GATE CS 2019 QGA-6 — General Aptitude

To find out who committed the crime, let us analyze the statements made by the four suspects under the given conditions:

1. Exactly one person committed the crime.
2. Exactly one statement is true (the other three are false).

Let's write down the statements in terms of who is guilty:

• $P$: $Q$ is guilty.
• $Q$: $S$ is guilty.
• $R$: $R$ is not guilty.
• $S$: $S$ is not guilty (since $S$ claims $Q$'s statement is false).

Step-by-Step Analysis:

Notice the relationship between the statements of $Q$ and $S$:

• If $S$ is guilty, then $Q$'s statement is True and $S$'s statement is False.
• If $S$ is not guilty, then $Q$'s statement is False and $S$'s statement is True.

Since these two statements are contradictory, exactly one of them must be True.

Because only one statement among all four can be true, the remaining statements by $P$ and $R$ must both be False:

• Since $P$'s statement is False, $Q$ is not guilty.
• Since $R$'s statement is False, the negation of "I did not do it" must be true. Therefore, $R$ is guilty.

Verification:

If $R$ is the criminal:

• $P$'s statement ($Q$ did it) is False.
• $Q$'s statement ($S$ did it) is False.
• $R$'s statement ($R$ did not do it) is False.
• $S$'s statement ($S$ did not do it) is True.

This perfectly satisfies the condition of having exactly one true statement and one criminal.

Thus, $R$ committed the crime.