Final answer:
Penelope proposes a test of strength and skill for the Suitors, requiring them to string Odysseus’ bow and shoot an arrow through twelve axe heads.
Step-by-step explanation:
The test that Penelope proposes for the Suitors is to string Odysseus’ great bow and then shoot an arrow through twelve axe heads. This test is significant as it is meant to find the suitor who is worthy of Penelope and strong enough to wield her husband's bow, not knowing that Odysseus himself is among them.
The task is daunting, as the bow is incredibly stiff and difficult to string, and the act of shooting an arrow through so many axe heads is a testament to both strength and precision.