The murderer is Miss Lily!
She went to the library, had an argument with Mr. Carter, and poisoned his tea. She then broke the cup to make it look like an accident. After tearing part of the letter away to hide the truth, she rushed back upstairs. However, her ink-stained hands and the missing teacup gave her away!
Clues:
Mr. Green saw her go downstairs before 9:15 PM, but she said she never left her room.
The torn letter was in her handwriting; she had written something to Mr. Carter, perhaps threatening him.
The teacup was missing from the kitchen and found broken near the body.
The red thread suggested she was wearing something with red fabric.
The window was open; perhaps she considered escaping but changed her mind.
The clock stopped at 9:15 PM, meaning the murder happened at that time.
The smell of perfume in the air suggested a woman had been in the room recently.
Lesduur is: 50 min
