In Homer's Odyssey, why doesn't Odysseus kill the giant Polyphemus when he has the chance?
This is a thought-provoking question. Homer makes the dramatic situation in the cave so intense that most readers probably wouldn't think of the alternative of killing the giant rather than blinding him. Here is a description of the giant from eNotes Study Guide:
A son of Poseidon and a Cyclops, a one-eyed giant. He lives on an island which is usually thought to be Sicily. He is presented as a member of a lawless race that does not acknowledge the gods, but which also lives in an area that provides for all their needs without effort on their part.
Polyphemus, in Homer, is depicted as a particularly savage giant, who eats human beings raw and washes them down with either milk or wine. He briefly captures Odysseus and his men; they get him drunk and blind him, after which they escape from his cave by clinging to the bellies of his sheep and goats. The blinded giant counts his livestock by feeling their backs, but is unaware of the escaping men sneaking out under the animals. Polyphemus asks his father Poseidon for revenge against Odysseus, which he gets.
Odysseus must have thought about killing Polyphemus but rejected the idea for at least two reasons. The main reason would have been that it was nearly impossible to murder the giant because they had no adequate means of doing it. If they had tried to stab Polyphemus with a sharpened stake, they might have only wounded him. It was fortunate for Odysseus and his men that the giant had only one eye; otherwise, they wouldn't have even been able to blind him. He was just too big for them to kill him. Even if they had all had swords and had all attacked him at once, they still would have had a hard time killing him. But they didn't have any weapons.
Another reason for not trying to kill Polyphemus was that every night the giant blocked the entrance to his cave with an enormous boulder. The combined strength of Odysseus and his men would have been inadequate to move that boulder (which gives a good idea of the size and power of Polyphemus). If they succeeded in killing him, they would have no way of getting out of the cave. In the meantime, the giant's friends would wonder why he hadn't come out to graze his sheep. They would roll the boulder aside and Odysseus would have a whole tribe of giants to deal with instead of just one.