LOgic Math question!!! NEED HELP?

Question

There are some days when Karol tells only the truth and there are some days when he always lies. Today Karol said 4 from the following sentences.

a)The number of my friends is a prime number
b)I have as many girl friends as boy friends
c)My name is Karol.
d)I always tell the truth
e)Three of my friends are older than I am.

Which of the above sentences did Karol not say today?

Answer 1

Well a and b can't both be true. If he has the same number of girl friends as boy friends, then the number of friends has would have to be even; therefore, it would have 2 as a factor; therefore, it would not be prime (unless it was 2...but because of letter e, we know he has at least 3 friends).

Also, we know d) is false, because we know there are days where he always lies.

So we know that at least 2 of his statements must be false; therefore, it must be a "lying" day.

As a result, we can conclude that he didn't say c), "My name is Karol", because that's the only statement of the 5 that is definitely true.

Answer 2

Let us first assume that Karol is telling only the truth today.

That means that if he says the number of his friends is a prime number is true, then he cannot have as many girl friends as boy friends. (That would mean that the number of his friends is divisible by 2, and therefore, A would be a lie). That would mean that B is the only one that Karol did not say. C is obviously true, but D is not becasue we know that Karol sometimes lies and sometimes tells the truth. We do not know about E.
If Karol only tells the truth today, then there are two statements that are not consistent with him telling the truth: B and C. Therefore, Karol is not telling the truth today.

If Karol is lying today, then we can assume that he says A, B, D, and E, because those four can be lies. C is not a lie because we know his name is Karol. The number of his friends does not have to be a prime number, it is possible that he does not have as many guy friends as girl friends, we know that he does not always tell the truth, and it is possible that not exactly three of his friends are older than he is.

Therefore, today, Karol is only lying and he said A, B, D, and E.

Answer 3

This is an interesting one.
Let's pretend Karol is telling the truth today and see what happens:
a) the number of friends is prime.
b) number of friends is an even number since there are an equal number of boys and girls.
Therefore, Karol has two friends: one girl and one boy
c) this is the truth
d) this is a lie
e) this is a lie
Therefore there are at least two things that have to be lies, so today is NOT the day Karol is telling the truth.
That means EVERYTHING he says is a lie,
Therefore, c) must be the one sentence Karol did not speak, since it is true.