Dive into a comprehensive 57-minute masterclass on the Law of Quadratic Reciprocity, exploring one of the most beautiful and surprising facts about prime numbers. Learn about mini rings, squares, and the quadratic reciprocity formula before delving into an ingenious card trick proof. Discover the intricacies of picking up and putting down cards in various patterns, leading to Zolotarev's lemma and a grand finale. Gain insights into why mathematicians have developed over 200 different proofs for this theorem, and experience a unique attempt to make this complex mathematical concept accessible to non-mathematicians.