Odysseus and his men land on the cyclops’s island looking for xenia — the sacred guest-friendship every host owes a stranger. What they find is a giant who eats his guests two at a time.
Trapped in the cave, Odysseus thinks. He cannot kill Polyphemus outright — the boulder over the cave is too heavy for the survivors to roll away. So he gives the cyclops wine, asks his name, and offers his own: Outis. Nobody.
When Polyphemus, drunk, falls asleep, Odysseus and his men sharpen an olive stake in the fire and drive it into his single eye. Polyphemus shrieks. The other cyclopes come running, ask through the wall who is hurting him. Nobody is hurting me! They leave.
The men escape clinging to the underside of the cyclops’s sheep at dawn. As they sail, Odysseus — pride boiling over — shouts back his real name. Polyphemus prays to his father Poseidon, and the rest of the Odyssey unfolds from that one act of vanity.