Odysseus passes Penelope's tests by:
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Penelope's final test is a tricky one: She pretends to ask the servants to move their bed from their marital chambers. However, known only to Odysseus and her is that the bed is carved into an olive tree and could not be moved without badly damaging the bed. Odysseus angrily replies that the bed should not have been moved, and describes the bed and it's workmanship, proving his identity.
Penelope's final test for Odysseus was to see if he could move and string his bow, and then shoot an arrow through twelve axe heads. This test was set up as a way for Penelope to confirm if the beggar claiming to be Odysseus was truly her husband returned from his journey. Odysseus passed the test and was reunited with Penelope.
By telling the men whoever could string the bow and shoot it through all the axes was odysseyus.