Discuss episodes in which Odysseus triumphed through his own ingenuity and episodes in which he received help from immortals.

At times, Odysseus survived on his own wits rather than the favor of the gods. One thing he did was have his sailors block their ears with beeswax so that they could not respond to the songs of the Sirens. At the same time, he asked them to tie him to the ship mast so that he could not respond either. His foresight and planning helped them escape where many men would have died.
Another time he was able to escape was when he blinded Polyphemus, the Cyclops. He told his men to bind themselves to the bottoms of sheep so that they would not be detected as they exited the cave. He was also clever enough to tell the Cyclops his name was "Nobody" so that when Polyphemus called for help, the other cyclops would think he was not being serious. He got the Cyclops drunk and used his own staff to blind him.
Other times, Odysseus only survived because the gods favored him. At times when he could not escape of his own accord, they would step in and provide assistance.
Hermes comes to Odysseus and tells him to eat a plant so that he will be immune to Circe. He also says that when she lunges, he should strike. Without Hermes's help, Odysseus would not have known how to defeat Circe. His show of strength causes her to become attracted to him; they sleep together, and Odysseus's men and him stay on the island in relative safety for a year.
Athena also helps Odysseus. When he returns home to Ithaca, he has to face many suitors who have disrespected his wife and home for a long time. She disguises him as a beggar in a ruse far beyond what he himself could have concocted. In this way, he is able to enter his home undetected. Athena also approaches Zeus for help getting Odysseus home after Odysseus angers Poseidon.
Zeus agrees with Athena and intervenes to help Odysseus get home. Without his intervention, Poseidon may have kept Odysseus away from home for a much longer time.
By his own wits and the favor of the gods, Odysseus was able to make his way through many dangers and return to his family in Ithaca.