One of the most understated, but important, examples of Odysseus's intelligence (metis, a combination of cunning and intelligence) is in book 3 when Nestor, known in The Iliad as the wisest of Agamemnon's counselors, tells Odysseus's son, Telemachus, about the Achaeans' disagreement over when to sail from Troy after Troy's destruction:
And all the time we were there, / not once did Odysseus and I, in assembly or council, / speak on opposite sides. We were both of one mind, / and we always agreed about how to advise the Achaeans. (3: 109–12)
That Nestor and Odysseus are always of one mind in matters of strategy speaks to Odysseus's intelligence in all matters requiring a strategic decision. Nestor, whose counsel Agamemnon relies upon completely, relies on the confirming counsel of Odysseus, a clear indication that the wisest among the Achaeans looks to Odysseus for intelligent counsel.
When, in book 6, Odysseus is washed up on the shore of the Phaeacians and encounters Nausicaa, the daughter of King Alcinous, he is covered with branches to hide his nakedness. After having frightened her attendants, he must decide whether to approach her and supplicate himself to her mercy—that is, bow before her, get on his knees, and grasp her knees, a sign of abject distress:
In the end, it seemed to him better to stay where he was / and speak to her from a distance, because she might take / offence if he clasped her knees. (6: 145–47)
He then asks her if she is a goddess or a woman and praises her for her beauty and bearing, comparing Nausicaa to Zeus's daughter Artemis. He ends this long, flattering speech by asking for her help. Nausicaa is suitably impressed and replies:
Stranger, your manners are proof/that you are neither a man of low birth nor a fool...since you have come to our country, you shall not lack clothing or anything else a supplicant needs...(6: 190–96)
Odysseus's intelligent decision not to appear initially as a supplicant has given him the opportunity to use his words as a way to both introduce himself as a high-born person and also to express his admiration for Nausicaa in courtly language, going so far as to pretend to mistake her for a goddess. He recognizes that he looks like a wild man, but, through his words, he is able to change that impression and put Nausicaa at ease.
Perhaps the quintessential example of Odysseus's intelligence—although many scholars call this his "cunning intelligence," his cleverness—is in book 9 when Odysseus and his men have put out one of the Cyclopes' eyes, and, hearing Polyphemus's screams of agony, Cyclopes who live nearby ask him why he is screaming, to which he answers:
"Noman, my friends, has tricked me and tried to kill me." / And they said to him, "Well, if no man is trying to hurt you, / obviously some illness that comes from Zeus / is causing your pain..." (9: 404–07)
Odysseus has created perhaps the world's first "Who's on first" trope, and he congratulates himself for having frustrated Polyphemus's attempts to get help. Later, of course, Odysseus comes up with an ingenious way of escaping from Polyphemus—strapping his men to the underside of large sheep who are taken out of the cave to pasture by Polyphemus. In this case, unfortunately, Odysseus's love of kleos (fame) is stronger than his common sense, and he tells Polyphemus his real name, which dooms him to another few years of voyaging.
An interesting moment that showcases Odysseus's intelligence as well as his profound curiosity is when he encounters the Sirens—those beautiful but deadly creatures who use their heavenly singing voices to lure unsuspecting sailors to their deaths upon the jagged rocks of their island. While Odysseus doesn't want to meet a similar fate, but at the same time he's curious to find out what the Sirens' song sounds like.
Luckily, he was instructed by the goddess Circe that, in order to survive hearing the Sirens' song, he would need to be strapped to the mast while the ship passes the sirens. Circe also tells him to have his crewmates plug up their ears with beeswax so that they don't fall under the Sirens' deadly spell. Wisely heeding the goddess's instructions, he follows them to the letter, showing the intelligence to defer to her knowledge of these matters. Odysseus then gives his men strict instructions not to untie him from the mast, no matter how much he begs them to.
Odysseus, as expected, is tortured by the exquisite beauty of the Sirens' song, and begs his men to untie him from the mast. But mindful of their orders and with their ears stuffed with beeswax, they ignore his plaintive cries and sail on as intended. The plan works to perfection, and thanks to Odysseus's intelligence and to Circe's warnings, he and his men survive unscathed from their encounter with these dangerous, legendary creatures.
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.