Arimaa/Print version


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. Relative Value of Pieces
  13. Advanced Tactics
  14. Setup
  15. Race Positions
  16. Trap Attacks
  17. Lone Elephant Attacks
  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 until 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. Face-to-face, it might 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 lasted as many as fourteen rounds. Jean Daligault was the world champion six times from 2007 to 2014, and then Mathew Brown was champion five times from 2015 to 2021.
  • From 2004 to 2015, the Computer Championship matched the top Arimaa bots in an elimination tournament. David Fotland's program Bomb placed first each year from 2004 to 2008, even though Fotland didn't update it after 2005. Stronger bots emerged in 2009; from then on, there was no one dominant bot until David Wu's Sharp went undefeated in 2015.
  • 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.

After DeepMind's AlphaZero mastered Go, Chess, and Shogi simply by playing itself, neural networking Arimaa bots were developed. Rusty_Zero is currently the top bot, with an Elo rating near 3000.

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. 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. Diagonals have no place in the rules of Arimaa.

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, and must always 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 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 portrays a position which could occur later in a game.

A piece which is adjacent to a stronger enemy piece is frozen, unless it is also adjacent to a friendly piece. While frozen, a 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. Gold could unfreeze the b7 rabbit by moving the gold dog from a6 to a7, or by displacing the silver dog from b6. 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.

In a 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. In a 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.

A frozen piece can be pushed or pulled by the opponent. A non-frozen piece can also be pushed or pulled — an adjacent friendly piece does not protect against this. For example, the d1 cat does not protect the c1 cat from being pushed to b1 or pulled to c2. Whether a piece is frozen is unrelated to whether it can be dislodged.

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, as occurred on 83g of this game, the game continues.

Setting up

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Gold has used the classic 99of9 setup; Silver has switched the dogs and cats.

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.

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, so that it could defend either home trap and potentially punish 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 step forward, 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 and thus protecting other gold pieces from a quick capture at home.

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 dog might later advance through the trap and be replaced by a cat or rabbit.

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 play Arimaa 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 ec3x Hd4s. 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.

When an actual game is cited, a turn is referred to by a number and "g" (gold) or "s" (silver). 1g and 1s are the setups, 2g is Gold's first regular turn, and 2s is Silver's first regular turn.

While this notation may seem verbose, each step must be accounted for with no ambiguity.


Introduction to Tactics

A tactic is a narrow plan. The most basic tactics involve a one-turn goal or capture. When the opponent has thousands of options, it is difficult to precisely plan beyond the current turn. Some moves, however, might severely limit the opponent's viable options. One facing an immediate goal threat must stop the goal or else lose the game. A capture threat may leave the opponent with only a few ways to avoid a substantial disadvantage. One creating a threat might thus have a response planned for any possible countermove by the opponent.

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; the dog 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.

With friendly support, a rabbit one or two ranks from the goal line might reach goal even if its path is blocked. 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 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 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 could either move his elephant onto c3 or use the c2 horse to push the rabbit west.

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. After the opponent defends goal, one can either press the goal threat further or take advantage of any new weakness resulting from the goal defense. One should always be aware of possible goal paths on both sides, so as not to be caught off guard or miss an opportunity. On the other hand, one should be cautious about advancing rabbits, which can never step backward.

One-turn capture

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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.

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 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 push or pull the b3 dog, capturing the c3 horse.

The g3 dog can capture the g2 cat by pushing it west and then north. Alternatively, the dog could move through the trap and pull the cat along; with the gold horse on e3, a gold piece can move through f3. A third way to capture the cat would be to pull it to g3 and then push it into f3; the dog would move north and then south. A pull-push in which a single piece moves an enemy piece two squares is known as a flip.

Capture defense

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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 must then 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. 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; when a trap has two defenders, any single attacker would need more than four steps to dislodge one defender and capture the other. 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.

A capture can sometimes be blocked. A phalanx blocks the gold camel from g7, and thus the f7 horse is currently safe even though it is Silver's only direct defender of the f6 trap. In other cases, one might block a capture by pushing an enemy piece into the capture path.

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 f3 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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Although not immediately obvious, a one-turn capture is possible in each trap.

It is easy to become too confident about one's defense. In the northwest, the two silver defenders will not prevent a capture in c6, as Gold has a stronger piece next to each. If the gold elephant pushes the silver dog from b6 to c6, the newly unfrozen gold cat can then push the silver rabbit off of c7, 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 two defenders do not prevent a one-turn capture, 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 ensured that the b3 horse cannot be pushed. Gold has overlooked another possibility, however; after pushing aside the b4 rabbit, the silver elephant can pull the gold horse 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.

The fork threatens a single piece with capture in two different traps. On 62g of this game, Gold forked the silver dog between c3 and f3. The dog could not escape, and Silver did not have time to defend both traps, as the e4 rabbit threatened goal and could not be frozen in place. 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. Instead of capturing the c4 rabbit, Silver uses it to help blockade b5 so that the gold 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 offensive tactic. A piece is brought next to a trap, and hemmed in on two sides. Even if unfrozen from the fourth side, the piece could only step into the trap. On 14g of this game, Gold fenced the silver horse next to the c3 trap. Silver is to move, but can't save the horse. The silver elephant could step to c5 (incidentally 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, one might keep the fenced piece in place as a hostage, and perhaps blockade the trap square to stop the piece from moving through it. 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. Early in this game, a gold horse and camel both appeared to be at risk, 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. A threat may create an advantage which can gradually be built upon, even if no captures happen for a while.

Direct goal

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If he is not distracted by a camel or dog capture opportunity, Gold to move can easily stop the silver goal and threaten the advanced silver rabbit with capture. (Game)

The ultimate object of Arimaa is to get a friendly rabbit to goal. If the opponent pays any attention to defense, an early goal is unlikely. It is difficult to directly tear a hole into a well-defended goal line, as pushes and pulls can be blocked. Until there is a substantial opening in the goal line, a rabbit advance should not be regarded as progress toward a 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.

Capture

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

Capture threats are important, whether or not they lead to actual captures. 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, an elephant can camp out beside a trap where there is a threat; since nothing can dislodge an elephant, its friendly pieces can never be captured in a trap it stays next to.

Often, 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 dragged up a flank would only be threatened in one trap. Many flank pulls are not worth the time they use up.

Hostages

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In this game, the silver camel was 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 beside its home trap could normally pull a non-elephant enemy piece into the 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 might 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. Generally speaking, it is no good for an elephant to hold a horse hostage while the enemy camel is elsewhere. Conversely, it is usually fine for an elephant to defend a horse hostage held by the enemy elephant, as the defending elephant might hope to rotate out, or trade its horse for the enemy camel if that camel advances.

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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Silver has taken control of the c3 trap, creating a large problem for Gold.

If a piece cannot be taken as a strong hostage, it can safely advance along with its elephant. If the enemy camel can be cut off from one wing, a horse might quickly advance on that wing. In the game shown, the silver horses advanced with impunity because the gold camel was far away. Silver was not worried about the gold elephant taking a horse hostage, as this would not have given Gold an advantage.

To own a trap square is to be safe from capture therein while the opponent is not. A trap attack entails an attempt to gain ownership of an away trap and thus threaten several quick captures. Here, Silver has a strong attack on c3, where he can clean up unless the gold elephant moves to c4, which would leave the a6 dog to be captured in c6. An elephant can usually defend against a trap attack, but then will not be available elsewhere.

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 the gold elephant remained stuck defending against an attack which would otherwise result in a quick goal by Silver. A strong trap attack will create a space advantage, capture threats, and potential goal threats. Had Gold attacked f6 sooner, a capture race might have ensued.

Home and away games

There are two basic plans: try to take control of an away trap, or drag enemy pieces toward one's home traps. Since rabbits can't retreat, they can in principle 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.

Piece mobility

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The gold camel is in little danger of getting stuck against the edge, as it could be unfrozen from above or from below. (Game)


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 area. Before getting into a position that will require one's elephant to stay put, one must consider the enemy 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.

In the opening, camel mobility is of first concern. If a camel becomes stuck against the edge, even in home territory, the effect may be similar to that of a camel hostage. Sometimes a horse quickly advances on a flank, potentially threatening attack but also providing a way of escape for a friendly piece which might get stuck below it. Rabbits may likewise advance on the flanks to keep stronger pieces mobile.

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 be considered first. 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 threat in the east.


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 avoid losses at home. Gold does not occupy d3, as any gold piece on that square could be pulled away and perhaps threatened with capture.

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 now wants to create a second threat and force Silver to make a tough choice. In preparation, Gold has wisely advanced a horse ahead of his camel; until the silver elephant comes east, a gold horse cannot be threatened there. A material exchange is possible if Silver moves her elephant east and blocks the gold elephant's path to e6 or f5. A horse-for-camel trade would be acceptable for Gold, but an even camel trade would waste the hostage.

Advanced horses can also protect 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 could now 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 will have to come east at some point.

Silver to move could easily 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 largely on which enemy 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 thus 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. One should always consider whether a hostage would create an advantage in free pieces, whatever those pieces happen to be.

Multi-piece defense

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With the silver elephant free, this camel hostage is no longer an advantage for Gold.

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 must not perpetually defend a hostage. 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. Elephant rotation may turn the tables, as the hostage-holding elephant often cannot afford to leave the trap. On 63s of this game, the silver elephant became the strongest free piece; the gold elephant was restricted by capture threats as well as goal threats. Gold could temporarily 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 might have pulled 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 went north, the gold camel might soon dismantle Silver's defense of c3. The silver camel obviously isn't free, and the gold camel can't go anywhere without quickly being confronted by the silver elephant, which would be free to follow the camel north. 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 currently 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 moving 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. An elephant rotation does not always free the elephant; it may just tie 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.

After the gold elephant rotated out of hostage defense in this game, it still had to protect the pieces defending f6. Each time the silver camel returned north, the gold elephant went north also, so as to protect its northeastern army. If Silver now keeps her camel in place and moves the c2 horse east, that horse will be the strongest local piece and can work toward a goal threat. If the h5 rabbit advances, the h7 horse may advance behind it and join the southeastern attack. Silver would have done better not to have a horse blocked in to begin with, as that weakens a hostage-holder's position.

Of these four examples, the first camel hostage was quite weak, the second was not quite strong enough, the third was strong, and the fourth was somewhat unclear, though by 21s Silver appeared to have stronger goal prospects. If the defender can free his elephant in time to mitigate a second threat, he will likely have the advantage. The hostage-holder will usually have the advantage if he can create a strong second threat while the enemy elephant remains on defense. If both elephants and camels are tied up, the side with the most active horse may have the advantage.

Camel hostage value

A hostage-holder wants to create a second threat that will overload the opponent. Conversely, the defender usually wants to free his elephant, preferably without giving up the hostaged piece. If the defender aims for an elephant rotation, the two sides will likely advance pieces on opposite wings. The outcome of such a race will depend 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. There are other possibilities, 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.

If the hostage-holder does not have an advantage in free pieces, a hostage may be of negative value even if the defender is not poised for an elephant rotation. Being advanced and centralized, the defending elephant might eventually join an attack on the non-hostage wing. If such an attack threatens goal, the opponent might not have time to capture the camel.

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 could try to go around c3, but this would free the silver camel to attack the goal line, and Silver would likely have time to defend against any threat in f3. If a gold horse or camel is nearby, Gold might hope to reassert ownership of c3, but the silver elephant could likely thwart and perhaps punish such a move.
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Here a gold rabbit blockades the trap. Unable to step backward, the rabbit is stuck on c3 as long as b3, d3, and c4 are occupied.

Silver does not yet have a strong goal threat on this wing, so Gold has some flexibility. 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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If the silver elephant left, the gold horse might occupy c4 or d3, and any nearby silver piece would be at risk. To rotate out her elephant, Silver must either blockade both c4 and d3 or remove the gold horse from the area. 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. If Gold could displace the silver dog or horse, the silver elephant might have to return to the trap to prevent catastrophic losses. 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 displaced, 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, especially if the opponent is well-developed on the wing.

If d3 and d2 were unoccupied, the gold elephant could move to d3 and pull the silver camel to d2 as a central hostage. This is a possible advantage of holding a hostage behind a trap, but the opponent can usually block such a maneuver.

Other hostage patterns

Double hostage

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A double hostage is stronger than a simple camel hostage.

On 16g of this game, the gold elephant took 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 piece 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 the g5 cat south, 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. Silver might then have 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, allowing the hostage-holding elephant to remain centralized. This pattern is tricky to maintain, as multiple squares must be blocked to keep the hostage in place. The silver camel is a high hostage here; a gold horse and a phalanx of rabbits keep the camel frozen, but b4 is not secure in the long term. If Silver eventually unfreezes his camel from the side, the gold horse could then be lost in c6, so this might become a risky hostage 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 might turn into 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

Horse-by-elephant hostages

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As long as her elephant remains in the northeast, Silver can do little other than pull a gold rabbit on the a-file.

As with a camel hostage, a horse or smaller piece hostage will be effective only if it gives the hostage-holder a usable advantage in free pieces. This is usually not the case when an elephant holds a horse hostage, as occurred in this opening. Silver's alignment is poor; the gold elephant is a far greater threat to the silver camel than is the silver elephant to the gold camel. Gold would gladly trade his horse for the silver camel, which if it came east would be at risk in both c3 and f3. If a silver horse advanced first, it could be threatened by the gold camel. If the silver elephant simply left the f6 trap, Gold's elephant-horse attack might decimate Silver's eastern forces.

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 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 should be prepared to go after an aggressive silver piece or clean up in the east, but if Silver plays gingerly, Gold might swarm the f6 trap and rotate out his elephant.


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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 threatened both gold horses, and Gold did not have time to defend both traps.


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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 could be strong or weak. The "defending" elephant can usually dislodge a hostage-holding camel, at least temporarily; what happens next depends on what other pieces are nearby. If the hostage-holder's elephant can roam free while the enemy elephant is stuck defending its horse, this may be a large elephant mobility advantage for the hostage-holder.


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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, the b2 cat allows for capture of the hostage if the silver elephant goes to b4. Gold should leave d3 clear, so that the camel could finish that capture on d3 rather than c4, where it would then be at risk in c6. 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, a hostage on b2 could be pushed to b1 if the camel needed to back away from the elephant. The camel might then move the hostage to c1, or back to b2 when feasible.

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In the third diagram, the hostage is held behind the trap. On c2, the gold camel is less exposed and thus perhaps easier to support. A horse hostage behind the trap can thus be strong, if the area is reasonably well-fortified and the opposing army is not well-developed on the wing.

In all of these cases, the defender should consider bringing in more pieces if possible. In the last example, Silver could own the trap if she could get her other horse or a dog safely onto b3. In the first, the silver camel might attack the a4 horse, weakening the hostage pattern.


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Gold has no good way to break this hostage.


When a camel holds a hostage behind a trap, the defender might be no better off if he pushes the camel away. In this game, Gold's 13g left him even more vulnerable in the west, where Silver then advanced a horse and dog. When the gold elephant left f7, the silver camel retook the hostage. When an elephant goes to a corner to try to rescue a hostage, there is also a risk that both pieces will get blockaded.


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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 while other strong pieces were tied up in the west, 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 allowed him to defend goal in the west, as no additional silver piece could reach the southwestern corner. 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. A dog-by-camel hostage is not ideal, but may still give the hostage-holder an elephant mobility advantage.


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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 for an elephant-camel 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.

A pinned piece is more stuck than a hostage defender, which can at least move between key squares or otherwise step away during a turn. 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 often give the framer a 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 is broken when a framing piece is dislodged. A frame might be broken from the side or from behind, but here Silver can likely prevent either maneuver. With the gold elephant pinned, 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; with the gold elephant immobile, this would require a gold piece to advance and begin a turn active on e7 or d8. If this piece could pull away a rabbit, the c6 horse could then push its way out. Silver can strengthen the phalanx by sliding the d7, c7, b7, and a7 pieces east.

A frame's value depends on each side's free pieces. This frame should be very strong once the silver elephant rotates out. By contrast, a breakable frame may backfire on the framer, since the formerly framed piece could become an attacker. On a depleted board, a horse frame may not be worth the material it uses, as the framer could become thin on the other wing. A dog or cat frame could be wasteful at any juncture; a framer must always weigh the cost of keeping material tied up in the frame.

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 c6 trap.

If the framed piece is not at imminent risk of capture in f6, however, c5 is an acceptable square for the pinned gold elephant. In fact, this may make it easier for a gold piece to break 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 the framer cannot rotate out his elephant or camel, the horses may decide the game. Due to elephant mobility, the framer's horses might be freer than their enemy counterparts, but the framed side might counter this by partially blockading the framing elephant and/or setting up a strong attack should it leave.

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.

In the course of a trap attack, it is sometimes possible to frame an enemy piece in its own home trap. On 13g of this game, Gold used a pull-and-replace to frame a silver horse in f6. If the pinned silver elephant were to move, the horse would be lost instantly, and more silver pieces could quickly follow.

The one potential threat to the frame is the silver camel. If the silver camel went east, however, the gold camel could dominate the west. Gold might get a strong attack on c6 before Silver could break the f6 frame and free his elephant.

Gold should not attempt to directly rotate anything out of this frame; a phalanx would not be viable, and replacing the gold elephant with the camel might allow Silver to get a horse-for-camel trade if the gold elephant didn't stand guard. However, Gold can advance more pieces in the east and perhaps eventually give up the frame but keep shared control of f6 even with his elephant elsewhere.

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.

A dog or cat frame which ties up the framer's elephant is usually not worthwhile. In this game, however, Gold's cat frame was strong, pinning down both the silver elephant and camel. With the silver camel pinned to the c3 cat, 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. If the silver camel were instead pinned on b3 while protected by a b4 phalanx and a piece on c5 or d4, Silver's shared trap control could be fairly solid. See, for instance, the blockade on 29s of this game.


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Gold to move can frame and then capture the e3 horse.

On 28g of this game, Gold played a pull-and-replace to create a frame which then forced a horse capture, as Silver could not keep the pinned dog in place or add a defender.

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 risk to his elephant. Even on 13s, Silver did not realize that getting his camel onto e5 or d4 was the only way to break the blockade in time for the elephant to escape. 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.

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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Gold to move can prevent a frame.

To force a frame, a basket-holding elephant must usually be next to a trap. With f5 open, the gold elephant could take two steps and prevent a horse frame. The silver elephant might then have to move west to avoid a blockade.

Frame-breaking tactics

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A breakable frame may end badly for the framer. With Silver to move in the diagram at left, the silver horse can reach b3 via the pull-and-replace mb5sn Hb3n hc3w. Gold could restore the frame with a pull-and-replace of his own, but this would recreate a position which Gold has already created once. Silver could then repeat her previous move and force Gold to do something different. With the frame 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. The advanced silver rabbits might then be vulnerable.

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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. Gold could thus restore the frame and 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.

If a camel flip were possible, it would at least buy Gold some time. The d4 cat prevents a camel flip; 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.


Trap Control

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


Trap control is related to ownership. A trap might be fully controlled by one side, or might be shared. 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 usually an important step toward wresting control of that trap. Even if one doesn't take full control of an away trap, strong shared control can impact the rest of the board.

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, whose elephant stayed beside c3 and thus could not defend in the east.

Deadlocked Traps

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