp out beside a trap where there is a capture threat; since nothing can dislodge an elephant, its friendly pieces can never be captured in a trap it stays next to.

The key to progress is to make strong threats around two different traps. The enemy elephant can only defend one trap, and any other defender or rescuer could itself be at risk.

The simplest double threat is a fork between one's home traps. If a piece can be flipped in the center, it may be doomed to such a fork. This is one of many reasons both elephants might stay near the center. Although any piece can unfreeze a friend, an elephant can do so without putting itself at risk.

A rabbit near the center could be an easy target for a fork, since rabbits can't retreat homeward. This is one reason rabbits often do not begin in the middle. A piece might be dragged up a flank, but this would not in itself force a capture, as it would create a threat in only one trap. Many flank pulls are not worth the time they use up, though certain threats can be game-changing; if an elephant is stuck defending its camel, neither piece can counter the enemy camel elsewhere.

Hostages

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In this game, the silver camel has been taken hostage.

In the diagram, Gold has taken the silver camel hostage, threatening to capture it in c3 if the silver elephant leaves. The c4 cat could not defend the hostage alone, as the cat could itself be captured if the trap had no other silver defender. Since an elephant on b3 could normally pull a non-elephant defender into the c3 trap, this type of hostage can only be defended by an elephant or by a team of well-placed weaker pieces. A solid camel hostage will make the forces asymmetrical: with the gold elephant holding the silver camel hostage and the silver elephant defending it, the gold camel is the strongest free piece. Until something changes, there is not even a threat to a gold horse. With the gold camel and horse both active in the east, Gold might soon overload Silver's defenses. Having the strongest free piece is a large advantage, if that piece is well-positioned and supported by other pieces.

Without a strong alignment, holding a hostage can be costly. An elephant deadlock at one's home trap is a potential space disadvantage, since the opponent can safely advance pieces toward the deadlocked trap, but the home elephant is not available to ensure safe advances of its own friendly pieces. A home hostage-holder often can't easily leave, as the former hostage might then team up with its elephant to force captures in the trap, clearing space for a goal. In fact, the defending elephant often has better prospects for leaving; its friendly pieces might swarm the trap and soon defend it independently. That could be a devastating turn of events for a hostage-holder, as the enemy elephant would become the strongest free piece. To prevent an easy rotation by the defender, a hostage-holder should usually have an additional strong piece standing guard.

Given such costs, one must be selective about hostage-taking. In general, a long-term hostage should only be held by a just-stronger piece; while the silver camel remains, the gold elephant must always be somewhat prepared to confront it, and thus should not be stuck in a corner while that camel is elsewhere. Conversely, it is usually fine for one's own elephant to defend a horse hostage held by the enemy elephant; if the elephants are deadlocked and both camels are elsewhere, the side with the more centralized elephant will have a freer camel, as that camel will face little danger from an elephant in a corner.

A camel could take a horse hostage, but a hostage-holding camel might be vulnerable to the "defending" elephant. To maintain a hostage position, a non-elephant hostage-holder may need supporting pieces. A solid horse-by-camel hostage can be quite effective, however, if the hostage-holder's own elephant is free.

Attacking

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In this game, Silver is making threats in c3 and c6.

If a piece cannot be taken as a strong hostage, it can safely advance along with its elephant. When the enemy camel is decentralized, a horse might advance on the other wing. In the game shown, the silver horses advanced with impunity because the gold camel was far away; Gold could have decentralized his elephant to hostage a silver horse, but that would have been a bad move overall.

Trap control concerns one's ability to make capture threats and defend against capture threats in a particular trap. A trap attack entails an attempt to control an away trap and thereby overload the opponent's defenses. If nothing else, an elephant could defend its home trap, but that would often give the attacker opportunities elsewhere. Here, Silver has a strong attack on c3, where he can make multiple captures unless the gold elephant moves to c4, which would leave the a6 dog to be captured in c6.

Seeing his overall weakness in the west, Gold looked east and counterattacked f6, but this was too late; Silver captured two gold pieces in c3 and then forced a rabbit through to goal. Even had the gold elephant defended c3, Gold's long-term prospects would have been bleak; more silver pieces could have advanced in the west, and the silver elephant likely could have gone elsewhere while a team of weaker silver pieces kept shared control of c3. Away trap control creates a space advantage, capture threats, and potential goal threats. Had Gold attacked f6 sooner, there might have been a capture race.

Home and away games

There are two basic plans: try to take control of an away trap, or aim to create capture threats at home. Since rabbits can't retreat, they can be pulled out gradually. Rabbit pulling was once routine, but fell out of favor as trap attacks caught on. One who simply chases and drags pieces may quickly find a home trap under siege. Homeward pulling has its place, but should perhaps not be one's main strategy.

Instead of thinking in terms of home and away play, one should think about the whole board. For a while, any possible move could be met with a solid defense or counterattack. The object is to attain the stronger overall position.

Elephant mobility

The elephants are extremely important, but cannot do everything at once. A recurring theme in Arimaa strategy is elephant mobility. Sometimes an elephant gets blockaded, but more often it is restricted by the cost of leaving a particular trap. Before getting into a position that will require one's elephant to stay put, one must consider the opposing elephant's situation. If one elephant is even slightly more free than the other, this can snowball, since it affects how free other pieces are.

Strategy vs. tactics

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Gold has a far superior board, but will lose the game unless he immediately occupies b2. (Game)

Though strategy is vital, tactics should not be overlooked. A goal wins the game regardless of the rest of the board; a strong position may mean nothing if one forgets to defend goal. Beyond that, one must watch for hanging pieces and false protection. If a player is careless in this regard, the opponent might quickly gain ground.

Camel Hostage

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To capitalize on this camel hostage, Gold must create a second threat.


The position at right is a basic example of a camel hostage. If the silver elephant left the trap, Gold could capture the silver camel by flipping or pulling it onto c3. For now, Silver can only defend with the elephant; 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.

If Silver ever abandons her 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. The silver camel is not held on b3, as Gold must occupy that square to keep shared trap control and thus protect the gold cat. Any gold piece on d3 could be pulled away and perhaps threatened in c6.

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. The gold camel is thus a grave threat in the east.

Use of free pieces

Gold has wisely advanced a horse ahead of his camel. At this point in the game, even the strongest free piece can't accomplish much by itself. Moreover, Gold doesn't want to lose his own camel; the silver elephant could get a head-start east, and Silver might block the gold elephant's path for long enough to force a capture in f6. A horse-for-camel trade would be acceptable for Gold, but an even camel trade would waste the hostage.

Advanced horses can protect not only the camel, but also weaker pieces. To defend the camel hostage, the silver elephant need only finish each turn on c4, d3, or c2; the silver elephant could thus dart around the trap and perhaps dislodge a small gold piece, which might then be delivered to c6 or f6. The gold horses are prepared to defend those traps. The western gold horse must keep some distance from the silver elephant, to avoid being frozen on c5 and then captured in two steps.

Gold now wants to create a second threat and force Silver to make a tough choice. A silver horse can defend against a gold horse, so the gold camel will be key. Gold could attack the f6 trap, but if he wants to play it safe, he can keep his camel at home 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 silver elephant defends the silver camel, horses are the strongest silver pieces which could defend f3, but that would be difficult with the gold camel right there; any capture threat at f3 would likely overload Silver. With nothing to stop the gold camel and horse in the east, several silver pieces could ultimately be lost there; the silver elephant cannot simply stay in the west.

Silver to move could trade her camel 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. The silver elephant could then go wherever it was needed. The loss of a camel for a horse is nothing compared to what Silver could have soon faced if she didn't move her elephant. Alternatively, Silver could play hb5ws Da2e ma3s, unfreezing and burrowing her 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 not free it, but would buy Silver time, as Gold would then need two turns just to reestablish a solid camel hostage, this time with the camel on b2.

Gold to move can stop 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.

Note:
In Arimaa, the strength and function of a piece depend on which other pieces remain. 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 camels had initially. On a depleted board, a horse hostage may amount to a camel hostage, and dogs may function as horses would in a camel hostage position.

Multi-piece defense

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The silver elephant has regained mobility, so Gold has no advantage from holding a camel hostage.

When the opponent has the strongest free piece, this must be changed before he can capitalize. When the gold camel is active, the silver elephant cannot perpetually defend a hostaged silver camel. Fortunately, it might not need to. While one's elephant defends an away trap, friendly pieces can safely advance toward it. This often allows one to rotate the elephant out of hostage defense, replacing it with two weaker defenders supported by other friendly pieces. On 63s of this game, Silver rotated out his elephant. Suddenly the tables have turned, and Silver's elephant is the strongest free piece; the gold elephant can scarcely afford to move, especially with the silver rabbits threatening goal. Gold could retake e3 or f4 with a pull-and-replace, but Silver could punish such a move.

A nearby strong piece can strengthen a hostage position. A gold horse or camel could pull a silver dog away from the trap, making elephant rotation more difficult for Silver.


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Gold has just freed his elephant, but still has a fight ahead.

On 28g of this game, Gold completed an elephant rotation just in time. Silver had a strong attack in the southwest, but the silver strength concentrated there made the northeastern rotation easier for Gold. Gold had no goal threat and no piece on f6, so the silver elephant had some degree of freedom. Meanwhile, Silver's western goal threat remained formidable; Gold's elephant rotation hardly sealed the game.


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This hostage position makes the eastern gold horse the strongest free piece.

On 13g of this game, Gold held the silver camel hostage and was strong on both wings, yet could not force a capture in c3 or f3. The well-placed silver elephant prevented 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 left the c3 trap, Silver would soon wipe out Gold's southwestern forces. If the silver elephant left e3, the gold camel could go west or north and do tremendous damage. The silver camel obviously isn't free, and the gold camel can't go anywhere without quickly being confronted by the silver elephant, which could safely leave e3 if the gold camel advanced. The silver horses are not free, as they are defending the silver camel and each other. The western gold horse is not totally free, as it is stopping a potential Silver swarm of c3, which could give the silver elephant more freedom.

The eastern gold horse is the strongest free piece; Gold can soon create an away threat that will force captures somewhere. The gold camel should stay in place for now, but the f2 dog could advance and be replaced by a cat or rabbit. The silver elephant might then go north and make multiple captures, but Gold could more than make up for that by sending his camel west. Even if both silver horses escaped, Gold could capture the silver camel and minimize his own losses, as Silver would have lost time getting her horses to safety.

This example and the next show that an elephant not directly defending a hostage can still be restricted by it. Rather than free the defending elephant, an elephant rotation sometimes just ties up more material in the hostage position.

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Each side currently has an elephant, camel, horse, and dog tied up in this hostage position.

In this game, Gold's elephant rotation left the silver elephant blockaded. Despite this, neither side had a clear advantage in free pieces. The gold elephant is not actually free, as it is the only thing between its northeastern army and the silver camel. Silver might move her camel south to attack c3, but then the gold elephant could safely go south also, becoming the strongest free piece. Silver might do better to send the c2 horse east, where it would be the strongest local piece and could work toward a goal threat. Such a threat would be stronger if the second silver horse were free; having a friendly horse blocked in weakens a hostage-holder's position.

Camel hostage value

When Gold holds the silver camel hostage in the southwest, Silver likely wants to advance pieces in the west while Gold wants to advance in the east. Gold wants to create a second threat, while Silver wants 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. For Gold, a camel hostage will tend to be worthwhile if two strong gold pieces are already on the non-hostage wing, and Silver is not well-developed on the hostage wing. Ideally, the free camel and a friendly horse will work together on the non-hostage wing, and the other friendly horse will guard against a potential elephant rotation on the hostage wing. This is not the only alignment for an effective hostage, but the free pieces are always key, and there is limited time to maneuver them. If the hostage-holder cannot soon create a second threat, the hostage may be weak.

If Silver is not well-positioned for an elephant rotation, she might abandon her camel and seek compensation elsewhere. While creating a second threat, a camel hostage-holder will likely expose a friendly horse to capture; the hostage value is thus related to the value of a horse-for-camel trade. A camel is estimated to be worth a horse and cat as an initial trade, so one might say that a solid camel hostage is worth a cat. This should be kept in mind if one has to choose whether to take a hostage or do something else.

If Silver is well-positioned for a quick elephant rotation, a hostage might be of negative value for Gold. The purpose of a camel hostage is to tie down the enemy elephant and camel, thus freeing one's own camel and/or horse to attack. If the enemy elephant quickly becomes free while one's own elephant remains effectively stuck, this has backfired badly.

Rotation strength

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This is typical of a successful elephant rotation; a team of silver pieces has freed the silver elephant and cornered the gold elephant. The a4 rabbit ensures that the silver dog could step right back to c4 were it pulled to b4. The c3 rabbit blockades the trap; if c3 were clear, the gold elephant could push the horse to e3, perhaps threatening it in f3 if the overall position allowed for that. The gold elephant must stay beside the trap, as Silver could quickly goal if space were cleared. If a gold horse or camel is nearby, Gold might hope to reassert ownership of the trap, but the silver elephant could likely thwart and perhaps punish such a move.
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Here a gold rabbit forms part of the blockade. Since rabbits can't step backward, they can sometimes be used against their own friendly pieces. This is why players often do not place a rabbit behind a trap; if pushed or pulled onto c3, a gold rabbit can't step back to c2.

The gold elephant has some mobility, as Silver does not yet have a strong goal threat in this quadrant. Gold could push the camel to b1 and perhaps slide the c3 rabbit to b3, clearing space for further maneuvers. This is a possible advantage of holding a hostage on b2; the hostage could be buried if necessary.

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The gold horse in the trap makes rotation difficult for Silver. If the silver elephant leaves, Silver will have to keep the gold horse off of c4 or d3; if the gold horse got onto either square with the silver elephant elsewhere, the silver camel likely could not be saved unless the silver elephant returned.

Silver might consider flipping the gold horse away, but Gold could occupy d2 and e3 to prevent this.

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The trap is clear, but the gold elephant can't move through it, as the trap currently has no other gold defender. The silver elephant has left the quadrant, but may still be nearby; how free it is depends on the rest of the board. Gold could become very strong if he could threaten the silver dog or horse, as those pieces and the silver camel could potentially fall like dominoes. Once again, the a4 rabbit ensures that the silver dog could return to c4 if pulled to b4. The a2 and b2 gold rabbits are also key to Silver's position; if Gold could clear b2, the silver horse would not be safe on c2, from where it could be pulled to b2. That would give Gold a double hostage and also allow a gold defender to occupy c2.

If the silver horse and dog are safe, this is a strong position for Silver. If either piece can be dislodged, however, this could soon become a strong position for Gold, especially if the silver elephant is far away.

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A hostage held by a c2 elephant alters the usual alignment, since c2 is typically held by a weak piece. With the gold elephant on c2, b3 may be vulnerable. Here the gold horse may not be able to stop an elephant rotation, as Silver could occupy b3 and b4 if the gold horse stepped east. A hostage held behind the trap may therefore be weak.

If d3 and d2 were unoccupied, the gold elephant could move to d3 and pull the silver camel to d2 as a central hostage. Silver can usually prevent such maneuvers.

Other hostage patterns

Double hostage

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A double hostage is stronger than a simple camel hostage.

In this game, the gold elephant held a double hostage. With three strong silver pieces tied up, Silver's options are quite limited. Unless Gold is reckless, 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. Neither hostage can escape; the g2 horse is frozen, and the silver camel is blocked by a phalanx.

A silver rabbit or cat on g4 could weaken this double hostage; if the silver elephant then left, any capture the gold elephant could make would land it on a square other than g3, so the second hostage could escape or be defended again after an exchange. Realizing that Silver could complicate things by stepping a rabbit or cat to g4, Gold should take care not to expose his camel to quick capture, as he might only get an even camel trade which would free and strengthen the southeastern silver horse. In fact, this horse could then help create a formidable goal threat.

High hostage

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Gold has an unstable but valuable high hostage. (Game)

A high hostage is held on the fourth or fifth rank. This is a relatively rare pattern, as it is tricky to maintain, but it is always good to have one's elephant centralized if possible. A gold horse blocks Silver from unfreezing his camel via c5, but the western silver horse might eventually reach b4 and unfreeze the camel from there. The gold horse could then be lost in c6, so this is a risky position for Gold. It may be easier to hold a high hostage on a depleted board, as Silver did in this endgame.

Central hostage

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A central hostage may face capture threats in two different traps.

A central hostage is held on the d- or e-file. If the hostage-holder has good control of both home traps, a central hostage can soon become a fork: Silver can play ed6s Md7s cc8s df6e, and Gold cannot defend both c6 and f6. This hostage position is more tactical than strategic; Silver gave up a horse to trap the gold camel. If a capture cannot soon be forced, a central hostage may be worth little more than an ordinary hostage.

Other Hostages

As with a camel hostage, a horse hostage will be effective only if it gives the hostage-holder an advantage in free pieces.

Horse-by-elephant hostages

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Gold's camel is more free than Silver's.

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 occurred in this opening. If the silver elephant left the northeast 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, which if it came east might get forked between c3 and f3. The gold camel is much less restricted, as it would face no such fork threat even if the silver elephant were to abandon the northeast trap. The camels are the strongest free pieces, but Gold's camel is more free, thus the hostage does not benefit Silver. Ultimately, this is an issue of alignment; decentralizing one's elephant on account of an enemy horse is highly questionable.

For now, the gold elephant is better placed on e6 than on f5. 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. 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 free the silver elephant while the gold elephant remained stuck defending the horse.

Gold can begin to advance in the east, but should try to avoid having a second piece taken hostage while the gold elephant is still needed for defense. The more Gold has at stake in the northeast, the more aggressive Silver can afford to be elsewhere.

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The dogs are the strongest free pieces.

A horse-by-elephant hostage may be effective on a depleted board. In this endgame, such a hostage tied up the three strongest remaining pieces. Up two-to-one in dogs, Silver can dominate the rest of the board. If a second gold horse or dog remained, this hostage would likely be weak.

Horse-by-camel hostages

The camel is usually the piece that should fight an enemy horse long term. This may lead to a horse-by-camel hostage, which will ideally tie the opposing elephant to defense while the hostage-holder's elephant is free to roam. However, the "defending" elephant can often attack the hostage-holding camel, and thus turn the tables. If the hostage is defended by the enemy elephant, a hostage-holding camel will need friendly support. If a solid horse-by-camel hostage is feasible, it may be worth using considerable material on, since it can give one the only free elephant.


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These diagrams illustrate different horse-by-camel hostage configurations, with Gold holding such a hostage in the southwest. A gold piece is always on a4, to keep the gold camel mobile if it is pulled to b4. In the first diagram, note that the b2 cat allows capture of the hostage if the silver elephant goes to b4. Gold should leave d3 clear, so that the camel could finish on d3 after making that capture. The b2 cat and empty d3 square indirectly protect Gold on the b-file; if the silver elephant could afford to step west and begin the next turn on b4, things could quickly turn around. As things stand, this may be a solid hostage position for Gold, whose own elephant is the strongest free piece for the time being.

Without a gold piece securely on a4, the hostage would be weak. With ec4we Mb3n ha3e, Silver could have her elephant on c4 and her horse on b3; if the gold camel were frozen on b4, Silver would have strong capture threats in c3, and could also flip the gold camel to c5 with a threat to capture it in c6. The a4 square is thus crucial to such a hostage position, and Gold does well to have a horse on that square, as a weaker piece could be pulled away more easily.

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In the second diagram, the hostage horse is on b2 rather than a3; this has implications if the camel is dislodged. For instance, if the silver elephant moved to b4, it would then threaten to capture the camel due to false protection. On the other hand, Gold now has the option of pushing the horse to b1. If the silver elephant left, the horse could then be pulled back to b2.

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In the third diagram, the silver elephant cannot easily displace the gold camel. As long as Gold can hold b3, Silver could break this hostage position only by decentralizing her elephant. This can be the strongest type of horse hostage, provided there are enough pieces to support the camel while its elephant is active elsewhere.

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


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Gold cannot save both horses.

While a horse-by-elephant hostage is usually a bad long-term strategy, it may be an effective tactic if the hostage-holder can make a quick second threat. In this opening, Silver took both gold horses hostage, and Gold did not have time to defend both traps.


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To break this hostage, the gold elephant would have to go to f7. (Game)


When a hostage-holding camel is behind a trap, the defender might be no better off if he pushes the camel away. From here, Gold played Ce1n Ee6n mf7s Ee7e; Silver moved the d6 rabbit to e6, and Gold had no active strong piece. When the gold elephant left f7, the silver camel retook the hostage.


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Silver's horse-by-camel hostage allows silver dogs to control an away trap.

By tying up the gold elephant and horse, Silver's horse-by-camel hostage at right enabled silver dogs to control f3. This led to a 29-turn win.


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Gold holds a camel hostage in the east, and a double hostage in the center.

In this game, Gold's central horse-by-camel hostage stopped Silver from forcing a western goal; with the d2 horse frozen and other silver pieces blocked out of the southwest, there was nothing to help the silver elephant counter Gold's defense. This would be a strong position for Silver if he could displace the gold camel, but Gold to move can blockade c3 and perhaps move toward a northwestern goal threat. With the southern forces tying each other down, the western gold horse is the strongest free piece; the silver elephant must stay beside c3, and cannot freeze the gold horse in place. One way or another, this horse can soon accomplish something.

Cat and dog hostages

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Silver does not have an ideal alignment, but may eventually overload Gold. (Game)

A smaller piece may also make a valuable hostage, if its elephant is the only piece which can defend it. In this example, the silver camel holds a gold dog hostage next to c6. The silver elephant is free to pull gold pieces toward f6. The gold elephant can't defend both traps.

However, Silver's situation is less than ideal, for two reasons. First, the gold elephant could currently leave the c6 trap without losing the c8 dog on the next turn, since the silver cat would have to leave the trap square to allow for the capture. Second, the h5 gold rabbit will make it harder for the silver elephant to threaten the gold camel. Silver might have to capture the h5 rabbit before making any other threat in f6.

Silver would have a greater advantage if her horse held the dog hostage and her camel were free; with the gold elephant stuck defending against a silver horse, the silver elephant and camel together could make a strong threat elsewhere. Like a horse-by-elephant hostage, a dog-by-camel hostage is at best non-ideal in most situations.


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Gold's dog-by-horse hostage allowed the gold camel to safely advance to g6.

In this game, Gold's early dog-by-horse hostage allowed the gold camel to lead an attack on f6. An early camel advance would often result in a camel hostage, but the southeastern hostage gave Gold time to advance his camel and support it with rabbits.

Frames

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The gold elephant is pinned to the horse framed on c6.

A piece held on a trap square, securely surrounded on three sides, has been framed. In this game, Silver framed a gold horse on c6; a phalanx stops the framed horse from pushing onto c7. The gold elephant is pinned: if it takes one step, the framed horse will disappear.

Unlike a hostage defender, which must merely end each turn adjacent to the trap, a pinned piece is completely stuck. Here, it would be pointless for the pinned gold elephant to break the phalanx; even flipping the d7 rabbit would instantly lose the framed horse. If a pinned piece abandons a framed piece, the framer won't even use up a turn making the capture, since the piece is already gone. This strength of frames is balanced by the high material cost of maintaining them.

When possible, a framer should rotate out strong pieces, replacing them with the weakest ones which can reliably hold the frame. If the a5 horse moves to c5, the silver elephant will be the strongest free piece, although it may not be totally free if Silver wants to avoid an even horse trade.

Even if rotation is not feasible, a frame involving both elephants will usually give the framer an elephant mobility advantage. Gold would instantly lose the framed horse if he moved his elephant, whereas Silver could move his elephant at no cost other than giving up the frame. It would thus be risky for the gold camel to advance, whereas the silver camel is already in Gold's home territory.

A frame may be broken by the arrival of a piece strong enough to dislodge a framing piece. If a piece on a trap gains a second supporter, or gets off the trap square, it is no longer framed. A frame might be broken from the side or from behind, but here Silver can likely prevent either maneuver. With the gold elephant stuck, the gold camel is the only piece that could dislodge a silver horse, but Silver could likely stop such an intrusion in the west. Gold might instead aim to break the c7 phalanx. Since the gold elephant is immobile, breaking the phalanx would require a gold piece to be safely on e7 or d8; on Gold's next turn, this piece could pull away a rabbit, allowing the c6 horse to push its way out. Silver can strengthen the phalanx by sliding the d7, c7, b7, and a7 pieces east.

Frames vary widely in value, depending on each side's free pieces. A breakable frame might have negative value, since the formerly framed piece could become an attacker.

Frames are most effective when there is enough material to hold a frame and still accomplish something else. Even a solid frame may not be worthwhile on a depleted board.

Elephant position

When a gold piece is framed on c6, it is too late for the pinned gold piece to choose its square. In general, however, d6 is a strong square for an advanced gold elephant, and this often holds true when that elephant is pinned:

  • With the pinned gold elephant on d6, the frame requires a silver piece on c5 (except in the case of rabbit frames). If the silver elephant goes east, the c5 piece may be vulnerable to capture in c3.
  • The pinned gold elephant on d6 stops Silver from moving the framed piece onto d6 or e6 for a fork.
  • For the gold elephant, d6 is a better attacking square than c5. If the frame can be broken, Gold may then have a strong attack.

On the other hand, the pinned gold elephant may be better placed on c5 if the framed gold piece is not at imminent risk of capture in f6. With the pinned gold elephant on c5, a gold piece might have an easier time attacking the frame along the b-file.

Camel frames

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Gold combines a camel frame with a horse hostage. (Game)

When an elephant is pinned to a camel framed by the enemy camel and elephant, the four strongest pieces are all tied up. If all four horses are free, and the framer cannot rotate out his elephant or camel, the game may move slowly. Due to elephant mobility, the framer's horses might be freer than their enemy counterparts, but the framed side might counter this by advancing pieces to set up a strong attack should the framing elephant leave. The framed side could also partially blockade the framing elephant.

If the framing elephant or camel can be replaced by a phalanx of weaker pieces, the framer will have the strongest free piece while still rendering the two strongest enemy pieces completely immobile. The diagram shows another strong type of camel frame, in which a horse is held hostage by the framing camel. There is no single strongest free piece, but Gold has two free horses against Silver's one. Gold's progress may be slow, but Silver has little counterplay, having so much to lose if she abandons f3. A hybrid frame-hostage can be very strong, if balance is maintained.

Away frames

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An away horse frame, from this game

In the course of a trap attack, it is sometimes possible to frame an opponent's piece in his own home trap. Gold's frame at right is strong; the pinned silver elephant can't think of leaving, as there would be catastrophic losses in f6. From here, Gold can advance rabbits on the h-file to solidify the space advantage.

Gold would not benefit by rotating anything out of this frame, even if other gold pieces were closer. A phalanx would not be viable, and replacing the gold elephant with the camel might let Silver get a horse-for-camel trade if the gold elephant didn't stand guard.

The one potential threat to the frame is the silver camel. If it went east, however, the gold camel would be a strong threat in the west.

If Gold can advance enough pieces to support his horses, he might eventually want to move his elephant elsewhere; at that point, Gold should flip the silver horse out of f6 and then push it away, so that it couldn't simply step to e6 and help the silver elephant retake control of the trap. If both gold horses were securely beside f6 even with the gold elephant elsewhere, Gold could quickly overload Silver.

Non-elephant pins

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The silver camel is pinned to the c3 cat, and both pieces will be at risk if the gold elephant can step to d4. (Game)

In this position, the silver camel is pinned to the c3 cat, and the silver elephant must stay on d4 to defend both the cat and camel. This leaves the gold camel as the strongest free piece; if this will remain so, the silver camel should abandon the cat.

However, Silver might hope to replace the elephant with a phalanx; this might also require a piece on c5 or b4 to prevent a flip. A d4 phalanx would divide up the board; the gold elephant might be forced to give up the frame and make its way east, but then the position would be unclear. A similar position with the camel instead on b3 might be good for Silver, since Silver's trap control could be fairly solid. See, for instance, the blockade on 29s of this game.

Horse baskets

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The gold horse cannot escape, and might soon be framed.

In this position, the gold horse is in a basket: it is blocked on three sides, so that no escape is possible even if it is unfrozen from the fourth side. Silver to move could flip the gold horse into the trap, creating a frame which the silver elephant might soon rotate out of. Gold to move might possibly prevent a frame via congestion, if he can occupy both f4 and e5 while still defending the trap. Alternatively, Gold could delay the frame by flipping the e7 dog, although this might lose time if Silver plays correctly. With e7 empty, Silver should not flip the gold horse into the defended trap, as the horse could then push onto f7, where it would be well placed. Instead, Silver might pull the gold horse to f5 and move the h5 horse to g5, fencing the gold horse next to the trap. On Silver's next turn, the gold horse could be pushed into the trap in two steps, leaving another two steps to reestablish the phalanx.

Since rabbits can't retreat, a silver rabbit on the wrong square could end up blocking the intended frame. A silver rabbit on e7 could be flipped into the trap. A rabbit on f7 would make a fence less effective; with the gold horse on f5, the gold elephant could pull an f7 rabbit into the trap.

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If Gold moves his elephant to f5, his horse cannot soon be framed.

If a basket instead leaves f5 open, the gold elephant might move to that square and thus prevent the horse frame. There would then be no immediate threat in f6, and it might be dangerous for the silver elephant to remain on the flank, where it could get blockaded.

Frame-breaking tactics

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The above frames would all be hard for the defender to break. Some frames are precarious, however, and may even end badly for the framer. A camel with friendly support can sometimes break a frame from the side. In the diagram at left, Silver to move can play a pull-and-replace, getting the silver horse onto b3 (mb5sn Hb3n hc3w). Although Gold could do a pull-and-replace to restore the frame, Silver would win the repetition fight; by undoing Silver's move, Gold would restore a position he created previously, namely the position in the diagram. Once the frame is broken, Silver will have a good position: the formerly framed horse will be a strong attacker, and the camel might pull the gold horse to c6.

Gold to move could delay the frame-break with Ec4ns mb5e. If there is a silver piece on d5 or c6, however, the camel could return to b5 in a single step, thus undoing in one step what Gold did in three. Instead of wasting steps and letting Silver strengthen her position, Gold to move might flip the silver horse to b4; the horse might escape, but at least it wouldn't take b3. Gold might soon capture a silver rabbit.

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The second diagram shows a similar position, but with the silver camel on b4 rather than b5. Now, the frame can be broken in three steps (mb4n Hb3n hc3w), but there is a complication: Gold could restore the frame with a pull-and-replace (Ec4sn hb3e Hb4s), and this time Gold would create a new position, since the silver camel would not be where it was after Gold's last turn. For this same reason, Silver would then need four steps for her own pull-and-replace, and thus would create the exact position she created the first time she broke the frame: the silver camel on b5, the gold horse on b4, the silver horse on b3, and nothing else different since she had no extra step. Thus, Gold would win the repetition fight, unless Silver found a different way to break this frame:

  • mb4n Hb3n hc3w Cc2n pulls the gold cat into the trap to prevent Gold's pull-and-replace. This is the simplest, and thus most common, maneuver to break such a frame.
  • mb4n Hb3n hc3w ed3w: The silver elephant occupies the c3 trap, and the silver horse cannot be dislodged from b3. An elephant occupying a trap must be very careful not to get captured.
  • mb4n Hb3n ra3e ra4s: Silver rabbits block the gold horse out of b3.
  • mb4n Hb3n mb5e Hb4n: By moving her camel to c5 and pulling the gold horse to b5, Silver establishes a phalanx which stops the gold elephant from stepping forward and unfreezing its horse. Gold could unfreeze the horse from below, but then the horse couldn't return to b3.

In this second diagram, note how the d4 cat prevents a camel flip, which would buy Gold some time. A silver piece on c5 would do the same job.

In both diagrams, Silver's a4 rabbit keeps the camel unfrozen on b4. If instead a gold piece were on a4, the frame would be stronger.

Elephant frame

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Silver must break this frame and move his elephant, before the gold camel can dislodge the pinned silver horse.

An elephant frame is occasionally possible, if an elephant has chosen to occupy a trap square. If the pinned piece can be dislodged, the framed elephant will be captured. On 12s of this game, Silver likely did not see the danger he was putting his elephant in. Even on 13s, Silver did not quite see the two-turn forced elephant capture he faced if he didn't get his camel to e5 or d4. It is not always bad to place one's elephant on a trap square, but extreme caution is needed; an elephant should never be left in a trap that is not adequately defended.

Trap Control

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Silver's long-term threat to c3 is strong.


At the outset, each player essentially has full control of his two home traps. Trap control depends on the key squares, especially the decentralized ones (e.g. b6 and c7 in the northwest, b3 and c2 in the southwest). Getting a non-elephant safely onto a decentralized key square of an away trap is an important step toward wresting control of that trap.

In this opening, a horse procured a long-term trap control advantage for Silver. Although Silver had no immediate way to force a capture, Gold had no direct way to deal with the intruding silver horse: the silver elephant would stop the gold camel, and Gold could not afford to decentralize his elephant just to deal with a horse. From here, Silver swarmed the c3 trap and overloaded Gold.

Deadlocked Traps

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There is an elephant deadlock at f6 and a camel deadlock at c3. (Game)

A trap is deadlocked when its strongest gold defender and strongest silver defender are equal. If both elephants stay beside the same trap, it is usually because both sides have a large stake there. Remember that a hostage position can tie down both sides, as a hostage could become an attacker if the hostage-holder simply left. Here, the silver elephant is not holding a hostage, but is defending against an attack which more gold pieces might soon join.

A camel deadlock will hold only so long as neither elephant can be bothered to break it. Here, Silver is well up in material, but has more to lose than does Gold; the silver camel and hostaged horse would both be at risk if the gold elephant went west. What happens in the northeast will help decide what happens in the southwest.

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