Shelly, Susan, and Samantha are all working at an amusement park for the summer. The manager finds out that one of them left the key in the door to the security office. When he asked them who did it, they replied:


Shelly: Samantha did it.
Susan: Shelly did not do it.
Samantha: I didn’t do it.

The manager concluded that Samantha did it because she was accused and was denying it.

Use an indirect proof to show that the manager was wrong and that Samantha did not leave the key in the security office door.

If the girls are telling the truth -- then neither Samantha nor Shelly did it. Who's left?

In order to prove that Samantha did not leave the key in the security office door, we can use an indirect proof.

Let's assume the opposite, which is that Samantha did leave the key in the security office door.

Given the statements provided by Shelly, Susan, and Samantha, we can analyze them one by one:

1. Shelly says Samantha did it.

If Samantha actually left the key in the door, then Shelly's statement is true.

2. Susan says Shelly did not do it.

Considering our assumption that Samantha left the key in the door, Susan's statement contradicts Shelly's statement. If Samantha did it, then it means Shelly did not, which aligns with Susan's statement.

3. Samantha says she didn't do it.

Since we assumed Samantha did it, her denial is contradictory to our initial assumption.

Based on the contradiction between Susan and Samantha's statements, our assumption that Samantha left the key in the door cannot be true. Therefore, Samantha is not the one who left the key in the security office door.

The manager's conclusion that Samantha did it was incorrect.