What are examples of Odysseus in The Odyssey by Homer showing cunning when he encounters the Cyclops, Polyphemus?
The story of Polyphemus in The Odyssey by Homer is meant to illustrate how Odysseus uses his cleverness and ability as a persuasive speaker to overcome a powerful being he could not best in a straightforward battle.
First, rather than try a futile attack against Polyphemus, he realizes that he needs Polyphemus alive to roll the stone away from the mouth of the cave, and waits for the right moment to plot an escape.
The first part of his stratagem involves clandestinely sharpening a piece of timber to a point to make a spear. Next, he takes advantage of Polyphemus' taste for wine, and persuades Polyphemus to drink a powerful wine that causes him to fall into a deep drugged slumber. He and his men blind Polyphemus while the Cyclops is sleeping.
Odysseus and his men also use cunning by sneaking out of the cave clinging to the bellies of the sheep when Polyphemus lets the sheep out to graze.