Battle of the Granicus
By defeating Alexander with just their cavalry, they would have regained their pride.

Battle: There are three different accounts of the battle given by the ancient historians Arrian, Plutarch and Diodorus Siculus. Arrian's account is the longest and most detailed, Plutarch's account is shorter but corresponds with Arrian's narrative except for some small differences. The account of Diodorus Siculus however is fundamentally different from that of Arrian and Plutarch.

Account of Arrian: After both armies finished their deployment, there was a moment of silence. Alexander then ordered Amyntas, son of Arrhabeus, to attack the Persian left wing with the Companion cavalry squadron of Socrates (now led by Ptolemy, son of Philip) at the front, followed by the Paeonian cavalry, the prodromoi and an unspecified infantry unit. Arrian then gives the impression that Alexander advanced at nearly the same time with the remainder of the right wing and the infantry, but later makes it clear that Alexander's attack came after the troops of Amyntas were pushed back. Arrian's source may have done so to make Alexander look more heroic.
The Persians answered the charge of the vanguard with volleys of javelins. Amyntas's force was at a disadvantage because they were severely outnumbered and the Persians were defending higher ground at the top of the bank. The Macedonians suffered losses and retreated towards Alexander, who now attacked with the remaining Companion cavalry of the right wing.
A. M. Devine reasons that the failure of the attack led by Amyntas was not a mistake, but a ruse to draw the Persian cavalry out of formation as they pursued the retreating force of Amyntas into the riverbed. The disruption of the Persian formation would have made them vulnerable to the second attack led by Alexander in person. There have been different translations and interpretations of Arrian's description of how Alexander executed the attack with his cavalry.
Some historians have taken Arrian's description as an attack on the Persian center. J. F. C. Fuller thinks Alexander led his attack in a movement diagonally to the left to attack the left of the Persian center. Similarly, A. B. Bosworth considers this movement to have been directed downstream to the left towards the Persian center. The Macedonian forces would have moved in a diagonal line and eventually formed up in wedge formation as they ascended the eastern bank of the Granicus.
Other historians maintain that Alexander attacked the Persian left wing. Ernst Badian thinks that Arrian makes it clear that the Persians had massed their cavalry at the left flank and that Alexander attacked that same mass of cavalry. This would have meant that Alexander must have moved more or less straight ahead. Presumably he would have entered the riverbed at a narrow point, so that his cavalry was in column and vulnerable to attack on their flanks as they ascended the riverbank on the Persian side at a wider point. To prevent this, Alexander would have sent his men towards the left, in the direction of the current, while still in the riverbed. This would have widened the front of his formation so that it would match the Persian formation as they emerged on the other bank. N. G. L. Hammond thinks it was a sideways movement upstream to the right so that the troops would remain in line as they crossed the Granicus. The purpose of this would have been to avoid crossing in disorder and in column, while giving Alexander's cavalry mostly the same formation as the Persian cavalry. A. M. Devine argues that it was not merely an oblique movement, but also an oblique formation.

Continuing Arrian's narrative, Alexander and his men engaged the Persians and their leaders in a hard-fought battle. The Macedonians eventually got the upper hand because their lance (the xyston) was more effective in melee than the javelin used by the Persians. During the battle, Alexander himself broke his lance and asked his servant Aretis for another. As Aretis had also broken his during the fighting, Demaratus of Corinth handed his lance to Alexander. Alexander then saw Mithridates, the son-in-law of Darius, leading a cavalry unit. Alexander charged and killed him with a lance thrust to his face. Another Persian noble, Rhosaces, then hit Alexander on the helmet with a sword, but the helmet protected him. Alexander dispatched Rhosaces by thrusting him in the chest with his lance. Sphithridates was about to attack Alexander from behind with his sword, but his sword arm was cut off by Cleitus the Black. The other Macedonian cavalry had by then established a foothold on the river bank and reinforced Alexander's force.
At this point in the battle the Macedonian cavalry and the light infantry which was interspersed with them forced back the Persian cavalry. This began on the Persian left flank where Alexander was fighting. Then the center was broken and the Persian cavalry on both wings fled. Alexander did not pursue them far but turned his attention to the Greek mercenaries, who had mostly remained inactive at their original position. The mercenaries were surprised by the turn of events and were attacked by both the Foot Companions from the front and the cavalry from all other directions. They were soon defeated and only a few were able to escape.

Account of Plutarch: Plutarch's account focuses more on the aristeia of Alexander's personal combat with the Persian leaders and gives less attention to the movements and tactics of the armies. This account mostly corresponds to Arrian's version of events, but there are some differences. According to Plutarch the Persians had stationed both infantry and cavalry at their front line. Alexander lead the first attack against the Persians and there are minor differences in his combat with Rhosaces and Sphithridates. While the cavalry battle was ongoing, the Foot Companions crossed the river and engaged the Persian infantry. After a short confrontation, all except for the Greek mercenaries routed. The mercenaries retreated to a higher position. They asked Alexander to negotiate for their surrender, but Alexander refused this and attacked. The mercenaries fought hard and were responsible for most of the Macedonian casualties.

Account of Diodorus: In the account given by Diodorus, the Macedonian and Persian armies encamped on the opposite banks of the Granicus and did not move for the rest of the day. Alexander then crossed the Granicus unopposed at dawn on the next day. Both armies then formed up and started the battle with their cavalry. Diodorus then describes the personal combat of Alexander with the Persian nobles, which differs slightly from the account of Arrian and Plutarch. When many of the Persian commanders were slain and the Persian cavalry was pushed back everywhere, the Persian cavalry opposite Alexander was routed. The rest of the Persian cavalry joined them in flight. After this, the infantry engaged each other, which soon ended with the rout of the Persian infantry.

Modern interpretations: Historian Peter Green, in his 1974 book Alexander of Macedon, proposed a way to reconcile the accounts of Diodorus and Arrian. According to Green's interpretation, the riverbank was guarded by infantry, not cavalry, and Alexander's forces sustained heavy losses in the initial attempt to cross the river and were forced to retire. Alexander then grudgingly accepted Parmenion's advice and crossed the river during the night in an uncontested location, and fought the battle at dawn the next day. The Persian army hurried to the location of Alexander's crossing, with the cavalry reaching the scene of the battle first before the slower infantry, and then the battle continued largely as described by Arrian and Plutarch. Green accounts for the differences between his account and the ancient sources by suggesting that Alexander later covered up his initial failed crossing. Green devoted an entire appendix in support of his interpretation, taking the view that for political reasons, Alexander could not admit even a temporary defeat. Thus, the initial defeat was covered up by his propagandists by a very heroic (and Homeric) charge into the now well-deployed enemy. In his preface to the 2013 reprint, Green conceded that he no longer believed his theory was convincing and that the contradiction could not be explained.
Other historians have dismissed the account of Diodorus as incorrect and favor Arrian's and Plutarch's narratives. The description given by Diodorus of a separate cavalry and infantry battle would have been uncharacteristic for the Macedonians, who always coordinated their infantry and cavalry to work together. It is likely that a crossing of the Granicus at night or at dawn would have been detected and opposed by the Persians. Even if an uncontested crossing elsewhere was possible, it might have induced the Persian army to retreat. In that case Alexander would have lost the opportunity to quickly destroy the only Persian army present in Asia Minor at that time. An attack in the afternoon would also have meant that the Persians would have to fight with the sun in their eyes, whereas the Macedonians would have the sun in their eyes if they had attacked at dawn.

Casualties: 

Macedonian army: For the Macedonians, Arrian gives losses of 25 Companion cavalry in the preliminary attack, 60 from the rest of the cavalry and 30 of the infantry, a total of 115 killed. Plutarch gives 34 dead, of which 25 were cavalry and 9 were infantry. Justin mentions 9 infantry and 120 cavalry killed.