Final answer:
No, Algernon does not find out about Jack because of a cigarette case from Cecily Cardew.
Step-by-step explanation:
In The Importance of Being Earnest, Algernon does not find out about "Jack" because of a cigarette case from Cecily Cardew. In the play, Algernon discovers the truth about Jack's double life when he intercepts a letter from Jack to Cecily. This letter reveals Jack's true identity and his engagement to Gwendolen.