There are 4 cricketers, A , B, C, D, each having a different age among 24, 28, 32 and 36 years. Also, they have 4 different roles among a left-hand batsman, a right-hand batsman, a spinner and a fast bowler. It is known that only one of them is a truth-teller, and also, there is at least one each of an alternator and a liar. A truth-teller always speaks the truth, a liar always lies and an alternator always alternates between a truth and a lie. Each of them says 3-4 statements as recorded below.
A: 1) I am not a spinner.
2) I am 32 years old.
3) B is not a liar.
4) C is not a left-hand batsman.
B: 1) A is not a liar.
2) D is a liar.
3) C is 32 years old.
C: 1) I am a truth-teller.
2) I am 36 years old.
3) B is 32 years old.
4) A is a liar.
D: 1) C is not a liar.
2) C is not 36 years old.
3) I am 28 years old.
Based on the information given above, answer the questions that follow:

Q. For how many players can we uniquely identify their ages?
Correct answer is '4'. Can you explain this answer?

Question

There are 4 cricketers, A , B, C, D, each having a different age among 24, 28, 32 and 36 years. Also, they have 4 different roles among a left-hand batsman, a right-hand batsman, a spinner and a fast bowler. It is known that only one of them is a truth-teller, and also, there is at least one each of an alternator and a liar. A truth-teller always speaks the truth, a liar always lies and an alternator always alternates between a truth and a lie. Each of them says 3-4 statements as recorded below.

A: 1) I am not a spinner.
2) I am 32 years old.
3) B is not a liar.
4) C is not a left-hand batsman.

B: 1) A is not a liar.
2) D is a liar.
3) C is 32 years old.

C: 1) I am a truth-teller.
2) I am 36 years old.
3) B is 32 years old.
4) A is a liar.

D: 1) C is not a liar.
2) C is not 36 years old.
3) I am 28 years old.

Based on the information given above, answer the questions that follow:

Q. For how many players can we uniquely identify their ages?

Correct answer is '4'. Can you explain this answer?

Answer

In the given problem set, we have been provided with the information that there is one truth-teller, at least one liar and at least one alternator. Let us assume each of them as a truth-teller separately and follow the other hints to reach a final conclusion.

Case 1: Let A be the truth-teller.

In this particular case, all his statements are true, he is not a spinner, he is 32 years old, B is not a liar and C is not a left-hand batsman. Also, since B is not a liar, he must be an alternator since there is only one truth-teller, which is A. Hence, we can represent them as follows:

In the above representation, TT - truth-teller, L - liar, A - alternator, S - spinner, FB - Fast Bowler, LHB - left-hand batsman, RHB - right-hand batsman. ~ - not(negation)

Now, since B is an alternator, let us analyze his statements. B's first statement is that A is not a liar, which is true. Hence his second statement must be false and the third statement must be true. The third statement is C is 2 years old, which must be true, but in the arrangement, we already have A as 32 years old. Hence, this case is not valid.

Case 2: Let B be the truth-teller

In this particular case, B's statements are true. So, A is not a liar, D is a liar and C is 32 years old. If A is not a liar, he must be an alternator, because there can be only one truth-teller. Hence, we can represent them as follows:

Now, let us analyze A's statements. His third statement is B is not a liar which is true. Hence, his first statement should be true, and his second and fourth statements should be false. Hence, A is not a spinner, A is not 32 years old and C is a left-hand batsman.

Considering C's statements, C says that he is a truth-teller, which is not true since there is only one truth-teller. Hence the first statement is a lie. His second statement is he is 36 years old, but we already know that he is 32 years old from the arrangement, hence this is also a lie. Hence, C is a liar. His third statement is B is 3 years old is a lie as well and his fourth statement is that A is a liar, which is again a lie. So, we don't get any additional information apart from the fact that C is a liar.

Considering D's statements, the first statement is C is not a liar, which is a lie. His second statement is C is not 36 years old, which is true, hence he is an alternator. But the arrangement says that D is a liar. Hence, this case is not valid.

Case 3: Let C be the truth-teller.

C's statements are true. Therefore, C is 36 years old, B is 32 years old, A is a liar.

Since A is a liar, let us analyze his statements. A says he is not a spinner, so he is actually the spinner. He says he is 32 years old, so he is not 32 years old., he says B is not a liar, hence B is a liar. He also says C is not a left-handed batsman, hence C is a left-handed batsman.

There must be at least one alternator, so D is the alternator.

Since D is an alternator, and his first statement that C is not a liar is true, his second statement must be a lie and his third statement must be true. Hence his second statement that C is not 36 years old is a lie and his third statement that he is 28 years old is true.

Therefore, the age of A is 24 years.

This is a possible arrangement.

Case 4: Let D be the truth-teller.

D's statements must be true. Hence C is not a liar, C is not 36 years old and D is 28 years old.

Now, since C is an alternator, his first statement that he is a truth-teller is false. Hence his second statement that he is 36 years old is true. But the arrangement already shows that he is not 36 years old, hence this arrangement is invalidated.

So, we come up with the following arrangement:

We can uniquely identify the ages of all 4 players.

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