Why does Odysseus blind the Cyclops rather than kill him when the giant is asleep?

This is a good question. The only conceivable answer is that Odysseus had no way of being sure he could kill such a gigantic man. Even if Odysseus had a sword, it would have been relatively puny compared to the Cyclops. Odysseus and his men had to create a weapon out of a long wooden stake. It took several strong men to drive the sharpened stake into Cyclops's eye. If they had thought of trying to drive the stake into his heart, they might have found they didn't have the collective strength to penetrate his flesh. What other way would there have been to kill such a monster? Odysseus probably made the right choice when he decided to blind him. Furthermore, the stone blocking the entrance to the cave was so huge and heavy that only Cyclops was capable of moving it. If they had killed him, they would not have been able to escape from the cave. As it turned out, Cyclops had to move the stone to let his sheep out to graze. The men were able to escape among the sheep because Cyclops was blind.