Final answer:
Penelope challenges the suitors to string Odysseus's bow and shoot an arrow through twelve axe heads; a task only Odysseus himself can complete.
Step-by-step explanation:
The challenge that Penelope gives the suitors that none of them can complete is to string Odysseus's great bow and shoot an arrow through twelve axe heads aligned in a row.
This impossible task revealed the true identity of Odysseus amongst the suitors when he, disguised as a beggar, accomplished the feat with ease.
This test is a pivotal moment in Homer's epic poem, The Odyssey, exemplifying the themes of identity, recognition, and the reclaiming of one's rightful place.