Arimaa/Advanced Tactics

Short- and long-term planning are both essential to Arimaa. A position might be greatly improved in one or two turns, if one can spot the correct move.

Double threat
A turn might create multiple immediate threats that can't all be stopped. Such moves can be hard to anticipate, but solid home defense will leave one less vulnerable to a surprise double-threat.

Problem: Silver to move and win material Solution: Dd3w ee3w Hd4n ed3n (or ee3n Dd3e Hd4n ee4w, threatening the gold cat instead of the dog). Silver makes threats in both c3 and c6; Gold would need four steps just to save the now d5 horse, leaving the gold dog or cat to be captured in c3. With only two defenders, the c3 trap was vulnerable. This tactic would not have forced a capture had a gold rabbit been on b3.

Problem: Silver to move and win material Solution: Hb4n ec4w re7ss. Gold could prevent a capture only by moving his elephant to c5, but that would allow the now e5 rabbit to reach goal. A gold rabbit on e1 or e2 would have made a difference.

Counter-threat
After 18s of, Gold faced two strong capture threats. Currently, the gold horse on g5 faces capture in f6. The gold elephant is blocked from e5, so the silver camel is currently safe. The g3 dog could unfreeze the horse, but then the horse could only travel north, at best reaching h6 where it could be immediately threatened again.

The gold elephant could, in two steps, occupy e6 and thus protect f6, but then the gold camel could be captured in c6. The gold camel is frozen, and would thus need three steps to retreat. Rb4n rc4sx Mc5s or rc4sx Cd4w Mc5w would work, but neither would leave the requisite two steps for the gold elephant to protect f6.

Since the horse capture would currently require four steps, de6e Ed6ew would delay a capture. This is a three-for-one, as Gold would use three steps which Silver could undo in a single step. For Gold, the question is whether any fourth step could offset the lost time. Gold might advance the g3 dog in hopes of burrowing his horse on the next turn, but the silver horse could push the dog back and block any further attempt to unfreeze the gold horse; a step would remain for the silver dog to return to e6, clearing f6 and reestablishing a one-turn threat which Gold would need two steps to defend against. Unless Gold can substantially change the position, Silver could force a capture by pushing the gold camel onto c6; the camel could push its way off, but could still be captured if Gold used the other two steps to defend f6. Gold appears to have no fourth step which would prevent a camel or horse capture within a few turns. A three-for-one will be effective only if the opponent cannot spare even the one step to undo the three steps.

Facing the forced capture of a strong piece, Gold must create a counter-threat. If Gold could threaten the b6 horse with one-turn capture, he might either get a horse exchange or buy time to defend on both wings. Rb4n Mc5n cc7w Mc6n may look promising for Gold, but would in fact open a goal path for Silver, who could then advance the c4 rabbit to b2. With the western goal line thinly defended, this would at least delay Gold long enough for Silver to come out ahead.

Gold spotted a different way to threaten the b6 horse: rc4sx Cd4w Mc5w Ed6n. The silver rabbit was captured, and the gold camel was out of immediate danger, allowing the gold elephant to step north. Gold's resulting false protection threat offered Silver a horse exchange that would leave Gold up by a rabbit. Silver could decline this exchange and reestablish a double-threat, advancing his dog from a7 to a5 so that the gold camel would need two steps to move again. With the silver elephant having stepped west, however, the gold elephant might soon reach e5, threatening the silver camel if it remained on f5. Whatever the ultimate outcome, this horse capture threat appears to be by far Gold's best option for 19g, as any other move would cost him a horse or camel for little in return.

Congestion defences
Sometimes, occupying a particular square can stop the opponent from playing a particular move. A simple example is at right: Silver threatens to frame the gold horse with ef6s Hf7s cf8s re8e, but Gold to move can place his dog on f5 to block the pull. Since a pull can be blocked by any piece, this type of frame threat can usually be thwarted.

Congestion is commonly used to block an undesirable flip. A flip is only possible if the flipping piece is adjacent to two empty squares, one for it to pass through and one for the piece to be flipped onto. Here, Gold cannot flip the silver dog, and thus Silver's defense of c3 is intact for now, provided the gold camel is not in a position to dislodge the silver horse.

On 24s of, Silver established a one-turn goal threat which could not be blocked on the goal line. By stepping his horse east, however, Gold surrounded the silver horse, which could not then pull the g2 rabbit. Silver then moved his elephant to f1, leaving an "air bubble" on g1 which ensured a Silver goal. Had the elephant moved all the way to g1 on 25s, a gold dog could have occupied f1, once again blocking the pull needed for a goal.

Pinch
In the first diagram (31s from ), the silver camel is threatened. No other silver piece can defend the c3 trap, and other advanced silver pieces prevent a one-turn escape. However, Silver to move can defend with rb5e cb4n mb3n Ra3e, which sets up a formation known as a pinch. In the second diagram, note that all four key squares are occupied by gold pieces, and the gold elephant is surrounded on three sides, leaving no room for any capture. Gold can renew the threat by vacating a key square, but the silver camel and dog now have time to scatter.

A pinch may give a trap attacker time to secure key squares. At right, Gold would like to attack c6, but would need four steps just to get his elephant to d6; the silver horse could then retreat to b6, blocking the gold horse. Gold would prefer to advance his horse immediately, but could then lose it in c6, unless the silver elephant could be pinched. Ed3nn Dc3n Hb4n would do the job, and would also give the gold camel a chance to threaten the silver horse in c3.

A pinch defends against capture by blocking capture moves. Not all capture moves are equal; one can sometimes afford to lose a piece if the capturing piece would land on an inferior square. Silver would gladly trade her hostaged horse for a gold horse; ee3n De2n ee4nw can bring about an even horse trade. The dog pull is crucial, as it will force the gold camel to end its capture on g3 or f4. If the gold camel could instead finish on e3, Gold would have a follow-up threat to the d3 horse.

Capture congestion can strengthen a goal threat. At right, Silver cannot do a flip capture, and cannot afford to move her camel east; any pull or push capture would allow for a one- or two-turn goal by Gold.

Split capture
One sometimes has the choice of a split capture or a capture done entirely by one piece. On 13s of, the silver camel could have simply pushed the gold horse into the trap. By using his elephant for the second push, however, Silver created an immediate follow-up threat to the gold camel.

A split capture might not be the best option even if it appears to create a follow-up threat. In, the gold elephant got pinched after making a rabbit capture which could have been made entirely by the gold cat. Gold then lost crucial time moving his elephant and cat back to their previous squares.

Blocking the defender
Forcing a capture often depends on keeping a defender away from the trap. Even an elephant right next to a key square might be blocked by a phalanx. In the first diagram, Silver to move cannot save the g3 horse; Gold will capture it with Cf3s Ef4sn hg3wx. Since such a phalanx must include a piece in the trap, most instances of this tactic have the threatened piece next to the trap. The second diagram shows an exception, where Gold can play a split capture of the silver dog.

Opposite-trap blocking
An elephant might make a capture in c3 and then wish to defend f6, but this is four steps' distance even with a clear path. If there is not a clear path between diagonally opposite traps, an elephant may not be able to stop a multi-turn capture after making a capture.

Gold has just threatened a silver horse in c3, perhaps thinking that due to the silver dog on f6, Silver would not have time to capture the gold camel in return. This was incorrect; if Silver now plays ee4ew Mg4w mh4w and Gold captures the horse, Silver can then play Mf4n ee4e df6e dd6e, occupying e6 and pushing the gold camel to f5, where it cannot be unfrozen in time to move, as g5 is occupied by a silver cat. Since the camel capture will only take two more steps, Silver can use the other two steps to counter any follow-up threat by Gold.

After Silver plays ee4ew Mg4w mh4w, Gold can still save his camel if he abandons the threat to the silver horse. The horse might then threaten a gold rabbit in c6, however.

In this diagram, Silver has a single move which forces the capture of the gold horse: Hg4n ef4e cf7ss. Silver pieces on e6 and now f5 prevent Gold from defending the f6 trap in one turn. The subsequent horse capture would take a full four steps and leave the silver elephant decentralized, so Gold will have time to force a capture of his own if indeed Silver follows through with the horse capture. The now advanced silver cat is vulnerable, as is the silver dog, but Gold can do better, threatening the silver horse with Ec4nwe hb6s. Silver could then undo the horse pull in one step, but that one step would preclude the capture of the gold horse currently on g5, and the now c5 gold elephant could then defend f6. If Silver goes ahead with the horse capture, her own horse will remain on b5, from where it could be pushed to b4 and thus doomed to capture in c3; from g6, the silver elephant could not defend it in time. This tactic is rather delicate: if Silver had a rabbit on d8, she could instead reply with hb5n rd8sss, rescuing her horse while also placing two pushes, and thus five steps, between the c5 gold elephant and f5 or e6. The c7 cat couldn't be used for this, since the silver horse could then be captured in c6.

If from the diagrammed position Silver declines the forced horse capture, she will still be down two horses to one. With the camels gone, this may be worse than being down one horse to none, as two horses might easily overload a camel-less opponent. Silver should thus capture the horse, and try to get a further threat in the east if indeed Gold goes for the western horse capture.

False protection patterns
Capture threats in which a piece is held right next to a trap are not overly common, as one's own piece might have to occupy the square to keep control of the trap. Such a threat might be effective, however, if no defender could reach and hold a second key square. Sometimes, a false protection capture will be possible if the opponent attempts to defend against such a threat.

In the diagram, the gold cat must leave f3 before the silver camel can be captured, so Silver might have time to save the camel. If the gold elephant went to f4, or pulled the camel onto f4, Silver could unfreeze and retreat the camel. Gold to move could keep his elephant in place and block both f4 and e3 with Hd4ee Cf3w, but this would leave Silver well-positioned to get compensation for the doomed camel, especially since Gold would need another full turn to capture it. Gold could instead push the camel to g3, but would not then have time to adequately block both f4 and e3, and thus Silver could add a second defender to the trap (if the gold elephant pushed the camel south and then moved to f4, the camel could escape up the h-file). However, Gold can indeed push the camel to g3 and use his other two steps to doom the camel on the next turn. If Gold plays mg4s Eg5s Cf3w Hd4e, any possible Silver defense of f3 would not hold, due to false protection; if a silver rabbit or dog moved to f4, Gold could capture the camel and threaten the piece which came to defend it. If Silver foresaw this and didn't try to defend f3, Gold would then capture the camel in two steps, leaving more time to respond to any Silver counter-threat. Moving the gold horse to e4 also makes the horse safer than it would be on d4 or f4; this move should preserve the gold horse, although the b8 dog can be captured.

Problem: Gold to move and win the silver camel Solution: Mg3n rf3e Ee5s Dd2n. This wins the silver camel on the next turn, by forcing a false protection capture. By contrast, capturing the rabbit by dislodging the camel right away would allow the camel to retreat or be defended by the silver elephant. This camel-capture tactic would not work if Silver had a piece on f6, as Silver could then unfreeze her camel and move it through the trap to f2, since the silver rabbit would be on g3.

One facing a false protection capture might create a phalanx to block a push. A pull might still be possible, however. After being pinched on 17s of, Gold moved his horse onto f3 and pushed the silver camel onto g3. Silver moved a dog to f4 and blockaded that square, but the dog was then pulled into the trap after the silver camel was pulled away; the dog capture was accomplished via two pulls.

Repetition fights
Arimaa rules prohibit any move which results in a position identical to one which the same side has created twice before, at any point in the game. A repetition fight occurs when each side aims to restore a previous position or something close to it. When the three-repeat rule forces one side to deviate, the other has won the fight, for whatever that ends up being worth.

Pull and replace moves are a common source of repetition fights. In this diagram, Silver to move can break the horse frame with mb5sn Hb3n hc3w. Gold could reverse this move by pulling the silver horse back onto c3 and moving the gold horse back to b3. With this pull-and-replace, however, Gold would restore a position he created previously, and thus could never again end on that exact position. Silver, by contrast, created a new position the first time she broke the frame, and thus can repeat that move and force Gold to do something different. After a change elsewhere on the board, this repetition fight could potentially reoccur, since the overall position would be different.

Some repetition fights include several different positions; the outcome depends on who runs out of positions first. If a move can be undone and redone in three steps each, the fourth steps will be critical. With a nearly unlimited selection of fourth steps, such a repetition fight could go on and on if neither side chooses something different. A prolonged repetition fight might be a way to build reserve time, but one should look for a way out before the opponent finds one.

If a move creates a new position, and four steps are needed to restore the previous position, reversing the move would only be a way to stall. With Silver to move in this position, a sequence of four pull-and-replace moves is possible. Silver plays mh4we Hg3n hh3w (gold horse to g4, silver horse to g3). Gold then establishes a frame with Ef4sn hg3w Hg4s (silver horse to f3, gold horse to g3). Silver breaks the frame with mh4sn Hg3e hf3e (gold horse to h3, silver horse to g3). Gold can then retake g3 with Ef4ew hg3n Hh3w (silver horse to g4, gold horse to g3). Any of the last three moves was reversible, but could have then been repeated, while the reversal could not have been. In any event, Silver is the one forced to play a move other than a pull-and-replace.

A two-turn cycle occurs when each player essentially repeats his previous move, but with two pieces transposed. If Gold flips the silver cat to c6, Silver's only goal defense is to move her dog to c8 (dd6n cc6e dd7nw). If Gold then flips the dog to c6, Silver's only goal defense (cd6n dc6e cd7nw) will restore the initial position. After four such flips by Gold, Silver must deviate, allowing goal.

If the d6 piece were a cat, this would only be a one-turn cycle, since swapping the two cats would not change the position.

Here Gold wants to pull the silver horse to c4 to threaten it in c3, while Silver wants to free the horse and retain the threat to the gold camel. After pulling the horse with three steps, Gold must step his western dog or cat north, to stop the silver horse from escaping via eb5s hc4ee eb4n. In getting the horse back to d4, Silver will push the dog or the cat; the dog could be pushed to b3 or a4, while the cat could be pushed to d3, e4 or d5. Gold always moves the cat to d4 or the dog to b4, and thus Gold will run out of positions sooner despite having started this repetition fight. The silver horse can thus escape.

This position is based on the one after 8s of, but there d6 was vacant, and the silver dogs were both on the a-file. Gold could have captured the c7 rabbit, but preferred to threaten the silver horse instead, as Silver otherwise appeared poised for a strong camel hostage. Not wanting an additional gold attacker at c6, Silver opted not to push the gold cat to d5 until the repetitions precluded a push to d3 or e4. Once the gold cat was pushed to d5, it went to c7, capturing the rabbit and also threatening to free the gold camel. Silver could have avoided this by moving the a5 dog to b4 on 16s or even on 12s, as this would have changed the position, and the silver horse could not have then been frozen on c4.

In, Silver flipped the g4 gold horse to f5, hoping to capture it in return for the doomed silver camel. Gold undid the flip with Ee4n Hf5es Ee5s, but Silver repeated the flip and thus seemingly won the repetition fight. However, Gold then played the ko threat Mb5es hc6s Ha5e, threatening a silver horse in c3. After that horse retreated, the eastern gold horse moved back to g4. Since Gold's ko threat and Silver's response changed the position, Gold was now winning the eastern repetition fight.

"Ko threat" is a term borrowed from Go, which applies much less commonly to Arimaa. A ko threat need not be a good move locally; Gold's western position is of secondary importance here. The purpose is to restart the eastern repetition fight.