# When is the truth spoken?

A thief speaks truth two days a week alternatively ( T F T ). The days are the same for each week. The thief gives these statements on 3 consecutive days :

D1 : I lie on Monday and Tuesday. D2 : It's Thursday, Saturday or Sunday today. D3 : I lie on Wednesday or Friday.

Now, you being a detective, wish to know which are the days the thief speaks truth. The question is, What is the probability that the thief speaks truth on Thursday?
Hint : We face tough Conditions and so is this one. Think rationally, as a computer. PS : You love Thursdays.(So do I, was born on it 😁 )

• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it. If not, some responses to the answerers to help steer them in the right direction would be helpful. – Rubio Jul 4 '19 at 0:12

We know that;

D1 is false since he cannot lie two days in a row,

so

D1: F, D2: T and D3: F

and

we also know that from D3

is F so, he lies on Wednesday and Friday or he doesnt lie on both days. so in conclusion, this information does not give anything to us.

so with the only truth we have, we cannot conclude anything since

Saturday and Sunday are consecutive days. Because saying three days with OR does not actually gives anything to us.

50%

• The term "or" only implies at least one of the premises are true in logic, not necessarily only one. – Nautilus Jun 27 '19 at 12:29
• @Nautilus i am not sure to be honest, if i lie on both days, and if i say i lie on wednesday or friday, i would be saying a lie or not? :) – Oray Jun 27 '19 at 12:33
• No. If you lie on both days, you'll be telling the truth. It'll ONLY be a lie if you tell the truth on both days. – Nautilus Jun 27 '19 at 13:19

I think "speaks the truth two days a week alternately" means he tells the truth two days a week, with one day in-between when he lies. If he simply alternates, why would "two" be in there? Also, if he never tells the truth two days in a row or lies two days in a row, then he can't tell the truth the same days each week.

Given that,

If the last statement is false, he must tell the truth on Wed and Fri. That makes the first statement true, since he only tells the truth twice per week. So the statements must be TFF. This can only be the case for Fri-Sat-Sun, but that would make the second statement true. Ergo, we must conclude the last statement is true.

The options now are:

either TFT or FFT. If it's TFT, he lies on Mon and Tue, so the first statement must be Wed, Thu, or Fri (all other options have him tell the truth on Mon or Tue). If the first statement was made on Wed, that would make Wed/Fri the truthful days, and the final statement false. If it's Thu, that works out. If it's Fri, then the second statement was made on Sat, and would have been true. Thus, one option is that the truthful days are Thu/Sat, the statements are made Thu-Fri-Sat, and are TFT.

On the other hand,

if the statement truth-values are FFT, then he must tell the truth on one of Mon or Tue. The four options are: Truthful on Sat/Mon, statements on Thu/Fri/Sat, Truthful on Sun/Tue, statements on Fri/Sat/Sun, Truthful on Mon/Wed, statements on Sat/Sun/Mon, Truthful on Tue/Thu, statements on Sun/Mon/Tue. Of these options, the second and third do not work, because the second statement would be true. The others do work.

So the options remaining are:

Truthful on Thu/Sat, Spoke on Thu/Fri/Sat, truth values TFT, truthful on Sat/Mon, spoke on Thu/Fri/Sat, truth values FFT, or truthful on Tue/Thu, spoke on Sun/Mon/Tue, truth values FFT.

If each case is equally likely,

Then the odds are 2/3 that he speaks the truth on Thursdays.

The first statement is false in that the thief never tells the truth or lies twice in two days in a row, so the second is true and the third is also false. The last statement's falsehood also means he tells the truth on both Wednesday and Friday, so the thief will never speak the truth on Thursday.