Arimaa/Advanced Tactics

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Even an apparently strong position may have a readily exploitable weakness. One might be on guard for basic tactical tricks, yet still miss a two-turn threat until the opponent has played the first turn, which may guarantee him a profitable second turn.

Double threat[edit]

A fork is not the only way one piece might create a quick double threat.

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Use this link to make moves on the board.

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, and Gold would need four steps just to save the now d5 horse, leaving the gold dog or cat to be captured. With only two defenders, the c3 trap was vulnerable. Were a gold rabbit on b3, this tactic would not force a capture.

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plan window

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.

False protection[edit]

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Gold to move cannot protect both the c5 camel and g5 horse, but can move his camel and threaten to capture the b6 silver horse in exchange for the gold horse. (game)

In the position at right, Gold to move faces two strong capture threats. Currently, the gold horse on g5 faces capture in f6. The g3 dog could unfreeze the threatened horse, but then the horse could only travel north, where it would be no safer.

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 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.

The gold elephant could temporarily protect both c6 and f6 by pushing the e6 dog onto f6, and then returning to d6 with a step to spare. However, that would waste three steps doing what Silver could undo in one step. One could analyze each of Gold's possible fourth steps, and then consider Silver's responses. In a live game with limited time, however, it is best to assume that trading three steps for one step would not end well, since the opponent would have three steps left to improve his position. That assumption would be borne out in this case.

Silver's Achilles' heel is the b6 horse. If Gold could threaten it with one-turn capture, Silver could not immediately capture the gold horse without giving up her own horse. If Silver declined such an exchange, Gold might then safely defend f6. Rb4n Mc5n cc7w Mc6n may look promising for Gold, but it would open a fatal weakness: with c5 and b4 now empty, the c4 rabbit would nearly have a path to goal. The silver elephant could step to c5, and the unfrozen and unblocked c4 rabbit could advance to b2. With the western part of Gold's home rank empty, such a goal threat would at least delay Gold long enough for Silver to come out ahead.

Discounting each of these options, Gold might conclude that capturing the c4 rabbit in exchange for his horse is the best he can do. However, if he continues looking for ways to threaten the b6 horse, he might think of Ed6n rc4sx Cd4w Mc5w. This would create false protection, offering Silver a horse exchange after a rabbit capture. If Silver declined the exchange and defended c6, the gold elephant could defend f6 in two or three steps, and the still unfrozen gold camel could back away from the threat in c6. The gold elephant would then be a threat to the silver camel, so a camel exchange might soon occur.

This tactic is pivotal; instead of losing a horse for a rabbit, Gold at worst loses a cat for a rabbit in the next two turns.

Congestion defences[edit]

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Sometimes, occupying a particular square can stop the opponent from playing a particular move. A simple example is at right: Silver is threatening 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 is often ineffective.

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Game

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.

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On 24s of this game, Silver established a one-turn goal threat. Goal was prevented by the single step Hf4e; surrounded, the g3 horse could not pull the g2 rabbit. Silver then moved the elephant to f1, leaving an "air bubble" on g1 which ensured that Silver could win on the next turn. Had the elephant moved all the way to g1 on 25s, a gold dog could have occupied f1, once again preventing the pull needed for a goal.

Pinch[edit]

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In the first diagram (31s from this game), the silver camel is threatened. No other silver piece can defend the c3 trap, and the camel cannot escape along the b-file because of the silver pieces in the way. 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, so the elephant cannot pull the camel into the trap.
  • The gold elephant is surrounded, so there is no room for a flip.

Gold can renew the threat by vacating a key square, but the silver camel and dog now have time to scatter.

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Sometimes, it is useful to set up a pinch even if no piece is threatened. Here Gold would like to attack c6, but if the gold horse goes to b6 it will be captured, while if the gold elephant immediately goes to d6, the silver horse can retreat to b6. The most direct way for Gold to advance is the pinch Ed3nn Dc3n Hb4n, which restricts the silver horse. If allowed, Gold might continue with Mb3n db6e Hb5n Mb4n, getting trap control, or Mb3w Ra4e Ma3n, trying to get a horse hostage.

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A pinch defends against capture by blocking capture moves. One can sometimes afford to lose a piece if the capturing piece would end 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.

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Capture congestion can strengthen a goal threat. Here, for instance, Silver cannot do a flip capture, and would lose within two turns if she did a pull or push capture.

Blocking the defender[edit]

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Forcing a capture often depends on keeping a defender off a square adjacent to the trap. Even an elephant one square away from defending 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 Gold must occupy the trap as part of the phalanx, 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 dog.

Opposite-trap blocking[edit]

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24s from this game

A second interesting case is the obstruction of an elephant moving between diagonally opposite traps. During a capture race, an elephant might make a capture and then, on the next move, wish to move to the opposite trap to defend. Since this is four steps' distance, any obstruction could be costly.

Here Gold just threatened the silver horse. This was a mistake, as Silver can play ee4ew Mg4w mh4w. Now, if Gold captures the horse, Silver will win the camel in two turns by occupying e6 to block the gold elephant, for instance with Mf4n ee4e df6e dd6e. Since the camel capture will only take two more steps, Silver can easily use the remaining two steps to defend against any gold counterthreat. Note that the g5 cat fences the gold camel.

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plan window

In this diagram, Silver has a single move which forces the capture of the gold horse: Hg4n ef4e cf7ss. Weak pieces on e6 and now f5 prevent Gold from defending the f6 trap in one turn. This is less effective than Silver's play in the previous diagram, since the subsequent horse capture would take a full four steps and leave the silver elephant decentralised. Although Silver to move can definitely capture the gold horse in two turns, Gold will have counterplay options:

  • The gold elephant could go to e5; if indeed Silver captures the gold horse, Gold can then capture the now f5 cat.
  • The gold elephant could go to e5, but with a somewhat longer-range plan. After Silver captures the gold horse, Gold could flip the e6 dog to e4, and soon fork it between c3 and f3.
  • Gold might move his elephant to b5, threatening to flip and win the silver horse if the gold horse were captured. The gold elephant would be no closer to defending f6, so the silver horse could retreat to b7.
  • Gold's strongest response is probably 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, giving the now c5 gold elephant time to defend f6. Taking three steps which could be undone in one step is usually bad, but here that one step is critical. 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 gold elephant could not defend it in time. This tactic is rather delicate: if Silver had a rabbit on d8, she could reply 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.

False protection patterns[edit]

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plan window

When threatening an enemy piece, it is common to keep it two squares from the relevant trap; if it were held right next to the trap, and another enemy piece then occupied a key square, the trap would have two enemy defenders. One might preempt this, however, by having a second friendly piece in place to make a false protection capture if a weak defender arrives.

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 push the camel to g3 on the current turn, but then there would not be time to 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 allow the horse to be saved, although the b8 dog can be captured.

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plan window

Problem: Gold to move

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 either the camel to retreat or the silver elephant to reach e3. 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.

Repetition fights[edit]

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 is said to win the fight.


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Pull and replace moves are a common source of repetition fights. In this diagram, Silver can break the horse frame by pulling the gold horse to b4 and stepping the silver horse to b3. This move is reversible: the gold elephant could pull the silver horse back onto c3, and the gold horse could return 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. Such a repetition fight could go on indefinitely, but one side or the other will usually try something different before too long.

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An unusual pull and replace sequence, from this game

If a move creates a new position, and four steps are needed to undo it, reversing the move would only be a way to stall. Here a fight is ongoing over whose horse will occupy g3. A sequence of four pull-and-replace moves is possible. Silver plays mh4w mg4e Hg3n hh3w (gold horse to g4, silver horse to g3). Gold then establishes a frame with Ef4s Ef3n hg3w Hg4s (silver horse to f3, gold horse to g3). Silver breaks the frame with mh4s mh3n Hg3e hf3e (gold horse to h3, silver horse to g3). Gold can then regain trap control with Ef4e Eg4w 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.

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Two-move (i.e. four-ply) cycles are fairly common. 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-move cycle, since swapping the two cats would not change the position.

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Here Gold wants to pull the silver horse to c4 to threaten it in c3, while Silver wants to free the horse without giving up the camel hostage. After pulling the horse with three steps, Gold must either put the dog on b4 or the cat on d4, to prevent Silver playing eb5s hc4ee eb4n. In moving the horse back to d4, Silver can push the dog to either b3 or a4, and the cat to d3, e4 or d5. In this repetition fight, Silver therefore creates six distinct positions, while Gold creates five positions – two with the cat on d4, three with the dog on b4. Because of this, Silver wins this repetition fight: Gold, despite moving first, will run out of positions sooner.

This position is based on this game, but there d6 was vacant. Gold still preferred to threaten the silver horse rather than capture the c7 rabbit and concede a good camel hostage, but pushing the cat to d5 was less appealing for Silver. Gold played out the repetition until the cat reached d5, and then captured the rabbit with the cat ending on c7, weakening the hostage.

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In this game, Silver flipped the g4 gold horse to f5, hoping to capture it in return for the doomed silver camel. Gold could undo the flip with Ee4n Hf5es Ee5s, but Silver would win the repetition fight. However, Gold played the ko threat Mb5es hc6s Ha5e, threatening the silver horse in c3. After the horse retreated, the gold horse moved back to g4. Since the ko threat and its 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. Often the response to a threat would need fewer than four steps, and the remaining steps would make continuing with the repetition fight impossible. Here, though, Silver would need three steps to capture in the east, leaving no time to defend in the west.

A ko threat need not be a good move locally. Here it is unimportant whether Gold's western position is improved by pulling the horse and allowing it to retreat. All that matters is that Silver must take a full move to respond.