Arimaa/Introduction to Strategy/Elephant Mobility
As long as an elephant is not blockaded, it retains a theoretical freedom to roam the board. There are, however, fine distinctions in exactly how free a free elephant is. A whole spectrum of positional features may limit an elephant's ability to switch its field of operation. Extreme contraints include guarding against an imminent goal, or guarding against the capture of a camel hostage. The range runs all the way down to minor constraints such as guarding against a rabbit being captured, or even guarding against a small positional disadvantage the opponent might create in the elephant's absence.
At the start of each game, both elephants are free, but differences in elephant mobility soon emerge. Some observers are astonished that tiny advantages in Arimaa seem to snowball; it is easy to underestimate how a more mobile elephant can pose increasingly tricky defensive puzzles for a less mobile elephant to solve. As the game progresses, one or both elephants tend to become more and more committed to the defense of a particular trap. When an elephant finally can't leave a trap without its army suffering captures in that trap, that elephant has lost mobility in a clearly identifiable sense.
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 disappear. In the position at right, from this game, 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. At right Silver can rotate the a5-horse to c5, making the silver elephant the strongest free piece. If not enough weak pieces are available for rotation, though, 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 right 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. In contrast, the gold elephant can't threaten the silver camel without abandoning the framed gold horse to instant capture.
A pin is not as absolute as a 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. More subtly, if Silver rotates the a5-horse to c5 and then attacks the east wing with the silver elephant, the gold elephant may be able to abandon the framed horse and in the same move push the silver horse from c5 to c4, threatening to capture it in the following turn. It would be a huge relief for Gold to abandon the pin in a way that resulted in only an exchange of horses 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 gains a second supporter, it is no longer framed. At right 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, as well as considering the damage that the silver camel will do on the east wing in the mean time.
An elephant holding a camel hostage is only one example of a whole class of positions where a frozen piece is held in danger of capture, committing the opponent to defense. It is, however, the most effective. Suppose, for example, that an elephant holds a horse hostage instead. This commits the opposing elephant to defense, true, but who has the strongest free piece? The defending elephant can't leave without losing a horse, but the hostage-holding elephant often can't leave either, because the newly-freed horse would cooperate with the formerly-defending elephant to wreak havoc at that trap.
In the diagram at left, from this game, Silver is holding a hostage, but neither elephant wishes to leave the northeast trap square. Indeed, it is the gold elephant which has the greater mobility because it is two steps closer to the center, and is thus a greater threat to the silver camel than the silver elephant is a threat to the gold camel. Since the gold camel has more freedom of movement, Gold may be able to make progress in this position despite having a horse held hostage.
Silver's hope in this position must be either to frame the hostage horse and rotate out the silver elephant (see above), or to pass the hostage horse off to the silver camel (see below), which would also free the silver elephant.
After an elephant holding a camel hostage, the second-most effective hostage situation is a camel holding a horse hostage. Ideally this ties the opposing elephant to defense, while the friendly elephant is free to roam. However, the situation is often unstable because the "defending" elephant can attack the hostage-holding camel, freeing the hostage horse, and possibly creating offensive threats. This means that the "free" elephant must work as quickly as possible to score material gain, before the hostage situation collapses.
When a horse is held hostage by a camel, the deeper the hostage is held, the greater the threat. In the diagram at right, the gold elephant has no easy way to approach the silver camel and switch to offense. In contrast, if a hostage-holding camel has no friendly pieces in front of it, the opposing elephant can more easily free the horse. At right, the silver elephant can pull the gold camel to b4 in three steps (returning to c4 after the pull) and the final step can be used to move the silver horse to b3. This would threaten the gold cat with capture on the following turn. Also the silver elephant has the option of flipping the gold camel to c5 on the following turn, threatening another capture.
Cat and Dog Hostages
A smaller piece may also make a valuable hostage. In the diagram at left, from this game, Silver has an advantage due to the gold dog held hostage by the silver camel. The silver elephant is a free piece, which will try to pull something to capture in the f6 trap. The Gold elephant will be overloaded if it must simultaneously defend c6 and f6.
However, the situation is less than ideal for Silver, for two reasons. First, the silver cat on c6 means that the hostage dog isn't threatened with one-turn capture. If the gold elephant leaves, it will take five steps for Silver to capture the hostage dog, giving the gold elephant that much more mobility. Second, the gold rabbit on h5 gives the gold camel more freedom to advance without fear of being taken hostage by the silver elephant. Silver would like to send his elephant after the gold camel, but in practice will probably have to try to capture the h5-rabbit before this is feasible.
In general, the advantage of a small piece hostage over a horse hostage is twofold. First, the smaller piece can be held hostage by a horse, sparing both the friendly elephant and the friendly camel for other duty. Second, it is easier to frame a small piece than to frame a horse. In the diagrammed position, Silver would have a greater advantage if the c7-camel were swapped with the g6-horse. Then the silver camel could defend the f6 trap while the silver elephant either hunted for a second piece to drag for capture, or assisted in framing the hostage dog. As it stands, the silver elephant can't help frame the c8-dog, because of the damage the gold camel would do in the mean time.
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.
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. In the diagram at right, from this game, Gold is losing because the gold elephant can't access the f3 trap.
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. At right, if the gold elephant could break through the wall, some silver piece would likely perish in f3. 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 that elephant wants to be anyway.