What Does The Green Knight Set Out To Test?

Sir Gawain and the Green Knight follows a classic quest formula, with a knight receiving a challenge, going out on a journey to meet that challenge, and returning home to report on his quest.
When the Green Knight arrives at the beginning of the poem, he says he has not come to visit, but to test Arthur's knights, who are famed as the best on earth. He asks instead for a Christmas game: He will trade a blow for a blow. If any man is brave enough to strike the Green Knight with his ax, he will give that man the ax to keep. However, that man must agree to receive the same stroke back in a year's time. The knights do not respond, and the green rider jeers at them. Angered, Arthur accepts the challenge and takes the ax, but Gawain asks to be given the task, saying that it is unseemly for the king to do it. Arthur gives him the ax. The Green Knight reminds Gawain of the terms of their agreement. The knight kneels down, and Gawain chops off his head. The Green Knight picks up his head and gets back on his horse. He tells Gawain to look for the Green Chapel, and then rides out of the hall. Arthur masks his amazement by saying that the event was an entertaining interlude. They hang the ax on the wall and continue the feast.
Finally, in the end of the poem, Sir Gawain manages to prove himself by passing the three major tests: the challenge itself, the testing of his virtues, and the penance he accepted as he confessed clean of his sins, at the Green Knights reveal behind the challenge.