Respuesta :
Based on the the excerpt from The Dark Game, I think that the statement that best leads to the inference that the men had intercepted a very important message was: "That single name fired the men with excitement as they began working on the message from the beginning." The statement suggests that they have unlocked something important as they were excited and became persistent on further working on the cryptic message.
This excerpt focuses on a very important message that men are trying to decipher, in the way they achieve to figure out important an important detail (the 90000 family indicates important names) that leads them to finally get where they wanted to: the author of the message, Zimmermann.
We can infer that the men had intercepted that important message because of the shock and the excitement the men showed when they discovered that the message was sent by Zimmermann, and from that point on they eagerly started working on the message from the beginning.