|
|
|
|
|
Arimaa StrategyIn the board game Arimaa, it is relatively easy to master the basic tactics (see Arimaa Tactics). Fairly soon into one's study of the game, it becomes possible to avoid hanging pieces and other blunders which result in short-term, concrete loss. When both players have a certain tactical competence, a good positional understanding weighs more heavily than further mastery of the nuances of tactics. Intermediate Objectives If there were a direct way to bring a rabbit to goal early in the game, Arimaa would not be very deep. As it happens, not only is it impossible to force a rabbit through early against an intelligent defense, it is in fact generally unwise to advance rabbits at all in the opening, lest they be captured. Defending one's own goal pays greater dividends than attacking the opposing goal on a board full of pieces. Because quick goals are generally not to be had, the opening phase of the game usually focuses on attempting to capture opposing pieces in order to gain a material advantage. In general, goal threats don't come into play until after four or five opposing pieces have been captured. Once one establishes a material advantage, particularly among the heavy pieces (for example winning a camel for a horse) it becomes possible to create more threats than the opponent can simultaneously counter. Having a material advantage tends to lead to an even greater material advantage, which makes capturing pieces an excellent means towards the ultimate end. Yet even capturing pieces is a non-trivial objective at first, and difficult to achieve against expert defense, particularly when there are many pieces on the board which can protect each other. Furthermore, equal forces tend to stalemate each other. In fact, without the aid of some deeper strategic themes, one might not be able to capture any pieces at all! A weak player might blame this on tactical errors, but being blanked in this fashion is more likely due to strategic misapprehension. Elephant Mobility The positional struggle in Arimaa ends with the rabbits, but it begins with the elephants. No piece but an elephant can undertake any action without checking first whether it will be harrassed, held hostage, or captured by the opposing elephant. Even several pieces advancing together, if not protected by the friendly elephant, are less of a threat to the opponent than they are themselves in danger from the opposing elephant. Only the elephants are not vulnerable to attack, and can advance with impunity. Therefore the most popular opening strategy among experienced players is to attack with the elephant alone, keeping all other pieces huddled on the first two or three ranks. If one or two other pieces also participate in an early attack, they should be kept away from the center of the board, and near the friendly elephant for safety. On the other hand, the two elephants can accomplish nothing directly against each other. Neither can threaten the other. Furthermore, each can to some extent protect friendly pieces from the other's designs. Either elephant can make any trap square permanently safe for friendly pieces by taking up an adjacent position and never moving away. Indeed, if there were only one trap square on the board, both elephants would protect it, and the game would stalemate. There are two basic types of positions in Arimaa, depending on whether the elephants are operating in the same quadrant or not. If the elephants are not near each other, each will reign supreme in its region, and it becomes a race to see which elephant can do more damage more quickly. A critical question is whether to bring another piece or two in support of one's own elephant to accelerate the attack, given that those pieces may be exposed should the opposing elephant switch wings. Positions in which each elephant is prosecuting its own attack are usually very sharp and often short-lived, as the player who is making less progress with the elephant will give up attacking to come defend against the doings of the other elephant. If both elephants persist in their separate attacks, the game may rapidly come to a conclusion with each side threatening goal. The more typical and longer-lasting situation is for the two elephants to be near each other, each trying to frustrate the other. Sometimes they will eventually reach a deadlock around one trap square. When the two elephants are locked in a struggle that neither can abandon without disadvantage, the camel (or second-strongest piece remaining) comes to the fore in attacking elsewhere. In such situations it is extremely important which elephant will suffer least from abandoning its post. The friendly pieces of the more mobile elephant have less to fear, and therefore a greater freedom of action in the rest of the board. Even a slightly inferior force may make headway on the rest of the board if the background threat of elephant intervention weighs more heavily on the opposing pieces. Between tactically competent players, with both elephants operating near the same trap, it is usually possible for both sides to prevent any captures in the short term. The nature of the struggle is then in essence each side trying to limit the mobility of the opposing elephant, while preserving the mobility of its own elephant. Blockades right The most extreme differential in elephant mobility occurs when one elephant (with the help of friendly pieces) blockades the other. In the diagram at right, Silver has just blockaded the gold elephant so that it has no legal moves. Not only is there no empty square for the gold elephant to step into, there is no empty square into which it can push its tormentors. Yet it may not be immediately obvious what a severe disadvantage to Gold this represents. One might argue that since nine silver pieces are required to maintain the blockade, including both the silver elephant and camel, that for as long as the blockade lasts, it is more costly to Silver than to Gold. For starters, Gold might attack the northwest trap square with the camel, where the camel will dominate. As it happens, however, Silver has many more options for improving than Gold. The most important is rotation (or replacement) of the pieces participating in the blockade. When it comes to being in the way, a weak piece serves just as well as a strong one. Silver to move can free the camel for duty in only four steps, while maintaining the blockade: camel h6 south, rabbit h7 south, rabbit h8 south, and rabbit g8 east. (The gold elephant can't make use of g8 to dig its way to freedom, so Silver can fill in that hole next move.) It would take more than one move for Silver to rotate enough pieces to free up the silver elephant, but even freeing the camel is enough to give Silver the upper hand. Gold's best option is to try to erode the blockade as quickly as possible, before Silver can complete a favorable rotation of forces. Unfortunately for Gold, undertaking this rescue mission will necessarily expose gold pieces to danger. It's true that the gold camel alone could quickly force Silver to abandon the blockade, but this is small comfort if the camel itself will be lost in the process. Even so, Gold must swallow hard and accept the danger at once, because any delay will cause a rapidly worsening situation. For example the silver camel, once activated, could drag a gold piece up to the northwest trap square, and the gold camel couldn't defend without exposing itself to as much danger as intially attacking the blockade would have entailed. Gold can hope to lose more slowly by hanging back, but all the more surely for that. As of mid-2004, all Arimaa-playing computer programs were susceptible to being lured into a blockade, for example by offering a piece for capture. In late 2004 the top programs became increasingly aware of elephant blockades, reducing the effectiveness of one of the favorite anti-computer tactics in the human arsenal. To force an unwilling elephant into a blockade is difficult, even if the stronger side has a full set of pieces and the weaker side has no other pieces but the elephant. An elephant blockade can't be inflicted on a wary opponent, so it is rare to see one in top-level games. left Pins A piece which is on a trap square, surrounded on three sides by opposing pieces which prevent it from pushing its way off the trap square, has been framed. The lone friendly piece providing support is pinned, because as soon as it moves, the piece on the trap square will be captured. At left, the gold elephant is pinned to the defense of the gold horse which has been framed on c6. Pins are most effective in the opening when the board is crowded, so that strong pieces participating in the frame can be rotated out and replaced with the plentiful weaker pieces. If not enough weak pieces are available, then the side maintaining the frame will have to commit three pieces at least as strong as the piece being framed, which may not be worthwhile on a relatively empty board. At left the silver camel has a freedom to attack that the gold camel does not. The silver elephant can switch wings at any time to threaten the gold camel at no cost other than giving up the frame, but for the gold elephant to threaten the silver camel it would have to abandon the framed gold horse, which would be instantly captured. Silver can in fact increase the difference in mobility of the two elephants by rotating the silver horse from a5 to c5 and the silver elephant from c5 to d5. This stations the silver elephant in the center where it is one step closer to any activities of the gold camel. Also it limits the options of the gold elephant should it decide to move. Nevertheless, a pin is not as forceful as blockade, because the pinned piece may choose to leave at any time. If the silver camel becomes too exposed, the gold elephant may cross over in a single move, and possibly even force the capture of the silver camel, which is certainly worth the sacrifice of a horse. More subtly, if Silver moves the a5 horse to c5 and also attacks the east wing with the silver elephant, the gold elephant may be able to abandon the gold horse and in the same move push the silver horse from c5 to c4, capturing it the next turn if the silver elephant is too far afield to help. It would be a huge relief for Gold to abandon the pin in a way that resulted in only an exchange of material rather than a loss. A frame may be broken by the arrival of a piece strong enough to dislodge the framing piece or pieces on one side of the trap. If a piece on a trap has two adjacent friendly pieces, it is no longer framed, because either friendly piece may step aside and allow the piece on the trap to step off. At left only the gold camel would be strong enough to break the frame. Gold must judge whether breaking the frame is worth the danger of the gold camel being taken hostage (see below), as well as considering the damage that the silver camel will do on the east wing in the mean time. As of late 2004 most Arimaa computer programs defend their material tenaciously, making them easy targets for elephant pinning. On any given move the program will decide, for example, that it isn't worth losing a framed horse merely to save a cat from capture. Therefore one may often capture multiple smaller pieces one at a time, eventually building up a large material advantage over a program with a pinned elephant. Hostages right A hostage is a frozen piece which is threatened with capture. At right the gold elephant is holding the silver camel hostage. Hostages are often kept exactly two squares away from a trap. A hostage three squares from a trap can't be captured in a single move, while a hostage directly adjacent to a trap can be defended by a second piece weaker than the attacker. (See mutual protection.) When a hostage is held exactly two squares from a trap, it requires two friendly pieces to protect the trap, or one piece at least as strong as the attacker. A protracted hostage situation typically occurs only between elephants, i.e. one elephant is holding a hostage while the other is defending the trap. Other hostage situations are less stable: If the piece holding the hostage is the strongest piece near the trap, it can, unless it is badly outnumbered, usually overcome all the defenders and begin capturing them one by one. Conversely, if the defending piece is stronger than the piece holding the hostage, the hostage can typically be liberated, resolving the standoff. Finally, if the defender and the hostage-holder are of equal strength but aren't the strongest pieces on the board, the situation will usually be decided one way or the other by the arrival of a stronger piece from elsewhere. An elephant which is tied to the defense of a hostage suffers some loss of mobility, but less than in the case of a pin. When the defending elephant moves away the hostage is not immediately captured. The hostage-holding elephant must take an entire move to capture the hostage, which gives the departing elephant an extra move, compared to the case of the pin, to create a threat of its own. Furthermore, even if the defending elephant can't afford to move away, it retains mobility in the neighborhood of the trap, which allows it to dart around dislodging nearby pieces while ending each move adjacent to the trap. For these reasons an elephant holding a weak hostage will often try to frame that hostage (see above) to take away some of the freedom of the defending elephant. The hostage situation most disadvantageous to the defender is when the friendly camel is held hostage by the opposing elephant. Although the camel is too strong to be framed in most circumstances, its absence is keenly felt on the rest of the board. In the diagram at right, the gold horses can advance relatively fearlessly. Normally they would fear an attack from the silver camel or elephant, but the silver camel is frozen and the silver elephant can't abandon its post to capture a horse without losing the camel. Silver can choose among several defensive strategies: - Try to unfreeze the camel with a friendly piece so that it can run away, or at least take longer to capture
- Accept an opportunity to trade a horse for the camel, thereby regaining elephant mobility.
- Bring up a wave of smaller pieces to protect the trap so that the silver elephant can move away. If enough pieces can be brought forward, they may also threaten to blockade the gold elephant.
To defend passively in this situation holds no hope for Silver, because the gold horses won't take long to pull little pieces down to the southeast trap square where they will be captured. Oddly enough, when a horse is held hostage by the opposing elephant in the opening, it may be only mildly disadvantageous. The horse is strong enough, and the friendly elephant has enough mobility, that the hostage horse usually can't be framed without the help of the opposing camel. Meanwhile the friendly camel can roam in relative freedom, since the elephant holding the hostage can't easily leave the trap at the mercy of the combined power of elephant and horse. It isn't always clear which elephant suffers the greater loss of mobility. In the early going, it is more disadvantageous to have one's dog taken hostage, since a dog can be framed more easily than a horse. left The Center A more subtle, positional aspect to elephant mobility is the ability of the elephant to access the areas of most importance at the current stage of the game. From the centre four squares of the board the elephant has access to all four traps, so as a general rule it is better for the elephant to be centrally located. Also, a centralized elephant is almost impossible to blockade. On the other hand, in order to attack or defend a trap, an elephant must decentralize itself at least to one of the eight squares in the ring around the center, and perhaps further if it is hunting down a fleeing piece. The possibilities available to an elephant in the center usually remain mere possibilities until the elephant leaves the center to pursue one of them. When defending a trap square, an elephant should almost always stay on the two sides nearer the center. To defend from the outside is to be two steps further from intervening in another part of the board. In the diagram at left, neither elephant wishes to leave the northeast trap square, especially not the gold elephant, but the gold elephant is two steps closer to the center, and thus a much greater threat to the silver camel than the silver elephant is a threat to the gold camel. The northeast trap square is deadlocked, but Gold has a greater freedom to make progress on the rest of the board, and therefore doesn't stand too badly in this position in spite of having a horse held hostage. It is occasionally possible to station a clump of friendly pieces in the center of the board, which can cut off the opposing elephant's ability to switch wings at will. This strategy must be used with great caution, however, because if the opponent is able to erode the dividing wall, there will probably not be time to retreat all of the participating pieces to safety. In general, pieces other than the elephant should stay out of the center. It is asking for trouble to give the opposing elephant a target in the center, where it wants to be. Trap Control To control a trap square is to be safe from capture in it, and further to be able to capture opposing pieces in it. At the beginning of the game each player controls the two nearest trap squares. The most basic offensive strategy is to use one's elephant to dislodge an opposing piece towards the two trap squares that one controls, in hopes of capturing that piece. There is no risk in going hunting with the lone elephant, except elephant blockades which can be avoided. Moreover, even an unsupported elephant can usually bring home something eventually, a rabbit if nothing else. A more ambitious offensive strategy is to try to take control of one of the trap squares nearer the opponent. It is more difficult and dangerous to try to invade enemy territory in this fashion, but if accomplished, the rewards are greater. An enemy trap usually has plenty of enemy pieces nearby, which can be captured in quick succession, as opposed to the laborious process of repeatedly prying loose a single enemy piece and dragging it all the way to a home trap for capture. An equally important benefit of taking control of an enemy trap is that if the opposing pieces scatter, it leaves a hole through which a friendly rabbit can march to the goal. Shared Traps Often each player will have partial control of the same trap square, which is to say that both are safe from capture there, and neither can bring about a capture there. Shared control can happen in two ways. First, if the strongest friendly piece near a trap is the same as the strongest opposing piece near that trap, then the only way either can make progress is in terms of blockades, pins, and hostages as described above. Tactics around a trap shared in this fashion can serve to tie up opposing pieces into awkward positions, but not to capture them. A second type of shared trap control occurs when one player has the strongest adjacent piece but the other has several weaker pieces to protect one another from capture and unfreeze one another when taken hostage. The player with the strongest local piece must be considered the attacker, since he is guaranteed at least shared control, and may be able to take control outright. Whether control will be shared indefinitely or the attacker can eventually take over depends on the exact disposition of the pieces but a general rule of thumb is: - Against a single attacking piece, three weak defenders suffice.
- Against a strong attacking piece supported by a second piece, the defender needs
- two pieces stronger than the supporting attacker and one as strong
- one piece stronger than the supporting attacker, two as strong, and one weaker
- two pieces as strong as the supporting attacker and three weaker
- six weak pieces
If the defender doesn't have control of the neighboring traps, defense is more difficult because the strong attacking piece may threaten and possibly even capture a defender in a neighboring trap without breaking stride in the fight for trap control. Conversely if the defender controls both neighboring traps, a slightly weaker force may be sufficient to defend. When fighting to gain control of a trap where one theoretically has the superior force, tactical competence is important, because in most situations there will be a second contested trap at which one has a theoretically inferior force. It is then critical which player can make progress most efficiently, so as to be able to begin capturing pieces. The Second Front When the two elephants are deadlocked, contesting control of a trap, and both are very unwilling to leave, then a second and perhaps third trap will probably be contested by the camels and horses, with tactics which are superficially similar to those described under the heading of Elephant Mobility. Pieces other than elephants may be blockaded, pinned, or forced to guard hostages, which may initially tempt one to import methods and strategies from the elephant front wholesale to the secondary fronts. However, any struggle which involves the elephants is fundamentally different from a struggle which doesn't involve the elephants, because the latter will be radically altered by the arrival of an elephant. The attacking side, if a friendly elephant arrives, can usually begin capturing pieces in short order. Similarly, the defending side can always be rescued by a friendly elephant which can dig out a blockade, relieve a pin, or free a hostage. In fact the defending side may become the attacking side after a move or two. On the second front, in addition to considering all the issues relevant when elephants butt heads, one must also constantly consider the effect of either elephant should it come join the fight. If at any point either elephant can make a greater difference on a second front than it will cost to abandon the primary front, then it will switch quadrants, usually causing the game to be sharp and unsettled for at least a short time. To forget about the elephants because they are tied up and to focus on a local struggle elsewhere is to invite the opposing elephant to profitably intervene. Distribution of Force Generally speaking, it is wise to keep one's forces balanced across the board rather than concentrated in one region. This is because, both on attack and defense, the strongest participating piece is more important than the second-strongest piece. For example, a camel and a dog can take control of a trap against two horses, but not vice versa. The camel-dog tandem will typically be more valuable than two horses in other situations as well unless the distinctions in strength are irrelevant, e.g. versus an elephant or versus three rabbits. An example of this principle in the opening is that it is strategically less wise to attack with an elephant and a camel in conjunction than with an elephant and a horse in conjunction. Either attack can be frustrated by the opposing elephant alone, but the latter leaves the camel free to operate on a second front. A second example is goal defense in the endgame. A cat and two rabbits on each side of the board makes for a stronger defense than two cats on one side and four rabbits on the other. A defense with only rabbits has nothing to freeze or dislodge an opposing rabbit, so it is helpless against as little as a cat and a rabbit. With the forces split evenly, it takes at least an opposing dog to usher a rabbit home, and even then there are more defensive options to delay until help arrives. Rabbit Advancement Advancing rabbits is a double-edged sword. Since the objective of the game is to get a rabbit to the other end of the board, it may seem that any rabbit advance is a good one, but it is not so. Unless one has taken control of an opposing trap, any advanced weak piece is in danger of eventual capture. Rabbits are particularly susceptible to capture in opposing traps because they cannot retreat, and even cats can push, pull, and freeze them. A less obvious disadvantage of advanced rabbits is that they may restrict the mobility of friendly heavy pieces, particulary the elephant. Unlike an opposing piece, a friendly piece can't be dislodged. Since rabbits can't retreat, they sometimes can't get out of the way even if they are not frozen. Often it is easier to blockade an elephant with the help of a rabbit or two which are nominally on the elephant's team. For the first phase of the game rabbits are rarely moved forward by choice. In fact, a common starting configuration is to have all the rabbits on the back row. These are left there as a defensive wall, which is difficult to penetrate quickly. By the same token it is considered a strategic victory to pull opposing rabbits off the back rank, to make them less useful defensively and more vulnerable to later attack. Even if the exposed rabbit(s) aren't captured immediately, their inability to retreat usually makes them a long-term liability. On the other hand, if one controls an opposing trap, it is very profitable to have an advanced rabbit in the neighborhood. The threat of goal prevents the defender from scattering to buy time. Until the defender can regain control of the trap, the goal threat will force the defender to sacrifice pieces one by one. Even if one has only shared control of an opposing trap, advanced rabbits may have some strategic value. If the elephants are deadlocked around one trap and neither wishes to leave, advancing a rabbit or two on that wing makes each elephant even more unwilling to leave. The home elephant will be more eager to stay due to the threat of an eventual goal, whereas the visiting elephant will be more eager to stay to prevent the capture of the exposed rabbit(s). It is a strategic judgement whether one wants to cement the deadlock at a given trap, or preserve as much felxibility as possible to retreat. If one has shared control of both opposing traps, it may pay to advance rabbits on both wings. A general advance of rabbits limits the opponent's options both by partially blockading opposing pieces and by threatening goal on both wings. Trap control is essential for this strategy to work: if the opponent breaks through or regains complete control of a home trap, the advanced rabbits will become a liability, both because they can themselves be captured and because they aren't available for home defense. left Once a significant amount of material has been traded, or several traps have been contested across the board, holes start to appear in each player's defenses. The emptier and more ragged the board becomes, the more profitable it becomes to advance rabbits. Even if there is no direct win available, a rabbit advanced to around the 4th rank becomes a constant threat, particularly on the wings where it forces opposing pieces to decentralize themselves in order to defend. A rabbit on the edge of the seventh rank is a nuisance to the opponent even if it is blocked, because a strong friendly piece may come usher it home at any time, yet to drag it into a trap takes two moves which might not be available if the position is tactically sharp. In the diagram at left, Gold to move is at an enormous material disadvantage, but the presence of an advanced gold rabbit on each wing generates significant winning chances, particularly with the gold elephant assisting and the silver elephant temporarily out of position. If Gold begins by moving the elephant to f7, pushing the silver rabbit back to f8, Silver faces multiple threats. In theory there may be a correct defense and a consequent win for Silver, but in practice a position like this favors Gold, because good defensive moves are more difficult to find under time pressure than good attacking moves. Two advanced rabbits which are near each other are not much more threat than one, so unless one is aiming for a partial blockade around a shared trap, it is wise to advance at most one rabbit on each edge, keeping the rest back for defense. In the relatively unusual case that less than half the pieces remain on the board, someone is probably imminently going to break through to goal, at which time a third and fourth advanced rabbit can be very useful. The opponent's concentration will often drop and allow one of these rabbits to sneak through. On the occasion that nearly all of the strong pieces in a game are concentrated on one side of the board, rabbit advancement on the other side is often valuable, even in the middle game. This is because a rabbit can advance quickly through open space, and thus may rapidly create a goal threat. In order to defend against a rocket rabbit, the opponent may have to spend more moves to divert material to counter a threat than one spends to create that threat. Even if the advanced rabbit is eventually lost, the extra time gained may allow one to win a tactical victory on the crowded side of the board. Initial Piece Placement right There is some agreement on how to best arrange the pieces during the setup phase. Gold almost always places the elephant front and center, and Silver does likewise, making sure not to have the elephants directly opposite. If the elephants directly face each other to begin, Gold can put Silver in an awkward spot by advancing the gold elephant four squares to fence in the silver elephant. Most or all of the rabbits are placed on the back row where they are not vulnerable to attack. Similarly cats usually go behind the traps to be less exposed, or in the back row if a rabbit or two are placed forward. Most players will split their horses, dogs, and cats to have one on each side of the board, in keeping with the principle that balanced forces are most effective. There is no agreement about the placement of the camel. Some players prefer it in the center where is has more options, and others put in on a wing so that, if the opposing elephant wishes to attack it, at least it will have to come out of the center of the board to do so. The diagram at right illustrates two typical setups. Gold has taken a rather aggressive posture, trading the vulnerability of the flank rabbits for a latent threat to advance one on either wing as soon as possible. The camel is kept central for attack and defense, despite being more exposed to the silver elephant. Silver, in contrast, has tucked away the camel on one wing, and kept all the rabbits in the back row for maximum safety.
|
 |
|
| Copyright 2005-2009 OnPedia.com. All Rights Reserved |
|
|