Now, is there some reason that Arya could not choose Tywin Lannister? He is the primary military commander of the enemies of House Stark, the most formidable, and the one likely to give her brother the biggest challenge. H'ghar sees Arya and offers to kill any three people of her choosing in order to make up for her saving the three of them from the fire earlier in the novel. A few days later Rorge, Biter and Jaqen H'ghar come rolling in to Harrenhal. Okay, having just re-read CoK, I am wondering about the part in which Arya is brought to Harrenhal by the Mountain and is put to work serving the current proprietors, which is Lord Tywin's host and allies.
