Arimaa/Introduction to Tactics
In Arimaa, a tactic is usually a one- or two-turn plan which can be calculated precisely. Since there are up to four steps per turn, plans which aim farther than two turns ahead are usually impossible to calculate exactly, and therefore generally belong to the realm of strategy. This is in contrast to chess where precise tactics of three moves and longer (i.e. combinations) are quite feasible.
The most basic tactics to master are those that bring a friendly rabbit to goal or capture an enemy piece within four steps. If a player can't achieve victory or capture in a single turn, the opponent may bring in reinforcements which change the situation dramatically. Such contingencies require strategic rather than tactical planning.
Most simply, a rabbit may be able to move to the goal within one turn, even if the path appears blocked by a trap square and/or opposing pieces which would seem to freeze the rabbit. At left, the gold rabbit on b5 can step to victory via b6, c6, c7, and c8. The rabbit is safe on the trap due to the gold dog on c5, and is never frozen because it is always next to a friendly piece on the way to goal. The rabbit will theoretically be frozen once it reaches c8, but Gold will have won regardless.
Pulling away an opposing piece may allow a blocked rabbit to advance. At left, if the gold cat on h7 moves to h6 while pulling the silver rabbit from g7 to h7, the gold rabbit on g6 can then reach the goal in the turn's final two steps. Note that the gold camel on f6 is lost when the rabbit advances, but reaching the goal is worth any sacrifice. Many beginners seem to prefer pushing over pulling, but note that a push is ineffectual in this case. If the gold cat pushes the silver rabbit, the cat will itself be in the way of the friendly rabbit, which won't then have time to reach the goal in the same move.
Near the goal line, a rabbit which is frozen but not blocked is a constant threat. Silver to move could push the gold horse on b3 to c3 with the silver camel moving from b4 to b3. This would unfreeze the silver rabbit on b2, which could then step to victory via a2 and a1. Occasionally a blocked and frozen rabbit may be unblocked and unfrozen at the same time. Silver to move could pull the gold rabbit from g2 to g3 with the silver elephant sliding from g3 to f3, unfreezing the silver rabbit on f2, which could then step to the vacated g2 and then to g1, winning the game. The silver elephant would be lost as soon as the silver rabbit stepped away from the trap, but since Silver wins immediately, the elephant loss is of no consequence. With the silver rabbit beginning the turn blocked in all directions and also frozen, this one-turn goal is very easy to overlook.
The rules of Arimaa dictate that a piece is captured if it is located on a trap square and not next to any friendly pieces. In the case of the c6 trap, the relevant squares are b6, c5, c7, and d6. As long as Gold is able to maintain at least one piece on one of those four squares, Silver will never be able to capture any of Gold's pieces in that trap. If both players are able to maintain at least one piece next to the trap, neither player will be able to make a capture.
The decision of which traps are worth fighting for and how to allocate forces is strategic, but the mechanics of fighting for control of a trap are in the realm of tactics. In particular, at any given point in time a trap is either owned by Gold, owned by Silver, or contested.
A piece which is two squares away from an undefended trap square can often be dislodged twice and captured. In the diagram at right, if it is Silver's turn, the silver dog on b6 could push the gold cat on b7 to c7 (or pull it to b6) and then push it to c6. A piece which is only one square away from an otherwise undefended trap square is even more vulnerable, because it is threatened even by non-adjacent enemy pieces. If it is Gold's turn, the gold elephant on d5 can step to c5, then b5, and still have time to push the silver dog on b6 to c6.
Sometimes a player may need to push an obstructing piece out of the way to get at a vulnerable piece. Gold to move could push the silver cat on g7 to h7 with the gold camel on g6, then push the silver horse on f7 to f6, capturing it.
A piece on a trap square with only one adjacent friendly piece is extremely vulnerable. Silver to move could capture the gold horse on c3 by stepping the silver elephant on d4 to c4, then b4, and then dislodging the gold dog on b3.
If one has two friendly pieces near a trap square, and the opponent has only one, it may be possible to capture by stepping through the trap square and pulling in the opposing piece. Silver to move could step the silver camel on f4 south, where it is safe due to the silver dog on g3, then pull the gold dog on f2 north to f3, capturing it.
One-move capture defense
The most common and fundamental capture defense is to station an elephant next to a trap square. Since nothing can dislodge an elephant, nothing can be captured in that trap square until the elephant voluntarily moves away. At left, no gold pieces can be captured in the northwest trap square as long as the gold elephant remains on c5. If all stronger enemy pieces are tied up elsewhere, a single piece such as a horse may defend a trap square alone, but must beware of a changing situation which liberates any stronger opposing piece to move nearer and threaten it.
The second-most common and fundamental capture defense, mutual protection, is to station two or more pieces next to a trap square. This allows weaker pieces with numerical superiority to defend against one, and sometimes two strong attackers. However, mutual protection cannot defend against three attackers because they can surround three sides of a trap. At left, Silver has defended the c6 trap square with a dog on b6 and a rabbit on c7. If Gold pushes the dog away, there will not be enough steps left to capture the rabbit. Gold to move can at most set up the threat of a capture for the following move, which gives Silver time to defend.
Occasionally players may wish to defend a trap square without bringing a second piece adjacent to it. In this case a player generally obstructs the path of the attacking piece with friendly pieces. The gold camel on g6 can't capture the vulnerable silver horse on f7, because the silver cat on g7 is in the way, and the silver rabbits on h7 and g8 prevent the cat from being pushed. Gold could pull the cat to g6, but that would only make it a second defender of the northeast trap square.
The situation around the southwest trap square is straightforward. Because of the gold rabbit on b4, the silver elephant can't get adjacent to the gold dog on b3 with two steps left to dislodge it.
One counter-intuitive way to block the path of an opposing piece is to station a friendly piece on a trap square. Silver to move could push or pull the gold cat on f3 in a variety of ways, but always to a square from which it defends the f3 trap square long enough to prevent the capture of the gold dog on f2 for at least the present move. The disadvantage of this strategy is that the cat itself will be captured if the dog can be dislodged.
A last resort capture defense is scattering, i.e. retreating threatened pieces away from a trap to the edges and corners of the board. Scattering usually only delays captures, because the weak pieces can be frozen and eventually pulled into the trap which has been stripped of its defense. Furthermore, scattering away from a home trap may leave a hole through which an opposing rabbit can march to the goal. On the other hand, delaying captures for a few moves may buy time which is critical to making progress elsewhere on the board. This defensive technique is most often useful late in the game when defenders are few (which makes mutual protection less feasible) and goal threats are imminent (which leaves less time to hunt down fleeing pieces).
After one has learned the basic ways of defending a trap square, one may be lured into a false sense of security by an illusion of defense. At right, Silver has two pieces next to the northwest trap square, but since both pieces can be dislodged, Gold has a one-move capture. First the gold elephant on b5 can push the silver dog from b6 to c6, and then the newly unfrozen gold cat on b7 can push the silver rabbit from c7 to c8, capturing the dog. Alternatively Gold can capture the rabbit by pushing or pulling the dog away from b6, then pushing the rabbit from c7 into c6.
Around the northeast trap square, the gold camel is blocked from pushing the silver horse into the trap square in one move, but not from pulling it there. Gold could first step the gold rabbit on g5 to f5, making it safe for the gold camel to step to f6, after which the gold camel could step either east or west while pulling the silver horse south and capturing it.
In the southwest corner Gold has calculated that even if the silver elephant pushes the gold rabbit from b4 to a4, the gold dog on b3 is safe from a push. This is correct but insufficient, because the silver elephant may still pull the gold dog from b3 to b4, capturing the gold horse on c3.
The situation around the southeast trap square demonstrates that an enemy piece in a trap square provides less protection than a friendly one. Silver may voluntarily step east or west with the dog on f3, and still have three steps left to enter the trap square with the camel on f4, then pull the gold dog from f2 to f3, capturing it.
The fork (or hostage fork) threatens a single frozen piece with capture in two different traps. It is the most basic of the two-move tactics: On the first move an enemy piece is threatened, and on the second move it is inevitably captured. In the diagram at left, from this game, Gold has threatened to capture the silver dog in either the c3 or the f3 trap square. Silver to move would need one step to defend the c3 trap square with the elephant, and three steps to defend the f3 trap square with the cat. Unfortunately for Silver, the gold rabbit on e4 is threatening to reach goal next move, so Silver must spend at least one step defending the goal, and the silver dog is lost.
Forks resulting in immediate material gain are more common later in the game when fewer defenders are available, but even if both relevant traps can be defended, holding a hostage near one or two traps can be an effective strategy to commit enemy pieces to defensive positions.
The fence (or hostage fence) is a two-move tactic somewhat less common than the fork. On the first move a hostage is brought next to a trap, and hemmed in on two sides. The piece may be unfrozen from a third side, but then its only direction of retreat would be into the trap. In the diagram at right, from this game, Gold has fenced the silver horse next to the c3 trap square. Silver is on move, but can't prevent the capture of the horse. The silver elephant could indeed unfreeze the fenced horse by moving to c5 (incidentally capturing the gold rabbit in the c6 trap square), but the technically unfrozen silver horse can't then escape, as its only available move would be suicide. Furthermore, the silver elephant can't get adjacent to the c3 trap in four steps, a blocking motif common in the opening when the board is still crowded.
Limitations of tactics
Beyond the above-listed tactics, there are not many ways to achieve immediate material gain in Arimaa. Particularly at the beginning of the game, defenders are plentiful, and captures can be postponed. The position at left arose from a relatively sharp opening in this game. Gold advanced all the heavy pieces, and Silver on the last move froze three of them. The horse on f4 and camel on e5 are both threatened with capture in the northeast trap square. Also the gold horse on d4 can be pushed to d6 with a hostage fork, if the gold elephant evacuates d6. Yet in spite of these multiple threats, Gold can move to prevent any captures in at least the next three moves of Silver, certainly beyond the ability of humans or computers to analyze exhaustively.
In fact, it may not be obvious to inexperienced Arimaa players that Silver is winning. Material is even, and Gold, with more pieces activated, may appear more able to make the first capture. Gold could press the attack by stepping cat to g4, horse to f5, and camel to e6, with a step to spare. (Perhaps dog to h3 to prepare to defend the southeast trap square again.) It is entirely a strategic judgement that the vulnerability of the advanced gold pieces is, in this particular position, a greater liability than their attacking power is an asset.
After becoming acquainted with the most basic Arimaa tactics, you should turn your attention to long-term strategy, which will better enable you to get into positions where tactics are of use.