Question

Four students P, Q, R and S see a box containing some white and some red balls. Exactly one of them is a liar. Their statements are:
P: There are equal number of red and white balls.
Q: P is a liar and there are 3 red balls and 2 white balls.
R: Q is not a liar and there are some red balls and some white balls in the box.
S: R is not a liar.
Who is the liar?

• A. P is the liar
• B. Q is the liar
• C. R is the liar
• D. S is the liar

Hint:

When exactly one liar is guaranteed, try making one of the relational statements (like “Q is not a liar”) true and see if all others fall into place uniquely.

Answer 1

The Correct Option is A

Step 1: Try assuming R tells the truth.
If R is truthful, then (i) Q is not a liar, and (ii) the box has some red and some white (at least one of each).
Step 2: Consequences for Q and P.
Since Q is truthful, both parts of Q’s statement must be true: P is a liar and the counts are \((3\,\text{red}, 2\,\text{white})\). Those counts also satisfy R’s “some red and some white”.
Step 3: Check S.
S says “R is not a liar.” If our assumption (R truthful) holds, S’s statement is also true.
Step 4: Count liars.
Under this assignment: \(P\) is false (because numbers are \(3\neq2\)), while \(Q,R,S\) are true. Exactly one liar → \(⇒\) consistent.
Step 5: Conclude.
Hence, the only liar is P. Final Answer: \[ \boxed{\text{P is the liar}} \]