Arimaa/Introduction to Strategy/Frames
|In this game, the gold elephant is pinned to the defense of the horse on c6, which is framed.|
A piece held on a trap square, surrounded on three sides by opposing pieces which prevent it from pushing its way off, 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, the gold elephant is pinned to the defense of the gold horse framed on c6.
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; thus the pinned elephant is no threat to the c7 phalanx. If the framed piece is ever abandoned, the opponent won't even use up a turn making the capture, since the piece will already be 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 has little hope of dislodging a silver horse, to say nothing of what the silver camel could do in the east if the gold camel left. 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 in the opening, 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 if, for instance, there is an important fight happening around the c3 trap.
|In this game, Gold has both a camel frame and a horse hostage|
When a camel is framed, the framing side must often commit both elephant and camel in order to hold the frame. Such a frame's value is quite small, since it does not result in a strongest free piece, but only an advantage in elephant mobility. Also, it is usually a space disadvantage to have a deadlock around one's own home trap; if nothing can be captured in that trap, the opponent is free to advance rabbits on that wing. This factor can be especially relevant to camel frames, since the deadlock could last a long time.
There are two main situations where a camel frame is strong. Firstly, if either the elephant or the camel can be replaced by a phalanx of weak pieces, the frame will be very effective, since it now gives the framer the strongest free piece while 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 on the rest of the board 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 immediately advance rabbits on the h-file to further strengthen trap control, and the pinned silver elephant can do nothing. Even if Silver were willing to abandon the framed horse, there would be additional captures if the silver elephant left the trap.
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. Building a space advantage in the east is the only way for Gold to further strengthen the frame.
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, 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.
When deciding whether to frame an enemy piece, or whether to let one's own piece get framed, one must consider how solid the frame would be. If the side with a framed piece would still have a strong piece available to attack the frame, it might or might not be broken easily. In the diagram at left, Silver to move can play a pull-and-replace, getting the silver horse onto b3 (mb5s mb4n Hb3n hc3w). Although Gold himself 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 back to c6.
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 (Ec4s Ec4n 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.