Arimaa/Trap Control

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At the outset, each player essentially has full control of their two home traps. Each elephant can quickly share control of one away trap, but that leaves the question of what comes next. A common aim is to get a friendly horse onto a decentralized key square of an away trap (e.g. b6 or c7 when attacking the c6 trap). If that is accomplished and there is nothing to immediately force that horse away, its side at least has shared control of that trap even without the friendly elephant next to it. Later in the game, such an attacker might be a dog instead of a horse.

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The b2 horse effectively gives Silver shared control of the c3 trap. (game)

There are several advantages to taking shared control of an away trap:

  • It might be possible to strengthen control to the point that captures are possible in the trap. An enemy trap usually has plenty of enemy pieces nearby, which might be captured in quick succession, in contrast to the laborious process of repeatedly prying loose a single enemy piece and dragging it all the way to a home trap for capture. Pieces threatened in their own home trap might delay or avoid capture by scattering, but this is even better for an attacker, since it leaves a hole through which a rabbit might reach goal.
  • It reduces the opponent's potential capture threats. A player who owns no traps likely cannot make a strong double-threat, since any capture would entail taking ownership of a trap.
  • Since friendly pieces won't be captured on this wing, they are free to advance. This can lead to a long-term space advantage and more freedom to maneuver than the opponent.
  • Rabbits can advance on this wing, creating potential goal threats that may force a strong enemy piece to stay beside its home trap rather than attack (see Arimaa/Rabbit Advancement#Cement deadlock).
  • If the opponent is unwilling to give up the trap totally, he must commit material to its defense, leaving his forces thinner elsewhere.

The drawbacks of attacking for trap control are:

  • Attacking pieces may get hostaged, framed, or blockaded, particularly if attacking against a well-defended trap. If it is not possible to quickly gain complete control and capitalize on it, stronger defenders may arrive later, depending on the whole-board fight.
  • The home traps may be left weak.
  • If attackers cannot retreat, the loss of flexibility may be a problem. In particular, rabbit advances require care.

In the position shown, a silver horse has penetrated enemy lines in the opening, and Silver is ahead because of a long-term trap control advantage. Although Silver has no immediate way to take complete control of the c3 trap, Gold has no direct way to deal with the intruding horse: the gold camel is far away, and the gold elephant must remain available to deal with the silver camel. Silver is free to advance weak pieces on the western wing.

Deadlocked Traps[edit]

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There is an elephant deadlock at f3, and a camel deadlock at c6, but Silver to move can end both and come out ahead.

A trap is deadlocked if the strongest gold defender is equal to the strongest silver defender. A deadlock is considered tight if neither defender can easily leave. This is often the case when the elephants are deadlocked in a hostage position at the hostage-holder's home trap; one side would stand to lose the hostage piece, and the other side would stand to lose control of a home trap. If the elephants are deadlocked in a frame, the framing elephant likely has more freedom to leave, although the pinned elephant may not be as stuck as it appears.

If an elephant deadlock leaves both camels free, they might then deadlock a second trap. In the diagram, the elephants are deadlocked in a horse frame at the f3 trap, while the camels are deadlocked in a horse hostage at the c6 trap. Unlike an elephant in its position, however, the gold camel is vulnerable. Silver to move can put her elephant on c4 and dog on d5, which instantly gives up the silver horse but also dooms the gold camel. Two deadlocks can be broken at once.

When a deadlock is broken by the arrival of a stronger piece, the stronger side will often be able to own the trap. However, the course of this fight will depend on the former strength of each side's control. For example, if a camel is pinned in a camel deadlock, and the opposing elephant arrives, it will capture the framed piece on the first turn and the pinned camel on the second turn. In contrast, if a camel is pinned in a camel deadlock and the friendly elephant arrives, it will first relieve the pin, then extract the formerly framed piece from the trap, and then perhaps gain ownership of the trap and start capturing. Even if the end result is the same, the amount of time taken can be important.

Sometimes, a player willingly allows a friendly piece to be framed or hostaged, due to the space advantage of deadlocking elephants around an away trap. Pieces can safely advance toward a trap defended by the friendly elephant; by contrast, the home elephant would have to leave the deadlock if it wanted to ensure safe advances of its own pieces. The side defending a hostage or frame can make it increasingly difficult for the home elephant to leave its trap; if rabbits advance toward it, they are potential goal threats.

One whose elephant is stuck in a home deadlock must force the opponent to do something other than work that deadlock to their advantage. Often, the answer is to attack an away trap, forcing the opponent to prioritize its defense. Taking control of an away trap usually creates a goal threat, so the enemy elephant may be forced back to its own territory, even if leaving the deadlock would abandon multiple pieces to capture. One who abandons an away deadlock can sometimes come out ahead, as in the position above.

Contested Traps[edit]

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All four traps are contested, since each strongest local piece faces multiple weaker pieces which protect each other.

A trap is contested when one side has the strongest local piece, but the other has multiple weaker pieces which protect one another from capture.

In this game, diagrammed at right, all four traps are contested. Until one side can establish total control of some trap, no captures will be possible. Whoever gets ownership of a trap first will start capturing pieces, gaining a huge advantage unless the opponent can quickly respond in kind.

Multiple trap control fights essentially amount to a positional race, as opposed to a capture or goal race. One poised to lose control of a home trap must either defend that trap or take full control of an away trap, balancing the position. Here, Gold can soon take over the f6 trap, unless the silver elephant goes north. If the silver elephant left the south, however, Gold could send his camel west and likely wipe out Silver's forces around c3, or at least capture the silver camel and then take advantage of the time Silver lost getting her horses to safety. Silver's problem is that Gold has an advantage in free pieces, which is huge in a positional race. Silver's best hope might be in an attack on f3, but Gold will likely get control of f6 first, and then advance more pieces toward it.

Each local fight must be considered in the global context. When a piece crosses quadrants to attack or defend a trap, the area that piece left might be affected.

Losing control[edit]

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The gold camel must take control of the f3 trap.

In the position at left (from this game), Silver has deadlocked both of Gold's home traps. It is tempting for Gold to target the silver camel on b3, but its friendly horse can prevent Gold from hostaging it effectively.

In the diagrammed position, it is most important for the gold camel to re-assert ownership of f3. Silver has stuck his neck out so far that Gold has no need (and no time!) to try to secure a more subtle positional advantage. The fight around the f3 trap is critical. If Silver can maintain shared control of both c3 and f3, then Gold's position is hopeless, but if Gold regains total control of at least one home trap, Silver must be careful lest some advanced piece be captured.

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Gold shares control of f6, but Silver has taken full control of f3.

In this game, Gold attacked the f6 trap, but Silver counterattacked the f3 trap and now has full control of it, whereas Gold only shares control of the f6 trap. Gold to move can't even defend f3 in one turn, as silver pieces place five steps between the gold elephant and e3 or f4. Gold could now try attacking c6, but the silver elephant could defend it without giving up control of f3. The gold elephant must contest the f3 trap as soon as possible, or else Silver will clear space for a goal.

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Silver owns no traps, and can't even move pieces through her home traps. (Game)

In the position at left, most of Gold's army has advanced into Silver territory, and Silver is severely cramped, with silver pieces unable to move through their own home traps. The silver elephant is currently stuck contesting f6. Silver appears to share control of c6, but the silver camel could be pulled away, giving Gold capture threats in both c6 and c3, plus the threats in f6. Gold defends all four traps, and Silver has nothing close to a capture threat anywhere. Gold pieces block nearly anything Silver might try.

Silver's only hope is in a southeastern attack led by the h6 horse, though even that would be delayed by the h5 rabbit. One at a clear trap control disadvantage might still surprise the opponent with a goal threat, which could be strong if the opponent's direct home defense is sparse.