Final answer:
Boo Radley is the one who put the blanket on Scout, and Atticus deduces this by considering the events and people present that night.
Step-by-step explanation:
The person who put the blanket around Scout's shoulders in Harper Lee's To Kill a Mockingbird is Boo Radley.
Atticus Finch, Scout's father, reaches this conclusion by piecing together the events of the night and realizing that Boo was the only person who could have silently slipped the blanket on her without her notice.
This act by Boo is significant as it showcases his protective instincts and his caring nature, despite being an elusive figure in the novel.