How did the team manage to send messages to the mainland?

The team managed to send messages to the mainland by using Morse Code, which is a method of encoding textual information through a series of dots and dashes representing letters and numbers. In the context of "The Mysterious Benedict Society," the use of Morse Code showcases the children's resourcefulness and cleverness, as it allows them to communicate effectively over distances without relying on modern technology or equipment.

Morse Code is particularly relevant in the story as it highlights the theme of creativity in problem-solving that runs throughout the narrative. The characters often rely on their wits and ingenuity, which is perfectly illustrated by their ability to send coded messages.

The other options do not fit the narrative or context of how the characters communicate effectively. Shouting at the shore would be impractical for relaying detailed messages, sending physical letters would take too long and wouldn't ensure the message reached its destination quickly, and a satellite connection is not mentioned or relevant in the setting of the story.