Murder Mystery Question from an Entrance Exam

Question

Here is a question I've been having some trouble with. I'd love any of your thoughts. It is Question 3 in this entrance exam.

Mr Cadbury was murdered last night.

Exactly one of six suspects is responsible, and each has made three statements. Each has made at least one true statement.

Moreover, three of them like biscuits, and three do not, but you do not know which is which, only that people who like biscuits always give an odd number of true statements, and people who don’t always give an even number.

Here are the statements made:

Miss Burton:

• “Dr Lyons dislikes biscuits.”
• “Prof. Peek-Frean is the murderer.”
• “Col. Huntley-Palmer, Mrs McVitie and I are all innocent.”

Rev. Mr Fox:

• “Col. Huntley-Palmer killed him.”
• “Prof. Peek-Frean killed him.”
• “Miss Burton killed him.”

Col. Huntley-Palmer:

• “I don’t like biscuits.”
• “Neither Mrs McVitie nor Miss Burton likes biscuits.”
• “Prof. Peek-Frean and the Rev. Mr Fox both like biscuits.”

Dr Lyons:

• “I like biscuits.”
• “Mrs McVitie did not commit the murder.”
• “Miss Burton did not commit the murder.”

Mrs McVitie:

• “I did not commit the murder.”
• “Prof. Peek-Frean did not commit the murder.”
• “The Rev. Mr Fox dislikes biscuits.”

Prof. Peek-Frean:

• “I did not commit the murder.”
• “Dr Lyons and Miss Burton either both like biscuits or both dislike them.”
• “The murderer dislikes biscuits.”

I am able to show that the murderer must be one of Col. Huntley-Palmer (hereby referred to as H) and Prof. Peek-Frean (hereby referred to as P). I am able to show that H's second statement H2 is false and that H3 is true. I also found out that P & F both like biscuits and that H & L don't like biscuits. I am struggling with finding out the identity of the murderer.

Supposing that H is the murderer, I find that B likes biscuits (B has only one true statement) while M dislikes biscuits. Hence, P has two true statements, which contradicts our finding that P likes biscuits. This inconsistency forces us to conclude that P must be the murderer. But here too, there is a contradiction.

If P is the murderer then (I find that) B likes biscuits (B has three true statements) and so does M. (This is already a contradiction since there are only three people who like biscuits.) We further find that P has zero true statements, which is contradictory to the question's premise that each person has at least one true statement.

Answer 1

If P committed the murder:

We know that B's second and third statements are true. We know that P's first statement is false. If P's third statement is true, then they dislike biscuits and therefore made two true statements, and thus their second statement must be true. If P's third statement is false, they like biscuits, hence must have made an odd number of true statements, and hence their second statement is true. In either case, we conclude B and L either both like or dislike biscuits.

However,

If B likes biscuits, they made an odd number of true statements, so their first statement is true, and L does not like biscuits. If B dislikes biscuits, they made and even number of true statements, so L does like biscuits. Either case contradicts the second statement of P.

Therefore,

We conclude P is not the murderer.

Now,

If L likes biscuits, then their first statment is true, and they made an odd number of true statements, so their second and third are either both true or both false. Likewise, if L dislikes biscuits, their first statement is false and they made an even number of true statements, so their second and third are both true or both false. If they're both false, there are two murderers, which is a contradiciton. Therefore, regardless of any other considerations, we conclude that L's second and third statements are true, and both V and B are innocent.

So, if everyone told at least one true statement:

We have already eliminated B, V, and P, so the only statement by Fox that could be true is that H is the murderer.

Updated: thanks to comment

Furthermore,

We know B's second and third statements are false, and since everyont told at least one true statement, their first statement must be true, and they must like biscuits. This means L does not like biscuits. This means L's first statement is false, and they do not like biscuits.

This tells us that H's second statement is false and therefore, their third must be true. That tells us P likes biscuits and thus made an odd number of true statements, so their third statement is false. Thus, we know the murderer (H) likes biscuits and their first statement is false.

So, in summary:

H spoke true, false, false, likes biscuits.
F spoke true, false, false, likes biscuits.
H spoke false, false, true, likes biscuits. And is the murderer.
L spoke false, true, true, dislikes biscuits.
V spoke true, true, false, dislikes biscuits.
P spoke true, false, false, likes biscuits.

Updated Conclusion:

The question also states that 3 people like biscuits and 3 people dislike biscuits.

The logic above concludes that the only solution consistent with the rest of the puzzle is that 4 people like biscuits and 2 do not. Therefore the puzzle is itself a contradiction and has no solution.