How is Odysseus intelligent?
One way in which Odysseus is intelligent is that he knows how to ask for help and where to seek vital assistance in his long journey back to Ithaca. He knows when to flatter people, and when to use his brawn to get what he needs. For example, in Book VI, he visits the Phaeacians on the island of Scheria. When he sees the young princess, the lovely Nausicaa, he knows exactly how to befriend her: "Plead now, with a subtle, winning word, and stand well back, don't clasp her knees, the girl might bridle" (Fagles translation). In other words, Odysseus knows that he must approach her with gentle, flattering words to win her over and that grasping her will only frighten her. To win her over, he compares her to the goddess Artemis. This comparison to a deity does the trick, and she assists him in bathing and in directing him to her parents. She tells him to "grasp my mother's knees--if you want to see the day of your return." This is exactly what Odysseus does. He throws himself on the mercy of Queen Arete, and King Alcinous immediately decides to help Odysseus and gives him a seat of honor at his feast. Odysseus then launches into the story of his travels to convince the Phaeacians to help him.
In Book VIII, Odysseus manages to impress the Phaeacians with a combination of brawn, flattery, and shrewdness. When one of the Phaeacians insults Odysseus's ability to compete at sports, Odysseus hurls a discus farther than anyone else and proves that he is a powerful man. Later, Odysseus praises the bard, Demodocus, and tells him "I respect you, Demodocus, more than any man alive." He again compares the person he wants to flatter to a god, stating that a god such as Apollo himself must have taught the bard to play and sing. This is very savvy. Demodocus then, unsurprisingly, sings a song with a lyre that boasts of Odysseus's feats. This performance gets the audience into a great frame of mind to appreciate Odysseus as he launches into his long saga of his travels. Surely, the Phaeacians will help him now, as he's buttered them up with his savvy sense of how to appeal to their emotions and made them think he is a great man.
In the encounter with the Cyclops who claimed to be the son of Poseidon, Odysseus shows his intelligence. First Odysseus takes the name 'Nobody' so, after he blinds, Cyclops, Cyclops drives his neighbors away by saying "Nobody has blinded me." He also shows intelligence by blinding Cyclops instead of killing him. If Odysseus had killed Cyclops, they would have been trapped in the cave. Later on Odysseus refuses Calypso's offer of eternal life without offending her by flattering her beauty in comparison to the wife he chooses over immortality. And he also shows intelligence in the way he proves his true identity to Penelope. I'm sure you can now find even more examples in The Odyssey of Odysseus' intelligence and cleverness.