Penelope, the daughter of Icarius and Periboa, is the wife of Odysseus and the mother of Telemachus. In many ways, she is like her husband, in that she has a very nimble mind. Whereas Odysseus is frequently given the epithet polutlas ("much enduring") or polumetis ("crafty" or "shrewd"), Homer often calls Penelope periphron, which means "very thoughtful" or "very careful".
Early in the epic, we find evidence of Penelope's mental prowess when we hear that she managed to stave off her 108 suitors for three years by pretending to weave a funeral shroud for Odysseus' father Laertes.
In Book 19, we see that she is a "good" Greek because she offers Odysseus, who is disguised as a beggar, hospitality. Odysseus himself is very interested in the custom of hospitality, as we can see, for example, from his encounter with the Cyclops Polyphemus in Book 9, and with his encounter with the swineherd Eumaeus in Books 13-16.
Penelope's clever mind also is on display at the end of Book 19 as she tells the disguised Odysseus about the contest that she is going to propose for her suitors. This contest involves an archery skill that Odysseus himself used to perform and requires stringing a bow that Odysseus was accustomed to carry.