Arimaa/Sample Games

Fritzlein vs. Omar, 2005 Postal Championship

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• Introduction: This game was one of the last games of the 2005 Arimaa Postal Championship to finish, and it ultimately determined the tournament champion. If Omar had won, he would have tied Fritzlein for first with a 9-1 record. As it turned out, however, Fritzlein prevailed, and finished the tournament undefeated.

• 1w: Omar deviates from the standard setup popularized by 99of9 by putting Dogs behind his traps and Cats back and center. A Dog behind the trap makes it easier to frame an opposing Dog. Also, as the game progresses, Cats are more likely to want to stay home for goal defense while Dogs are more likely to want to go on offense; Omar's setup anticipates this.

• 1b: Fritzlein chooses 99of9's opening as originally conceived. Having only a Cat next to a home trap makes it less costly to abandon that trap. This gives him a slightly greater freedom to counter-attack in the opening rather than defending, i.e. a tactical advantage rather than the strategic advantage Omar chose.

• 2w: Omar aggressively advances his Elephant four squares, which was not only the most common opening of that time, but was also invited by Fritzlein's central Camel.

• 2b: Fritzlein, by vitrue of having Cats behind his traps, could now respond by advancing his own Elephant four squares, because he would be threatening to capture a Dog for a Cat. Omar could then enter an extremely sharp line in which each Elephant captures several pieces, and it is not clear whether the intiative of the first move compensates for losing Dog for Cat. In any event the counter-attack is very playable for Silver, but Fritzlein plays very defensively, protecting his Camel and warding off an EH attack on f6.

• 3w: Due to the danger of an Elephant smother, Omar can't create any weaknesses by poking around with his Elephant on the seventh rank. He opts instead for a safe developing move. The fact that the Gold Elephant might as well step back one against a solid defense has dented the popularity of the Elephant-forward-four opening since the time of this game. Nowadays the most popular move is Elephant forward three and a Horse forward one on the same flank.

• 3b: Fritzlein defends c6 and moves his Elephant nearer Omar's Camel.

• 4w: Omar drags Fritzlein's Horse towards the f3 trap. Of course, capturing the Horse any time soon will be impossible with Fritzlein's Elephant in the neighborhood. The threat is strategic rather than tactical: Omar's Camel is near f3, waiting to hold the Horse hostage if the opportunity arises.

• 4b: The value of an Elephant taking a Horse hostage was very much in contention in this tournament, and still is. In some circumstances the hostage-giver appears to be in better shape than the hostage-taker. Nevertheless, Fritzlein decides to play defensively and free his Horse. The nearby Cat and Rabbit could also unfreeze the Horse, but not without exposing themselves. Giving up a Cat or Rabbit hostage at this stage of the game would be worse than giving up a Horse hostage.

• 5w: Omar is also unwilling to give up a Horse hostage. Current opening theory suggests that it is bad to give up a Horse hostage on a wing where the opposing Camel waits, because the hostage Horse is then easy to frame or to pass off to the waiting Camel. So far the opening has been defensive, with neither side willing to expose a piece other than the Elephant.

• 5b: Fritzlein finally takes a tiny risk to go on offense. It is not clear whether the tactical disadvantage of decentralizing his Elephant outweighs the strategic advantage of pulling a Rabbit, but since Omar has no large immediate threat, it may be a favorable tradeoff. A plausible alternative attack would have been to push Omar's e3 Horse to d3, unbalancing Omar's forces by putting both Horses on the same wing, but that wouldn't be to Omar's disadvantage unless Fritzlein switched wings with his Camel as well. Exposing a Rabbit is more tangible and immediate.

• 6w: Omar does not have any play against the Silver pieces on the east wing, he can't pull a Rabbit, and he can't stop Fritzlein from pulling the h-Rabbit further. Therefore, he astutely switches wings to make a bigger threat. Fritzlein can hardly continue hunting a Rabbit when his b6-Horse is in jeopardy, but his Elephant is too far away to properly defend the west wing, which creates a tactical dilemma.

• 6b: Fritzlein's b6-Horse can't stay safe without an advanced Rabbit to unfreeze it. Rather than advancing a Rabbit and accepting a game of competing Rabbit pulls, Fritzlein plays more ambitiously with a counter-threat to Omar's e3-Horse.

• 7w: If Omar now flips the b6-Horse to b4, Fritzlein can flip the e3-Horse to e5. Yet if Omar brings his e3-Horse to safety, Fritzlein's Elephant will arrive on the west wing in time to defend his threatened Horse. Omar chooses the conservative plan of centralizing his Elephant while pulling the b6-Horse to c5.

• 7b: Fritzlein's Elephant is now near enough to bring his Horse to safety. The exposed Rabbit on h4 leaves him at a slight strategic advantage, because nothing more important is going on anywhere else. In an opening where neither player attempts to take control of the other player's traps, exposed Rabbits are the main strategic marker.

• 8w: Omar makes an exciting move to take the initiative. Had Omar continued to play with his lone Elephant on offense, he would have been accepting a slightly disadvantegous position, because none of Fritzlein's Rabbits are exposed and none of Fritzlein's pieces can be taken hostage while Fritzlein's Elephant is centralized on defense. Note how Omar uses his Horse on the wing away from Fritzlein's Camel.

• 8b: The exposed Horse on a6 is a tempting target, but Fritzlein's Camel is on the wrong wing. (This is a frequent dilemma on defense in the opening: a centralized Camel is a target for the opposing Elephant, but a Camel on one wing leaves the other wing open for a Horse to attack.) Fritzlein would love to take the a6-Horse hostage with his Camel, but Omar's centralized Elephant could hinder the Camel's crossing long enough for the Horse to get away with a Rabbit pull. Fritzlein must therefore either take the encroaching Horse hostage with his Elephant, or ignore it.

• Had Fritzlein taken the hostage, Omar would have defended c6 with his Elephant, leading into a class of positions that is very poorly understood to this day. Opinion varies wildly both as to whether Gold or Silver would stand better in the resulting position, and as to what strategy each player should pursue.

• Instead, Fritzlein decides to drag the h4-Rabbit further forward, in order to be the first to threaten a capture.

• 9w: Omar could use his Horse to pull out Fritzlein's a1-Rabbit, but Omar doesn't have great prospects in a Rabbit-pulling race in which he is already behind. A more forceful option would be an E-H attack on the c6 trap, which would essentially force Fritzlein to take the attacking Horse hostage with his Elephant. The threat would be slow to develop, however, and would probably be worth less than a Rabbit lost in the mean time.

• Instead Omar declares an intention to defend his h-Rabbit with his Camel. Arimaa theory at the time of this game strongly frowned on exposing one's Camel so early in the game, lest it be taken hostage by the opposing Elephant. Subsequent experience, however, has shown that the presence of an advanced Rabbit on the wing where the Camel is operating makes the danger to the Camel considerably less.

• 9b: Fritzlein drags the Gold Rabbit to h6, an excellent square for holding a hostage, and brings his Elephant nearer to intimidate Omar's Camel from advancing further. Omar's a6-Horse is now roaming free of danger, which can't make Fritzlein happy, but the threat from Omar's Camel is too serious to be ignored entirely. As a general rule of thumb, if your opponent goes on the attack with his Camel, your Elephant has to leave whatever else it might do in order to go after the encroaching Camel.

• 10w: Omar correctly defends the f6 trap to save his Rabbit, and places his Elephant on e6, a better square than f5. From e6 Omar's Elephant exerts influence on the c6 trap, as well as indirectly threating Fritzlein's home rows. The shocking part of the move (at that time) was Omar's further Camel push, in defiance of Fritzlein's nearby Elephant.

• 10b: Fritzlein will not be able to capture anything in the f6 trap as long as Omar uses his Elephant to share control of it. Therefore Fritzlein is very eager to retain total control of the c6 trap. Without total control of at least one trap, Fritzlein will not be able to threaten captures. He immediately crosses with his Camel to scare off Omar's advanced Horse, and inches his Elephant closer to Omar's Camel.

• 11w: Omar's Horse retreats empty-handed, but his Camel advances undaunted, and threatens a capture in f6. Omar's Rabbit on h6 not only prevents his Camel from being frozen, it also prevents his Camel from being pushed to h6 as a hostage of Fritzlein's Elephant. Thus the g6-Camel and the h6-Rabbit, which each individually would be weak, support each other.

• 11b: Fritzlein decides to make room in his back ranks to threaten pushing Omar's Camel g7, where Omar's Rabbit can't unfreeze it any more. Fritzlein's Camel incidentally inches nearer Omar's Horse. Normally it would be bad to expose the Camel on d6 right next to Omar's Elephant on e6, but Fritzlein calculates that Omar's Elephant can't afford abandon the f6 trap any more. Fritzlein would be happy to lose his Camel while capturing both Omar's advanced Camel and Rabbit.

• 12w: Omar would have liked to use his Camel to pull the f8 Rabbit to f7 or the g8 Rabbit to g7, because either move would have deprived Fritzlein of a square to hold the Camel hostage in. Unfortunately, either pull would require all four steps, leaving the a5-Horse to be taken hostage by Fritzlein's Camel. Therefore Omar retreats his exposed Horse and delays Fritzlein's ability to set up his Elephant on g6 holding the Camel on g7.

• 12b: Omar has declined his last chance to retreat with his Camel; now Fritzlein can (and does) permanently secure the Camel as a hostage, albeit in a temporarily very awkward position. For Fritzlein to leave his Elephant on a trap in a live game would take guts, but in a postal game there is plenty of time to make sure it is not a blunder.

• Fritzlein's move is actually quite greedy, in that he retreats his Camel out of range of Omar's Elephant. In effect he is saying, "I am no longer willing to trade my Camel for your Camel plus Rabbit", in other words, "I think I am presently winning by more than a Rabbit." The further course of the game shows this judgement to be overly optimistic.

• 13w: Omar flips Fritzlein's Cat into danger, a move which incidentally would not have been possible had Fritzlein's Camel stayed on d6. Omar isn't trying to free his Camel so much as get adequate compensation for it, but it is important to note that he must keep his Elephant on e6 while flipping the Cat. If Fritlzein could station his Elephant on e6 holding Omar's Camel hostage on e7, Omar would be hard pressed to save his Camel from outright capture.

• 13b: If Fritzlein were in the ideal Camel-hostage position, he could send a Horse to defend f3. Because he still needs a couple of moves to get into proper hostage-holding position, he has no time to save his Cat.

• 14w, 14b, 15w: These moves are nearly forced on each side. Each player is getting what he wants, and oddly, each player is happy with the resulting position. Fritzlein believes that the Camel hostage will be worth more than a Cat, and Omar thinks the Cat will be worth more. The last word on this trade-off has not yet been written, but it is quite probable that Fritzlein, if he is winning at all, is winning by less than a Rabbit.

• 15b: Omar is very constrained by the need to keep his Elephant adjacent to f6. His only active plans are (1) Repeat his Cat-capturing maneuver with his Elephant to capture another of Fritzlein's pieces in f3, or (2) Flood the east side of the board with little pieces to hem in Fritzlein's Elephant and free Omar's Elephant to roam the board. Fritzlein therefore immediately disrupts both of Omar's potential plans by stationing a Horse adjacent to f3.

• 16w: Omar just as quickly abandons his hostage Camel for an opportunity to capture a Horse. This calls into question Fritzlein's judgement on move 15b. Should Fritzlein have made a move which would do less delay to a piece flood in the east, but also not give Omar the option of trading? Too few games of this nature have been played to answer this question definitively, but for the second time each player gets what he wants, and an unbalanced position results.

• 16b: The only question for Fritzlein is whether to end the move with his Elephant on e6, f5, or g6. Ending on g6 would threatening Omar's Rabbit immediately, but it would be better to have the Elephant centralized in case Omar attacks Fritzlein's heavy pieces around c6. Leaving the Elephant on e6 is more attractive, since that would deprive Omar of the best square for defending f6. The tradeoff would be leaving the g-file more open for Omar's Elephant to dive in to pull Fritzlein's Rabbits as well as generate a minor goal threat.

• 17w: Omar grabs the prime defensive spot on e6, where his Elephant can defend f6 as well as influencing c6 and disrupting Fritzlein's ability to rebalance his forces. The Horse step is very important, as Omar has his eye on g6 again.

• 17b: Fritzlein fails to understand Omar's designs on g6, and does nothing to correct the bad balance of his forces. He has a Camel and a Horse on one side, and two Dogs on the other. He would be much better off with a Dog on each side and the Camel and Horse split. Fritzlein's threat to the h6 Rabbit is premature and impotent.

• 18w: Omar advances his Horse, while bringing up other pieces to unfreeze it at need and to keep f3 safe.

• 18b: Fritzlein belatedly realizes his Dog on g6 is overmatched, and starts switching wings with his Camel.

• 19w: Omar takes advantage of the temporary misplacement of Fritzlein's pieces lauch an Elephant-Horse attack on f6. This attack removes the danger to Omar's h6 Rabbit, ties up Fritzlein's Elephant, and forces Fritzlein's Camel to come across to relieve the pressure. On the other hand, the attack poses no immediate danger to Fritzlein, and it effectively forces him to do a reblancing of forces he needed to do anyway. If Omar has to retreat his Horse soon, it will be a net loss of time.

• A final argument against this Horse advance is that Fritzlein can't afford to frame the h6-Rabbit anyway until after his Camel has finished crossing, because if he completes the frame with a Dog on g6, Omar can break the frame anyway. Therefore Omar might have done better to hold off advancing his Horse, instead putting his Elephant on f7, with a threat to pull the f8 Rabbit and/or attack Fritzlein's Camel if it comes closer.

• 19b: Fritzlein once again plays ambitiously, threatening to take Omar's exposed Horse hostage with a Camel. Omar's Dog and Rabbit on that wing have therefore also become indirectly threatened. However, in haste to complete this plan Fritzlein doesn't have a spare step to shore up the defense of the c6 trap.

• 20w: Omar divides his Horse from Fritzlein's Camel. Omar's Elephant may be subject to blockade threats behind enemy lines, but it also makes incidental threats to pull Rabbits. Using the fourth step to advance the western Horse may look irrelevant, but neither side has capture threats at the moment, so there is time to spare for a move with a strategic purpose. As long as Fritzlein's Elephant and Camel are both tied up around f6, Omar's other Horse will be safe to pull a Rabbit down the a-file.

• 20b: Fritzlein could occupy g6 with his Camel, pushing Omar's Horse to g5, but he doesn't want Omar to flip the f8 Rabbit to f6 in response. Fritzlein's actual move is rather sharp, as it leaves his Horse alone next to a trap, and exposes his Cat to being pushed. On the other hand, the threat to Omar's Horse is now severe.

• 21w: Omar wisely chooses to retreat his Horse before it loses the option to retreat. However it is an open question whether the fourth step might not have been better spent advancing his other Horse again rather than retreating all the way to h3.

• 21b: Fritzlein's Camel sits a bit awkwardly on h5, but he wants to make a threat large enough that Omar must choose to stop it rather than flipping the f8 Rabbit to f6.

• 22w: Omar resists having his Dog taken hostage, and incidentally gains the small strategic advantage of getting one of Fritzlein's Rabbits off the back rank. On the other hand, he has increased the danger of an Elephant blockade.

• 22b: Fritzlein now simultaneously threatens to blockade Omar's Elephant and threatens to frame the Dog in f6. This tactic leaves Omar with only one playable move, but maybe Fritzlein should have done something else, because the one move Omar can play is strong enough to free him without losses, and in a postal game Omar is unlikely to miss it. The tactics in this sequence were sufficiently complex that it isn't clear even in retrospect whether better moves by one player or the other could have tipped the balance.

• 23w: Omar escapes from the threatened blockade, and makes an incidental threat to Fritzlein's b6-Horse. Thus Fritzlein's earlier sharp play succeeded in generating threats, but also left him open to counter-threats.

• 23b: Fritzlein elects not to take the Dog and Rabbit in exchange for a Horse. Ordinarily this exchange would be nearly even, but experience has shown that even trades hurt the side with stronger pieces while helping the side with more numerous pieces. The FAME score agrees that Fritzlein would lose out to get DR for H, and would also lose out to get H for DR. Fritzlein needs to use the strength of his Camel to win pieces outright if the opening trade of M for HC is going to work out in his favor.

• 24w: Omar has only one option for a second move in a row. His actual move is the only one to prevent the capture or frame of his f5 Dog.

• 24b: Fritzlein is playing against Omar's exposed Rabbit while keeping Omar's Elephant away from his Camel. Since Omar can obviously defend his Rabbit no matter what, Fritzlein wants the Camel forward to assist in the creation of a second threat after the Rabbit is framed.

• 25w: Omar defends f6 with his Elephant, and enters a race of sorts. Fritzlein will be trying to create a second threat, so that Omar's Elephant can't defend both his advanced Rabbit and the other threat. Meanwhile Omar will try to drag a Rabbit down the a-file to threaten it in c3. But Omar sagely doesn't put all his eggs in one basket by using three steps in the west, because if Fritzlein crosses wings with his Camel Omar will have to give up on Rabbit dragging to attack g6 with a Horse again. Thus Omar's move is split between defending f6, threatening the a-Rabbits, defending c3 from attack, and preventing Fritzlein's Camel from easily switching wings.

• 25b: Fritzlein activates his Camel and pins Omar's Elephant to defense.

• 26w: Omar still does not want his Horse to be taken hostage. If Fritzlein were able to get his Camel back to g6 holding Omar's Horse on g7, it would indeed look grim for Omar. On the other hand, perhaps it isn't so critical for Omar to avoid a Dog hostage, and a couple of steps would be better spent charging up the a-file.

• 26b: Fritzlein voluntarily advances a Rabbit with some thought of generating a goal threat. This is misguided, however, because Omar's defense is sufficiently thick at the moment.

• 27w: Omar develops his threat apace.

• 27b: Fritzlein rotates a Dog forward with the idea of freeing his Elephant for attack. The Dog will cover f5 so that Omar can't put a small piece on f5, freeing Omar's Elephant.

• 28w: Omar uses only two steps to pull the a-Rabbit and spends two to bring his Dog out of danger. He feared that Fritzlein would attack it with either his Elephant or his Camel and eventually threaten it in c6. However, it is not clear that Fritzlein can safely leave the east with his Elephant or safely cross over with his Camel. Moreover, retreating the Dog makes it rather easier for Fritzlein to take control of f3. In hindsight it would perhaps have been better for Omar to pull Fritzlein's a-Rabbit with all four steps.

• 28b: Fritzlein rotates his Elephant into the attack. It appears that he can now forcibly gain control of f3 against any non-Elephant defense, so in theory his second threat has been established. Fritzlein's Rabbit push to h4 doesn't pose much additional threat, though. His own west wing is now too depleted to flip Omar's Horse to g4 and launch a goal attack; indeed he rather has to seal off Omar's pieces from advancing.

• 29w: Omar belatedly counters the threat to f3. However, given that he can no longer protect f3 in the long run, he would be better off to abandon his framed Rabbit immediately rather than letting the situation deteriorate further.

• 29b: Fritzlein not only threatens Omar's Dog on f3, but also attempts to seal off the Gold pieces from coming to the rescue of f6. If Omar could support f6, he would be able to score a goal more quickly than Fritzlein, regardless of any losses suffered in f3. However, it appears that if Omar doesn't defend f3, Fritzlein has time to capture a Dog and fall back in defense quickly enough to prevent goal.

• 30w: Suddenly Omar can't bring his Elephant back on defense without losing either his d3-Cat or his f3-Dog, so he must delay his Elephant retreat for another move.

• 30b: Fritzlein has won the race. Omar faces unacceptable losses in f3, and therefore must give up on f6. In spite of this, Fritzlein's "victory move" of flipping Omar's Dog to g4 was actually a serious strategic blunder. To instead flip Omar's Horse to g4 would have forced Omar's retreat just as effectively, while also creating the strategic plus of getting Omar's Horse out of position where it could later be threatened. In general Camels much prefer to attack Horses than Dogs.

• 31w: Omar must retreat his Elephant, but it is worth considering pushing Fritzlein's Dog to e4 rather than f5. Pushing to e4 would create a stronger threat to take Fritzlein's Camel hostage. On the other hand, it would give Fritzlein slightly greater potential to launch a double-trap attack, contesting control of both c3 and f3, while rolling forward Rabbits on both sides of the board. If Fritzlein were able to share control of both of Omar's home traps, it could in certain circumstances be enough of a strategic advantage to compensate for a Camel hostage. Timing would be critical in such a wild attack.

• Just as Omar was not particularly distraught to give up his own Camel hostage, he is not overly eager to take Fritzlein's Camel hostage. Omar's push to f5 allows Fritzlein's Camel to escape much more easily, but this does not necessarily lead to disadvantage. If the Camel runs away Fritzlein's advanced Rabbits are simply weak.

• 31b: Fritzlein seizes the opportunity to run with his Camel, carrying a Dog hostage, no less. A double-trap attack is not Fritzlein's style, so he wants to avoid a Camel hostage at all costs. A free-roaming Camel represents his only advantage over Omar. If Omar gets the Camel hostage, Fritzlein might not get as much compensation (a full Cat) as Omar did in the opening.

• 32w: Omar temporarily ignores the threat to his Dog in order to prevent Fritzlein from creating any threats elsewhere, in particular an Elephant-Horse attack on c3. With the given material imbalance, it would be devastating if Fritzlein got an Elephant-Horse attack rolling, because Omar has no Camel to defend with, and defending with his Elephant would leave Fritzlein's Camel to rampage unimpeded.

• 32b: Fritzlein gets his Camel a step further from Omar's Elephant and puts another Rabbit in between for safety. Incidentally he threatens Omar's Dog with capture.

• 33w: Omar defends f6 and shores up the defense of the east corner against any possible goal attack.

• 33b: Fritzlein threatens to frame Omar's Dog, which Omar could ill afford. Omar would have no way to break the frame, and Fritzlein's Elephant would then roam free.

• 34w: Omar prevents the frame and retreats into a compact position. His position now has no weaknesses other than the hostage Dog, which will make it very difficult for Fritzlein to create a second threat.

• 34b: Fritzlein plays an amusing move. His plan is to threaten Omar's Dog from behind the trap in such a way that the Camel will not be exposed to any danger of being taken hostage. This plan is far too slow, however, given that Omar can quickly set up threats to Fritzlein's Rabbits on a5 and h4.

• 35w: Omar threatens a Rabbit frame.

• 35b: Fritzlein occupies e3, a good defense point for his Elephant, and laboriously continues his action against Omar's Dog. Now that Fritzlein's Elephant is tied up defending a small threat rather than generating a big threat, it should be clear that Fritzlein has misunderstood the position.

• 36w: Omar pins Fritzlein's Elephant. Now it is his turn to attempt to overload Fritzlein's defense. If Omar creates a second threat, Fritzlein's Elephant will not be able to defend them both.

• 36b: Fritzlein gets his Cat out of the way and moves Omar's Dog a square closer to being threatened. The Cat comes forward to threaten to occupy f4, which would free Fritzlein's Elephant.

• 37w: Omar blocks the Cat and rotates forward his Horse. His Horse and Elephant combined represent a serious threat to take control of f6.

• 37b: Fritzlein finally threatens the Dog. Also he aligns his Cats and Rabbits so that neither Omar's Elephant nor his Horse can get to g6 in one move.

• 38w: Omar clears the path to g6.

• 38b: Fritzlein re-establishes the block to g6, but he has become very worried. It is finally clear that he doesn't have the resources to frame Omar's Dog, and there is no second threat in sight which would overload Omar's Elephant. He would like to pass the hostage Dog off to his Horse so that his Camel could operate against Omar's Horse, but that is another slow plan, and it could be disrupted by Omar's Elephant moving up the middle.

• 39w: Omar seals off the east wing to prevent any goal threats, and moves against Fritzlein's exposed Rabbit on a5. He is careful to support his advanced Horse so that Fritzlein has no opportunity to abandon his framed Rabbit, switch wings with his Elephant, and quickly generate a second threat.

• 39b: Unless he does something drastic, Fritzlein will lose this race. Omar will soon have a second threat, and Fritzlein will not. Fritzlein must abandon his framed Rabbit to activate his Elephant. Yet it is not clear how much play this will buy him, since Omar has minimized vulnerabilities in his own position, and can quickly threaten another of Fritzlein's h-Rabbits. In desperation to generate counterplay, Fritzlein leaves his west wing underdefended, and advances his Horse into Omar's territory.

• Fritzlein's advanced Horse serves a dual function. It threatens to burrow behind f3 to release the framed Rabbit (possibly even scoring a goal), and it threatens the c3 trap somewhat in conjunction with Fritzlein's Elephant crossing over. However, both of these plans will take longer to come to fruition than the damage Omar could do around the c6 trap.

• 40w: Omar makes his biggest mistake of the game. Had he played against c6 immediately, he would have launched a race to goal with time on his side. His threat to Fritzlein's Horse instead squandered a couple of steps in the goal race.

• 40b: Fritzlein removes his Horse so that Omar can't set up a one-turn capture threat against it. If Omar could threaten the Horse again in one turn, then that would trump the threat to Omar's Dog, and 40w would have been good strategy after all. Tactics and strategy are closely interwoven: the presence of a certain tactic (like Fritzlein hiding his Horse) can make an otherwise good strategy turn sour.

• 41w: Omar can't now afford to return to defending his Dog from capture in f6, because Fritzlein's Horse is two moves closer to digging out the f3-Rabbit from behind. Omar therefore launches the attack on c6 that he could have started on move 40b. The difference of delay is that now he must give up his Dog to get the attack rolling.

• 41b: Fritzlein can't defend c6, and can't generate a fast enough threat to get the f3-Rabbit to goal, so he snatches the Dog before his west wing implodes.

• 42w: Omar captures a Rabbit and pushes his Rabbit forward one step, only five steps from goal on a clear line. Fritzlein's west side is in tatters.

• 42b: Fritzlein has no choice but to defend with his Elephant, abandoning his framed Rabbit. The exchange of two Rabbits for a Dog is actually slightly in Omar's favor, but Fritzlein's Elephant is no longer pinned, and he no longer faces material threats. Indeed, it is Fritzlein who is in a better position to make the next material threats. Omar's only remaining trump card is having an extra Rabbit with which to make goal threats.

• According to FAME, Fritzlein was materially +25 to have captured M for HC in the opening, and now having captured MDR for HCRR he has extended the advantage to +66. On the other hand, his forces are temporarily very poorly placed, with all his officers on the east wing so the west wing still vulnerable to goal attack despite the Elephant defending. Omar may be able to capitalize on this weakness before Fritzlein can favorably re-align his slightly stronger army.

• 43w: Omar shores up the defense of c3 and prevents his Horse from being taken hostage. Superficially this is reasonable, because the consequences of a Horse hostage look dire. Omar's Elephant would be tied to the defense of c6 while Fritzlein's Camel and Horse could quickly generate a second material threat.

• What this calculation leaves out, however, is that goal threats are inherently more forceful than material threats. Omar has nothing to lose by advancing at least one Rabbit in the east without delay, as Fritzlein will be too busy defending to make progress elsewhere. His only parallel concern should be dividing his four steps between goal defense (against the a5 Rabbit) and goal attack.

• 43b: Fritzlein supports the defense of c6 with a Dog, and inches his Camel closer to Omar's Horse.

• 44w: Again Omar incorrectly plays the material game rather than the goal game. His Horse now can't be taken hostage, true, but the threat to Fritzlein's Rabbit is illusory given the proximity of Fritzlein's d1-Horse. Indeed, given that Fritzlein has the stronger army, it will in general be extremely difficult for Omar to create two simultaneous material threats. For him to gain ground materially, he needs at least one of his two threats to be a goal threat.

• 44b: Fritzlein defends his threatened Rabbit and prevents Omar's Elephant from centralizing.

• 45w: Omar correctly launches a goal attack (finally), although now Fritzlein has a Dog helping on defense which wasn't there two moves ago.

• 45b: Fritzlein is in a delicate situation. He can't easily create a goal threat, and his only hope of generating two material threats it to activate his Camel. On the other hand, activating his Camel will expose it to danger should Omar choose to switch wings with his Elephant. Fritzlein decides he has time to play the material game for one more move, time that his c7 Dog buys him. However, this requires accurate forecasting of the strength of Omar's attack. In a live game Fritzlein's move would be intolerably risky, as there would be no way to be sure his west wing defenses would be adequate.

• 46w: Omar correctly ignores Fritzlein's threats in the west and presses his goal attack in the east.

• 46b: Fritzlein does the minimum to defend while pressing his western attack. Again, this requires very precise calculation to be playable, but prevents him from being forced into a totally passive posture on defense.

• 47w: Omar continues to play strongly. Fritzlein's defense now hangs by a thread.

• 47b: Fritzlein plays the only move to save his position. Omar's threats are so sharp that Fritzlein's position would crumble if, for example, he took Omar's Horse hostage on a6 instead of b7. Furthermore, it was necessary for Fritzlein to have calculated this correct defense already on move 45b. Thinking five or six ply ahead is possible only postally, and even then only in extremity of need.

• 48w: Omar must now realize that his attack can't break through, but his continued advance regains mobility for his Elephant, saving his threatened Horse and creating alternative attacking possibilities.

• 48b: Fritzlein is forced to release the Horse he is holding hostage, which in itself proves the correctness of Omar offering it as a hostage on move 47w, and again calls into question Omar's refusing to offer it as a hostage on 45w.

• 49w: Omar's Rabbit is now far enough from goal that Fritzlein is threatening to enter a goal race by capturing Omar's b3 Cat. Omar has several reasonable options for dividing between attack and defense; the chosen move leans towards eventual disengagement.

• 49b: Fritzlein's move appears to be a threat to Omar's b6-Horse, but in fact, since the Horse can retreat, the move is aimed equally at the d5-Rabbit. Omar can't save one without losing the other.

• 50w: Rather than defend or delay, Omar opts to trade Horses. The advantage of making an even trade is that every trade weakens the importance of Fritzlein's Camel and raises the importance of Omar's extra Rabbit. The disadvantage of the trade is that Omar's Elephant loses time leaving the west wing when it will have to return to save his f4 Horse and/or threaten Fritzlein's Camel.

• 50b: Fritzlein has no choice but to accept the offered trade, since he can't defend his own Horse.

• 51w: Omar now offers the additional trade of Horse and Rabbit for Dog and Rabbit. This puts him even further behind in terms of strong pieces, but also thins out the board a bit more, which works to the advantage of the side with more numerous pieces.

• 51b, 52w: Fritzlein again has little alternative to Omar's offer, and Omar must stick to the course he has charted.

• 52b: Fritzlein captures the Rabbit before the Horse, since Omar's Horse is doomed anyway, and once the Rabbit is gone Omar's goal threats disappear. Putting the Elephant on c5 rather than d6 lends extra defense to the depleted west wing.

• 53w: Omar's Cat is scared by Fritzlein's Elephant, and captures to c2 so that Omar's Elephant can leave next move without hanging the Cat. However, given that Fritzlein's west wing has been denuded once again, Omar will probably want to launch another Rabbit up that side before long. Leaving the Cat on b3 after the capture might therefore have been more strategically sound.

• 53b: For the second time in the mid-game, Omar's failure to be aggressive in the west encourages Fritzlein to make a material-oriented play while leaving a minimal goal defense. He couldn't have afforded to centralize his Elephant if Omar's Cat had stayed on b3, but with the Cat on c2 it is just barely possible.

• The point of Fritzlein's Elephant steps is to eventually attain a favorable alignment of pieces. If Fritzlein's Dog ends up opposite Omar's Dog, then Fritzlein's Camel will be opposite Omar's Cat(s), and Omar's Elephant will do nothing but chase Fritzlein's Camel. In the worst case, Omar would take Fritzlein's Camel hostage and totally neutralize his material advantage. If, on the other hand, Fritzlein's Dog ends up opposite Omar's Cat(s) and Fritzlein's Camel ends up opposite Omar's Dog, Omar will be threatened on both wings at once, and his Elephant will eventually become overloaded.

• 54w: As before, Omar has no hope of creating two material threats, so he wisely starts setting up a goal threat.

• 54b: Fritzlein defends with his Dog rather than with his Elephant, for the piece-alignment reasons mentioned above.

• 55w: Omar intensifies the goal threat, threatening to keep Fritzlein's Dog on the wrong wing.

• 55b: Fritzlein stubbornly refuses to defend with his Elephant, which would release Omar's Dog to switch wings. Instead he brings his Cat to e6 to prevent his Dog from being pushed there. Fritzlein is starting to feel a pinch because his eastern Rabbits are too advanced to participate in defense of f6, yet not far enough advanced to be a direct goal threat.

• 56w: Omar re-creates the goal threat, and astutely advances a Rabbit to c3. This late in the game one doesn't have the luxury of keeping Rabbits on the back row for goal defense, because they are more necessary for trap defense and for the creation of goal threats.

• 56b: Fritzlein defends the goal threat, and attempts to complete the re-alignment of his forces.

• 57w: Omar makes a strategic mistake. It is necessary for him to push Fritzlein's Dog back into the c6 trap, keeping alive the possibility of goal threats in the east. His threats against Fritzlein's Camel are much less pressing. Early in the game nothing is more important than a Camel, but late in the game goal threats loom large. Given that Fritzlein's Dog is now favorably aligned, Fritzlein might gain the option of attacking in the east with Elephant and Dog to create a goal threat, sacrificing his Camel in the west to buy time.

• 57b: Fritzlein keeps Omar's Dog on the wrong side, and moves his Camel further west to force Omar's Elephant to choose between extremes.

• 58w: Omar plays consistent with his mistake on 57w. After this move he is strategically lost.

• 58b: Fritzlein finishes aligning his Dog to the west. He is happy to have his Cat linger on defense in the east while his Rabbits are forward to support his Camel and eventually create goal threats. Cats are the one type of piece which are strategically happiest to spend the entire game in the home half of the board.

• 59w: Omar continues to play against Fritzlein's Camel, now directly threatening it with capture. Also his Rabbit advance is tactically tricky to meet. However, given his poor strategic situation, tactical tricks are all that remain to Omar. At postal speeds, Fritzlein has plenty of time to be careful.

• 59b: Fritzlein now offers his Camel to be taken hostage for the price of only Omar's c4 Rabbit. It may seem odd that he was happy to get a Camel hostage in the opening for a whole Cat, while now he is happy to give up a Camel hostage for a mere Rabbit, but the relative piece values have changed immensely since then. The Camel has sharply declined in value relative to the little pieces.

• When one is defending a Camel hostage in the opening, one's Elephant has the option of leaving at any time in pursuit of some other prize. Usually that other prize is a Horse at best, i.e. something worth less than the hostage Camel. In an endgame, in contrast, an elepant can abandon a hostage Camel in pursuit of a goal, i.e. something worth more than a Camel. Thus not only does the value of the Camel itself decline, the value of the Camel hostage declines.

• 60w: Omar decides he can't afford to accept the offered hostage. His crossing underscores the importance of Fritzlein stepping his Rabbit to e4 on move 59b. It costs Omar one step in the goal race.

• 60b to end: Fritzlein's win is now a matter of patience and technique. He has time to calculate variations exactly to be sure that his goal threats will always land home one move sooner than Omar's, so Omar must always defend rather than racing. The fact that Fritzlein's threats are always stronger has nothing whatsoever to do with the relative tactical ability of the two players. On the contrary, it all traces back to the piece re-alignment of moves 53b to 58b. With a different position it would be Omar's extra Rabbit causing his goal threats to be eventually unstoppable.

• On move 63w Fritzlein's Dog is only optically threatened, as Omar would allow a goal in two moves by capturing it. When Omar finally concedes on move 64w that his Elephant must retreat to prevent goal, Fritzlein completely shuts down any counterplay and indicates an intention to win materially rather than force an immediate goal. Facing a painful mop-up operation, Omar decides with move 67w to go down shooting rather than be paralyzed.

• Conclusion In the end the stronger pieces prevailed over the more numerous pieces, but this was far from an inevitability. On the contrary, each side outplayed the other strategically in various phases of the game, causing the probable outcome to swing back and forth. Until move 58, the game remained rich and balanced, with deep opportunities for both sides to uncover within the position.

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