Final answer:
Penelope designs an archery contest for the suitors, where they must string Odysseus's bow and shoot an arrow through twelve axe heads.
Step-by-step explanation:
The contest that Penelope designs for the suitors in The Odyssey is an archery contest. The challenge is to string Odysseus's great bow and then shoot an arrow through twelve axe heads lined up in a row. This contest is crucial in the plot of Homer's epic as it provides a means for Odysseus, who has returned disguised as a beggar, to reveal himself to the suitors and reclaim his wife and kingdom.
The bow contest is symbolic, proving the hero's unique strength and skill and serving as a pivotal moment leading to the suitors' downfall.