Arimaa/Introduction to Strategy/Frames
|In this game, 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. The lone friendly defender is pinned: if it takes one step, the piece on the trap square will disappear. In the position at right, Silver has framed a gold horse on c6; a phalanx prevents the framed horse from pushing away the c7 dog.
A frame is somewhat like a hostage: the pinned piece is the defender. Unlike a hostage defender, which must merely end each turn adjacent to the trap, a pinned piece is completely stuck. Here, it would be pointless for the pinned 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 opponent 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, one holding a frame should rotate out strong pieces, replacing them with the weakest ones which can reliably hold the frame. At right Silver can replace the elephant with the a5 horse, making the silver elephant the strongest free piece, although it may not be totally free if Silver wants to avoid an even horse trade.
Even if rotation is impossible, the frame may be valuable. At right the silver camel has a freedom to attack that the gold camel does not, because of the difference in elephant mobility. 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 frame may be broken by the arrival of a piece strong enough to dislodge a framing piece. If a piece on a trap gains a second supporter, or gets off the trap square, it is no longer framed. In the position at right, Gold's options for breaking the frame are quite limited. With Silver's strength in the west, the gold camel would have little hope of dislodging a silver horse. To break this frame, Gold would have to break the c7 phalanx. Since the gold elephant is immobile, breaking the phalanx would require a gold piece to be safely on e7 or d8; on the next turn, this piece could pull away a rabbit, which would allow the c6 horse to push its way out. Silver can make this harder to accomplish, by sliding the d7, c7, b7, and a7 pieces east. This would also mildly improve Silver's main weakness: having both dogs stuck in between rabbits.
Frames vary widely in value, depending on each side's free pieces. A breakable frame can have negative value, since the formerly framed piece can become an attacker.
Frames are most effective when there is enough material to hold a frame and still accomplish something else. Even a solid frame may not be worthwhile on a relatively empty board.
Suppose, as above, Gold is defending a piece framed on c6. It is usually better for the pinned gold elephant to be on d6 than on c5:
- With the pinned elephant on d6, the frame requires a silver piece on c5 (except in the case of rabbit frames). If it's not the silver elephant, the c5 piece may be vulnerable to capture in c3. If Silver couldn't defend c3 within a turn, the gold elephant might trade its framed piece for the silver piece on c5 (or for a weaker silver piece which helps the former frame-holder escape.)
- The pinned elephant on d6 prevents Silver from moving the framed piece onto d6 or e6 for a fork.
- d6 is a better attacking square for the elephant than c5. If the frame can be broken, Gold may then have a solid attack.
On the other hand, the pinned elephant may be better placed on c5 for various particular reasons. For instance, it might make it easier for a friendly piece to attack the frame along the b-file.
|In this game, Gold has both a camel frame and a horse hostage.|
When an elephant is pinned to a camel framed by the enemy camel and elephant, the four strongest pieces are all tied up. Such a deadlock could become even tighter if the framed side advances rabbits toward it.
In itself, such a frame is not a big advantage for the framer; it may even be a disadvantage if the framed side advances on both wings. A camel frame can be strong, however, if it gives the framer an advantage in free pieces. If either the elephant or the camel can be replaced by a phalanx of weak pieces, the framer will have the strongest free piece while still rendering the two strongest enemy pieces completely immobile. The diagram shows another possibility: the gold camel is both participating in the frame and holding a horse hostage. 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. This frame-hostage gives Gold a strong advantage.
|An away horse frame, from this game|
In the course of an attack, it is sometimes possible to frame an opponent's piece in their own home trap. Gold's frame at right is very strong, since Silver can now do little in the east. From here, Gold can advance rabbits on the h-file to solidify the space advantage, and the pinned silver elephant can do nothing. This elephant is stuck next to its own home trap; leaving would lose the horse plus other pieces.
As things stand, Gold would not benefit by rotating anything out of the frame, even if other gold pieces were closer. A phalanx would not be viable, and replacing the gold elephant with the camel would just force the gold elephant to prevent Silver from trading the framed horse for the camel.
The one threat to the frame is the silver camel. Silver could try moving northwest pieces to the southeast, threatening gold pieces in f3 while also making room for the silver camel to dislodge the f7 horse. The gold camel should prepare to go east or north, to either punish such a counterattack or create a new threat in the west.
If Gold can advance enough pieces to support his horses, he might eventually want to move his elephant elsewhere, in which case he would have to flip the silver horse out of f6 and then push it away, so that it couldn't simply step to e6 and then help the silver elephant fight Gold's space advantage. As long as both gold horses remained securely by f6, and the silver elephant was the only strong silver piece in the northeast, it would have to stay there to prevent captures and then a goal.
|In this game, if Silver wants to preserve the framed cat, the silver camel and elephant must both stay in place.|
In this position, the silver camel is pinned to the c3 cat, and the silver elephant must stay on d4, both to defend the cat and to keep the camel safe in the centre. This leaves the gold camel as the strongest free piece; if this will remain so, Silver should abandon the cat.
However, the silver elephant is not completely stuck, and Silver might hope to replace the elephant with a phalanx on d4. If this happened, the position would become unclear; Gold would not have strongest free piece, but the silver camel would be awkwardly placed in the centre of the board. A similar type of position with the camel instead on b3 might be good for Silver, since Silver's trap control could be fairly solid. See, for instance, the blockade on 29s of this game.
Baskets and fences
To frame a horse, one must usually immobilize it first. In this game, a gold horse was blocked on three sides, so that no escape was possible even if it were unfrozen from the fourth side. On 14g, the gold elephant moved to e6, since that was the only way not to immediately lose the horse. Silver then flipped the horse into the trap, and on the next turn rotated his elephant out of the frame, creating a strong position which overcame a two-rabbit deficit. The horse basket diagrammed here led to the unbreakable frame.
Had the gold elephant been on e6 sooner, Gold could have prevented or delayed the frame by flipping the e7 dog, since that would have broken the phalanx behind the trap. With e7 empty, Silver should not flip the gold horse into the defended trap, since the horse could then push onto f7. Instead, Silver might have pulled the gold horse to f5 and moved the h5 horse to g5, fencing the gold horse so that it could be pushed into the trap in two steps, leaving another two steps to reestablish the phalanx. Without a horse basket or fence, it would be hard to force a horse frame.
In such a position, Silver shouldn't leave a rabbit on e7 or any square from which it could be moved into the trap. Since a rabbit can't retreat, a friendly rabbit in the trap would block the intended frame. If a fence is necessary, it may be best for Silver not to have any piece on e7 until the frame is in place.
If Silver faced an imminent threat elsewhere, the silver elephant could safely leave the northeast, since the gold horse would not be a direct threat to the f6 trap. Such an alignment should not be seen as an inefficient use of the silver elephant.
The value of a frame depends on how solid it is. The above frames would all be hard for the defender to break, at least without suffering a big loss elsewhere. However, a frame is sometimes vulnerable enough that the defender could break it and come out ahead. A camel with friendly support can sometimes break a frame from the side, turning things around dramatically. In the diagram at left, Silver to move can play a pull-and-replace, getting the silver horse onto b3 (mb5sn Hb3n hc3w). Although Gold could do a pull-and-replace to restore the frame, Silver would win the repetition fight; by undoing Silver's move, Gold would restore a position he created previously, namely the position in the diagram. Once the frame is broken, Silver will have a good position: the formerly framed horse will be a strong attacker, and the camel may be able to pull the gold horse to c6.
Gold to move could delay the frame-break with Ec4ns mb5e. However, if there is a silver piece on d5 or c6, the camel could return to b5 in a single step, thus undoing in one step what Gold did in three. If Gold is to move in the diagram, his best move might be to flip the silver horse to b4; it might escape, but at least Gold would retain control of the c3 trap, and might soon capture a silver rabbit.
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. Thus, Gold would win the repetition fight. However, Silver has other ways to break this frame:
- mb4n Hb3n hc3w Cc2n is the most common of these maneuvers, pulling the cat into the trap to prevent Gold's pull-and-replace.
- mb4n Hb3n hc3w ed3w: the silver elephant can occupy the trap to prevent a horse frame, but must be careful not to get framed itself.
- mb4n Hb3n ra3e ra4s: silver rabbits block the gold horse out of b3.
- mb4n Hb3n mb5e Hb4n: the camel moves to c5, pulling the horse twice. The silver pieces on c6 and d5 contribute to a phalanx on c5, preventing the gold elephant from stepping forward and unfreezing its horse which is now on b5; the gold elephant could step to b4 and unfreeze the horse from below, but then the horse couldn't return to b3.
In this second diagram, note how the silver cat on d4 prevents the camel from being flipped into the center, which would buy Gold some time. A silver piece on c5 would do the same job.
In both diagrams, it is important that Silver has a piece on a4 and on a5, so that the silver camel can move along the b-file without being frozen. With a gold piece on a4 especially, the frame would be much stronger.