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Four friends get together. One tells the truth all the time. One lies all the time. One tells the truth on odd-numbered days and lies on even-numbered days. One tells the truth on even-numbered days and lies on odd-numbered days.

One day in May, they made the following statements:

Abe: "I lied yesterday."

Blanca: "Today is the twelfth."

Carol: "Yesterday's date was even."

Doug: "Carol's statement is true."

Which friend tells the truth on even-numbered days and lies on odd-numbered days?

Any hints on how to better understand these problems?

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4  
I like the fact that "One day in May" is actually relevant information. – ugoren Mar 30 at 8:38
    
@ugoren How is that? I don't think I needed it in my answer. – cst1992 Mar 30 at 10:02
1  
caveat: Who says that the opposite of lieing is telling the truth? X could just've been silent the whole day ... – larkey Mar 30 at 11:09
5  
Out-of-puzzle logic: if a person lies all the time, how come he still has any friends? – svavil Mar 30 at 13:21
1  
@svavil, if a person lies all the time, then the truth value of his statements is quite easy to predict. Not something that can be said for most people. – ffao Mar 30 at 21:54
up vote 17 down vote accepted

We have no way of knowing what any of the characters are doing or what the roles the other characters play are, but we know for certain that Abe must tell the truth on even days and lie on odd days.

There are three things we need to note:
1. There are always two truth tellers and two liars on any given day.
2. Carol and Doug are either both lying or both telling the truth.
3. Abe is the only person who references his past actions.

This means that:
1. Abe is always doing the opposite of what Carol and Doug are doing.
2. Since C+D are lying on even days and truthtelling on odd days, Abe is truthtelling if the date was even and lying if the date was odd.

Now we know that:
1. If the date was even, Abe is telling the truth about lying on an odd day.
2. If the date was odd, Abe is lying about lying on an even day.

Regardless of who is actually lying or not, he is the one who tells the truth on even days and lies on odd days in both scenarios!

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There are many solutions.

Facts We Know:

Doug: "Carol's statement is true." => Means today, Doug and Carol agree.
So today, Abe and Blanca also agree.

Abe: "I lied yesterday." => Abe alternates truth and false.

Blanca: Today is the twelvth => Abe agrees, so he must tell the truth on even days and lie on odd (the opposite leads to paradox).

Blanca agrees with Abe today, so she can't alternate truth or false.
So lanca either always tells the truth or always lies.

Possibile Solutions:

Today could be even:

1. Abe tells the truth on even days (such as today), lies on odd days.
2. Blanca always tells the truth. As the says, today is May 12th.
3. Doug and Carol lie on even days (such as today).
4. Either Doug or Carol always lie, the other alternates truth and lies.

Today could be odd:

1. Abe tells the truth on even days, lies on odd days (such as today, so he lies about yesterday).
2. Blanca always lies. Today's not the 12th.
3. Doug and Carol tell the truth on odd days (such as today). As Carol says, yesterday's date was even.
4. Either Doug or Carol always tell the truth, the other alternates truth and lies.

Notes:

One day in May removes the possible complication of two consecutive odd days (May 1st is after April 30th).

Abe: "I lied yesterday." doesn't 100% necessarily means he alternates truth and lies. He could always lie, if yesterday he was silent.

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Answer is:

Abe

First, some terminology.

Assume:

T tells the truth all the time
L lies all the time.
TOLE tells the truth on odd and lies on even days
TELO lies on odd and tells the truth on even

Let's assume A B C D for the four appropriately named friends.
We get 24 combinations of the four states(T, L, TELO, TOLE) and (A, B, C, D). But from these, A can't be either T or L, first due to nature of the sentence, and second due to the Liar's Paradox. That eliminates 12 out of 24.
D always agrees with C. That means they either both lie or both tell the truth. Takes another 4 incorrect solutions out. That gives us a matrix of 8 possibles:

T    L    TOLE TELO

B    C    A    D
B    C    D    A
B    D    A    C
B    D    C    A

C    B    A    D
C    B    D    A


D    B    A    C
D    B    C    A

If we assume it's the 12'th, then C's as well as D's statement is false, so they can't be TELO. That only leaves A.
Now, the matrix looks like this:

T    L    TOLE TELO

B    C    D    A
B    D    C    A

C    B    A    D
C    B    D    A

D    B    A    C
D    B    C    A

If it's not the 12th and the day is odd, C is not lying and D can't be TELO. That also leaves A for TELO.

T    L    TOLE TELO
D    B    A    C
D    B    C    A

Now, as above, if D is telling the truth, the day is odd and C is not TELO. That also leaves A.

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The blocks make hiding the rest of the answer difficult. If anyone has a suggestion, let me know. – cst1992 Mar 30 at 9:38
    
I was kinda surprised how B was not TELO in a single one of the 8 remaining options, but after a bit of thinking, it made sense. – cst1992 Mar 30 at 9:56

Scenario 1:

Consider Carol to be telling the truth, so Doug is also telling the truth. Therefore; Blanca tells a lie. Abe lies because Carol was true on telling yesterday was even, and our truth quota is full after first 2 lines; proof is Abe told the truth yesterday, but today (on an odd day) he tells he lied.

So considering above facts I come to the answer;

Abe is telling the truth on even days, lies on odd days. And we can not decide if Carol or Doug is always telling the truth or only tells truth on odd days.

In a reverse scenario;

Carol lied so today is even. This can make Blanca telling the truth. Doug lies on an even day so he tells the truth on odd day, so Abe is telling the truth because yesterday was an odd day.

Conclusion:

We know for sure that Abe is even day truth teller. But we can't absolutely come to a conclusion about the others

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The answer is:

Abe: I lied yesterday.

says truth on even days and lies on odd days (saying truth as it is 12th today)

Blanca: Today is the twelfth.

says truth always

Carol: Yesterday’s date was even.

says truth on odd days and lies on even days (telling lie today that yesterday was even dated as it can't be because today is even date)

Doug: Carol’s statement is true.

lies always

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2  
Not necessarily, there a lot of valid possibilities you didn't give. – ffao Mar 30 at 5:13
    
Yes I agree there are many possible answers. But this is also one of the valid answers. – Developer107 Mar 30 at 5:26
    
Do you have any explanations for the answers in spoilers? – cst1992 Mar 30 at 10:20

The easiest solution is this:

Abe says: "I lied yesterday". Therefore: Abe is definitely not the "all days liar " nor "all days truthteller" why? Because if he was the all days liar, then he is lying about lying yesterday, which means he told the truth yesterday! and if he is the all days truthteller, it means that he lied yesterday. These are both contradictory.

Now that we know that, we have two cases:

  1. If today is actually even: Abe is, as stated before, either the even day liar or the odd day liar.
    Caroul is lying. So: Doug is lying too. Therefore in this case, one of carole and doug is definitely the all days liar and the other one is the one who lies on even days. Therefore: ABE is the one who says the truth on even days and lies on odd days

  2. If today is actually odd: Abe is, as stated before, either the even day liar or the odd day liar.
    Caroul is saying the truth. So: Doug is saying the truth too. Therefore in this case, one of carole and doug is definitely the all days truthteller and the other one is the one who tells truth on odd days. Therefore: ABE is the one who says the truth on even days and lies on odd days

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